Multi-carrier modulations over sparse channels: communication, channel estimation, and radar sensing

UNIVERSIT ` A DEGLI STUDI DI PARMA DOTTORATO DI RICERCA IN “TECNOLOGIE DELL’INFORMAZIONE” CICLO XXXIII Mul i-Ca ie Modula ions O e Spa se Channels: Communica ion, Channel Es ima ion, and Rada Sensing Coo dina o e: Chia .mo P o . Ma co Loca elli Tu o e: Chia .mo P o . Giulio Cola olpe Chia .mo P o . Giuseppe Cai e Do o ando: Lo enzo Gaudio Anni 2017/2020 a chi c ede in me Con en s In oduc ion 1 S a e o he a 5 1 Mul i Ca ie Modula ions 9 1.1 The Communica ion Channel . . . . . . . . . . . . . . . . . . . 9 1.2 O hogonal F equency Di ision Mul iplexing (OFDM) Modu- la ion ................................ 11 1.2.1 Inpu Ou pu Rela ion . . . . . . . . . . . . . . . . . . . 14 1.3 O hogonal Time F equency Space (OTFS) Modula ion . . . . 16 1.3.1 Sys em Model and De ini ions . . . . . . . . . . . . . . . 16 1.3.2 Modula ion and T ansmission o e he Channel . . . . . 18 1.3.3 Demodula ion........................ 20 1.3.4 Special Case: Rec angula Wa e o ms . . . . . . . . . . 25 1.3.5 Conside a ions on ma ix Ψ................ 27 1.3.6 Symbols shi wi hin he Dopple -delay g id . . . . . . . 32 1.3.7 Gene al Wa e o ms . . . . . . . . . . . . . . . . . . . . . 33 1.3.8 The C oss-Ambigui y Func ion . . . . . . . . . . . . . . 36 2 ML Me hods o Rada Pa ame e Es ima ion 39 2.1 Join S a e Sensing and Communica ion . . . . . . . . . . . . . 39 2.1.1 OFDM ........................... 39 i ii Con en s 2.1.1.1 Maximum Likelihood Es ima o . . . . . . . . 39 2.1.1.2 C ame´ -Rao lowe bound (CRLB) . . . . . . . 41 2.1.2 OTFS ............................ 43 2.1.2.1 Maximum Likelihood Es ima o . . . . . . . . 43 2.1.2.2 C am´e -Rao Lowe Bound (CRLB) . . . . . . 48 2.1.2.3 ML Wa e all Analysis o Single Pa h . . . . . 49 2.2 Rada Resolu ion and Mul i-Ta ge De ec ion . . . . . . . . . . 53 2.3 Simula ionResul s ......................... 55 2.3.1 Join S a e Sensing and Communica ion . . . . . . . . . 55 2.3.2 Join Rada and Communica ion Pe o mance . . . . . . 57 2.3.3 Sel -In e e ence . . . . . . . . . . . . . . . . . . . . . . 61 3 ML Rada Me hods in MIMO Con igu a ions 65 3.1 In oduc ion............................. 65 3.2 Physicalmodel ........................... 68 3.2.1 OTFS Inpu Ou pu Rela ion . . . . . . . . . . . . . . . 70 3.2.2 Beam o ming ma ices . . . . . . . . . . . . . . . . . . . 73 3.3 Join De ec ion and Pa ame e s Es ima ion . . . . . . . . . . . 74 3.3.1 Successi e In e e ence Cancella ion (SIC) and Join Ta - ge De ec ion and Pa ame e s Es ima ion Algo i hm . . 76 3.3.2 Reduced-Complexi y Pa ame e Es ima ion . . . . . . . 79 3.3.3 C am´e -Rao Lowe Bound (CRLB) . . . . . . . . . . . . 79 3.4 Nume ical Resul s . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.4.1 Simula ion Resul s . . . . . . . . . . . . . . . . . . . . . 83 4 OTFS De ec ion 91 4.1 In oduc ion............................. 91 4.2 TheDe ec o s............................ 93 4.2.1 P oposed MP-based de ec o (“Ma ix Galgo i hm” — MPG)............................ 93 4.2.2 Ano he MP-based algo i hm (“Ma ix Ψalgo i hm” — MPΨ)............................ 97 Con en s iii 4.2.3 Linea block-wise MMSE equaliza ion . . . . . . . . . . 98 4.3 Pe o mance o Sepa a ed De ec ion and Decoding . . . . . . . 99 5 Channel Es ima ion 105 5.1 In oduc ion............................. 105 5.2 OFDM Modula ion and he CS Algo i hm . . . . . . . . . . . . 109 5.2.1 The LASSO Sol e . . . . . . . . . . . . . . . . . . . . . 113 5.2.1.1 Complexi y o he LASSO Sol e and S ep Size Re inemen ....................114 5.2.1.2 So -Th esholding Ope a o . . . . . . . . . . . 115 5.2.1.3 Nes e o ’s Accele a ion Fac o . . . . . . . . . 117 5.2.2 Pilo Scheme ........................118 5.2.3 Recei ed Samples Exp ession — Real and App oxima ed Channel Condi ions . . . . . . . . . . . . . . . . . . . . 119 5.3 OTFS Modula ion and he P oposed Es ima ion Algo i hm . . 120 5.3.1 Pilo Scheme ........................122 5.3.2 Channel Es ima ion . . . . . . . . . . . . . . . . . . . . 124 5.4 Compa ison in Te ms o P ag ama ic Capaci y . . . . . . . . . 127 5.4.1 Simula ion Resul s . . . . . . . . . . . . . . . . . . . . . 132 5.5 Conclusions.............................137 Bibliog aphy 145 Lis o Figu es 1.1 Channeldomains........................... 10 1.2 OTFSsys emmodel ........................ 16 1.3 Di ichle unc ions . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Dopple -delay shi example . . . . . . . . . . . . . . . . . . . . 33 1.5 Two dimensional Di ichle unc ion . . . . . . . . . . . . . . . . 34 1.6 C oss-ambigui y unc ion examples . . . . . . . . . . . . . . . . 38 2.1 Wa e all beha io . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 RMSE cu es o ange and eloci y, and Gaussian capaci y . . 58 2.3 RMSE cu es o mul i-pa h case . . . . . . . . . . . . . . . . . 60 2.4 RMSE sel -in e e ence cu es . . . . . . . . . . . . . . . . . . . 62 3.1 B oadcas ing and acking scena ios . . . . . . . . . . . . . . . 68 3.2 Recei e beam con igu a ion . . . . . . . . . . . . . . . . . . . . 69 3.3 T ansmi e beam o ming examples . . . . . . . . . . . . . . . . 73 3.4 RMSE o a single a ge and de ec ion w. . . dis ance . . . . . 84 3.5 RMSE o wo a ge s including de ec ion . . . . . . . . . . . . . 86 3.6 Example o masking e ec . . . . . . . . . . . . . . . . . . . . . 87 3.7 T acking scena io wi h mul i-beams . . . . . . . . . . . . . . . 87 3.8 RMSE pe o mance o acking phase . . . . . . . . . . . . . . 88 4.1 Fac o g aph o ma ix G..................... 95 4.2 Fac o g aph o ma ix Ψ..................... 97 2 In oduc ion ion loss ypical o ha equencies. Unde his con ex , agains common and well known ada sys ems, able o e icien ly de ec and loca e a a ge wi hin he ange- eloci y plane, new join ada and communica ions echniques a e aking pa o he cu en li e a u e. These sys em a e mainly ocused on he ansmission o use ul in o ma ion owa ds he a ge s, which a e, hus, no only “passi ely” de ec ed, and a single equipmen is able o pe o m bo h ope a ional modes, a oiding o spli he unc ionali ies be ween wo dis inc subsys ems (wi h inc eased cos and complexi y). Hence, he e a e mainly wo app oaches o sol e he a o emen ioned p oblem. The i s one conside s he applica ion o common ada wa e o ms, adap ed o ca y use ul in o ma ion wi h hem. The second one, which is he one explo ed in his disse a ion, akes in o accoun ypical communica ion wa e o ms (single- o mul i-ca ie ), and, while communica ion asks come na u ally, he ada p ocessing is pe o med wi h no el me hods exploi ing he knowledge o he ansmi ed in o ma ion (known by bo h ansmi e and ecei e , i physically coloca ed), hus di e s om a mo e di ec , o “ ada -like”, h eshold analysis o he backsca e ed powe om he a ge . The choice o he communica ion wa e o m is subjec o a non i ial adeo . On one hand, he sys em aims he maximiza ion o he communica ion achie able a e, i.e., he amoun o in o ma ion sen in a ime- equency window. On he o he hand, ada asks ha e o be pe o med wi h as much p ecision as possible, in o de o co ec ly localize a a ge in all dimensions, i.e., ange, eloci y, and space (angula ) loca ion. Typically, pu e ada asks a e pe o med wi h chi p-like pulses, i.e., sho single-ca ie impulses wi h la ge bandwid h, such ha he o al ene gy deli e ed owa ds he a ge s is compensa ed by he band occupa ion o he signal. Thus, he join de ini ion o a sho pulse, oge he wi h la ge bandwid h, leads o a e y p ecise localiza ion o he a ge o e he h ee a o emen ioned domains. Howe e , he amoun o (possible) use ul in o ma ion, imp essed on op o such chi p, is poo . A solu ion o imp o e he communica ion a e is he use o mul i-ca ie digi al wa e o ms, modula ing in o ma ion symbols no only in ime domain (as single-ca ie ) bu also in he equency band, spli in In oduc ion 3 many subca ie s each occupied by a di e en modula ion symbol. Howe e , limi a ions a e linked o he de ini ion o he symbol ime and he subca ie spacing, which a e in a one- o-one ela ion, and a good localiza ion is no only challenging, o ins ance, in e ms o signal p ocessing algo i hms, bu also de ini ely sub-op imal wi h espec o single-ca ie solu ions, bu his is he cos o pay in o de o b ing communica ion ea u es oge he wi h ada asks. In conclusion, he cu en li e a u e is mo ing owa ds he de ini ion o new mul i-ca ie schemes able o b eak he limi s, in e ms on communica- ion a e, imposed by classical ada wa e o ms, and he op imiza ion o he adeo be ween he wo di e en asks is an open p oblem, whose op imal solu ions ha e no been de ined ye . The choice o he mul i-ca ie modula ion o join ada and communica- ion alls in o wo dis inc wa e o ms, i.e., o hogonal equency-di ision mul- iplexing (OFDM) modula ion and o hogonal ime equency space (OTFS) modula ion. OFDM is he mos popula mul i-ca ie modula ion o ecen yea s, widely s udied and s anda dized in mos o he cu en communica- ion s anda ds, including 5G. The mo i a ion o his choice is simple: hanks o he applica ion o a cyclic p e ix be ween symbols, i.e., a gua d in e al o p e en in e -symbol in e e ence, and unde he assump ion o absence o in e -ca ie in e e ence, which holds unde easonable amoun o he Dopple e ec and subca ie spacing, he communica ion channel can be diagonalized and symbol-by-symbol de ec ion pe o med. Clea ly, he appealing simplici y o de ec ion makes OFDM he bes choice o mode n digi al communica ions. On he o he hand, OTFS is a modula ion wa e o m wi h wo big di e ences wi h espec o i s di ec compe i o . Fi s , i does no necessi a e he inse - ion o he cyclic p e ix, achie ing a be e communica ions a e, i.e., mo e in o ma ion is sen o e a ime- equency window, bu a he cos o a mo e complex de ec ion app oach, wo king blockwise and no symbol-by-symbol. Second, OTFS is no sensi i e o delay and Dopple shi s, meaning ha i s pe o mance is kep cons an wha e e he dis ance and he speed be ween ansmi e and ( a ge ) ecei e . This ea u e is e y appealing o join ada 4 In oduc ion and communica ion asks, being he scena io e y dynamical, wi h possible e- ma kable Dopple shi s and delays, inc eased by conside ing he ound- ip ime be ween ada ansmi e and a ge . Based on he a o emen ioned anal- ysis, in his disse a ion we ake ca e o a ai compa ison be ween he wo digi al modula ion o ma s, om he poin o iew o ada , pa ame e es i- ma ion, achie able communica ion a e, channel es ima ion, and mo e o he asks, o de e mine hei posi i e and nega i e aspec s, such ha a sys em designe is able o choose he mos sui able wa e o m o a gi en scena io o applica ion. S a e o he A By ex ending he in oduc ion, some de ails a e p o ided he e, wi h he co e- sponding e e ences o li e a u e, bu some o he a e le o he in oduc ions o chap e s. The 5G communica ion s anda d will b ing some no el ies o o e come ou da ed and old echniques [1]. By mos ly ocusing on mul i-ca ie modula- ion o ma s, in pa icula OFDM [2, 3] and OTFS [4, 5, 6], his disse a ion p o ides a comple e analysis and pe o mance compa ison unde di e en sce- na ios, wi h he common denomina o o he spa se desc ip ion o he com- munica ion channel [7], whose cha ac e is ics depend on he su ounding en i- onmen . Unde his con ex , ypical channels a e cha ac e ized by ew e lec- o s wi h hei ela i e line-o -sigh and small numbe o addi ional mul i-pa h componen s (g ound e lec ion and some o he e lec ions om, e.g., me al su - aces) [8, 9]. Mo i a ed by eme ging ehicula applica ions (V2X) [10], join ada and communica ion sys ems ha e been s udied, in such a con igu a ion he wo unc ions sha e he same physical esou ces [11, 12]. Thus, di e - en ly om pu e ada asks, which aim o de ec a ge s wi h high esolu ion [12, 13], also an ac i e communica ion, i.e., he ansmission o use ul da a, is conside ed, such ha bo h unc ionali ies migh wo k oge he o join ly imp o e he pe o mance. No e ha his di e s om ypical beacon-based ini ial acquisi ion o communica ion s anda ds [14, 15, 16, 17, 18], whe e he alignmen be ween ansmi e and ecei e is achie ed h ough a so o hand- shake, i.e., he wo en i ies alk oge he o achie e he common ask. Gi en 6 S a e o he a he appealing o join ada and communica ion sys ems o u u e ehicle- o- e e y hing (V2X) communica ions, he ecen li e a u e p o ides many di e - en and de ailed solu ions [19, 20, 21, 11, 22, 23, 24, 25, 26], basically di ided in o wo classes: in o ma ion-embedded ada wa e o ms [11, 27, 24] and com- munica ion wa e o ms applied o ada de ec ion and pa ame e es ima ion [11, 19, 21, 26, 28, 29]. Mo eo e , no e ha he join ly app oach could b eak he limi s imposed by sepa a ed, o esou ce sha ing, me hods [30]. This disse - a ion, as said be o e, s udies he case o communica ion wa e o ms applied o ada , by e iewing some well-known signal p ocessing o OFDM (see [26, 21] and e e ences he ein), while exploi ing new me hods o OTFS, whose base- lines a e sha ed by o he wo ks in he li e a u e, bu in di e en shapes (see, e.g., [28, 29, 15]). In o de o demons a e ha he join app oach is supe- io (no always bu gi en some sys em se ups) wi h espec o he physical esou ces spli o one o he o he ask, he compa ison will also ake in o accoun ypical ada wa e o ms [13, 20]. When a communica ion wa e o m is employed, he p oblems o ada de- ec ion and pa ame e es ima ion a e based on he knowledge o he in o ma- ion ansmi ed and backsca e ed om a a ge , which esul s o be known i ada ansmi e and ecei e a e coloca ed. No e ha his is also pos- sible hanks o ull-duplex con igu a ions [31, 32, 33], which limi s he sel - in e e ence o he sys em, which esul s, on he o he hand, unable o p ope ly wo k i his condi ion is no ul illed. Thus, in o ma ion symbols a e ea ed as known in he condi ioning o p obabili y densi y unc ions du ing digi al signal p ocessing ope a ions, as, e.g., in [15], a he han unknown as in ypi- cal de ec ion p oblems [34]. Hence, he join ada sensing and communica ion pa adigm esul s simila o classical channel es ima ion schemes, because he inal goal is equi alen , i.e., he cha ac e iza ion o he su ounding channel. Being he channel s a e in o ma ion, i.e., he knowledge o he communica ion channel, equi ed o pe o m cohe en de ec ion in any scena io, he li e a- u e ea ing i s es ima ion is wide [34]. Gene ally, his in o ma ion is ac- cessed h ough symbols, i.e., pilo s, known a bo h ansmi e and ecei e S a e o he a 7 [35, 36, 37]. I is s aigh o wa d o unde s and ha , by aking in o accoun a ull block o symbols backsca e ed om a a ge , in a join ada and com- munica ion scheme, all symbols ake he ole o pilo s. Many echniques o sol e he channel es ima ion p oblem a e p esen in li e a u e. By aking in o accoun he spa se channel ep esen a ion in he Dopple -delay domain (see, e.g., [7, 5]), o OFDM, hese echniques migh use concep s om comp essed sensing li e a u e, e.g., [38, 39, 40, 41, 42, 35, 43], while OTFS p opose a a i- e y o solu ions, based on minimum-mean squa e e o es ima ion, maximum likelihood, comp essed sensing, and o he s [37, 44, 45, 46, 47, 48]. A las , by aking in o accoun he addi ional spa ial dimension, mul i an- enna sys ems a e s udied. The choice o conside ing mul iple-inpu mul iple- ou pu (MIMO) con igu a ions o ada is undamen al [49]. In ac , o he han opening o angle o a i al (o , equi alen ly, space) es ima ion o he a ge , i allows, h ough a ca e ul design o he beam pa e n [50], o sep- a a ely ack di e en objec s o a ge s [51, 52], while imp o ing he powe deli e ed owa ds a di ec ion, hanks o he addi ional an enna gain, which is e y ele an in V2X ada s [20] and opens o ansmission o e millime e wa e equency bands [50]. By conside ing he p oblem o ada de ec ion, a non i ial adeo appea s wi h espec o he angula co e age o he beam pa e n. On one hand, a wide angula sec o co e age enables o de ec po en- ially mo e a ge s simul aneously, i he ecei ed backsca e ed powe is high enough. On he o he hand, a mo e di ec ional alloca ion o he powe owa ds a na owe angula sec o , g an s a highe ecei ed signal- o-noise a io, a he cos o a ime-consuming sea ch (as classical ada successi ely swapping ad- jacen egions, see, e.g.,[13]). Di e en solu ions can be ound in he li e a u e (see, e.g., [52, 53, 15, 54]). Mo eo e , his disse a ion will ea he p oblem o misma ch be ween he numbe o an ennas and he numbe o adio e- quency chains. In ac , by conside ing MIMO con igu a ions o e millime e wa e equency bands, i is di icul o implemen a ully digi al beam o ming, o , equi alen ly, o associa e one adio equency chain pe an enna (includ- ing A/D con e sion, modula ion, and ampli ica ion) in a small o m ac o 8 S a e o he a and highly in eg a ed echnology o e a la ge signal bandwid h. The e o e, o millime e wa e au omo i e applica ions, we s udy hyb id digi al-analog beam o ming schemes (see, e.g., [55, 56] and e e ences he ein), hus, wi hou elying on op imal ull-duplex con igu a ions as gene ally done in li e a u e [49, 52, 54]. Mo e de ails abou he s a e-o - he-a will be gi en wi hin he ollowing chap e s. Chap e 1 Mul i Ca ie Modula ions 1.1 The Communica ion Channel The communica ion channel desc ibes how he ansmi ed signal is modi ied when a eling h ough he communica ion medium (e.g., an op ical ibe , he ai , a coppe line, e c.). Di e en impai men s and e ec s cha ac e ize each di e en scena io, and he associa ed channel is comple ely desc ibed by i s channel impulse esponse (CIR). These e ec s, including, o ins ance, ading luc ua ions, shadowing, delay, equency shi s, phase noise, e c., a e desc ibed by ma hema ical models, exploi ed du ing he algo i hmic design o de ec o s, es ima o s, and any o he digi al signal p ocessing (DSP) which could be pe o med by he communica ion ecei e (Rx). The channel conside ed in his disse a ion is ime- equency a ying, i.e., i s beha io changes wi h espec o (w. . .) he ime ins an and he ca - ie (o subca ie ) equency conside ed. In o de o simpli y i s ea men , he channel can be ep esen ed in dis inc domains, each owning i s di e en (bu beha io ally equi alen ) desc ip ion unc ion. In ac , clea ly, he channel beha io mus emain he same while i s ep esen a ion changes. In o de o swi ch be ween di e en domains, a di ec o in e se Fou ie ans o m (i.e., F o F−1, espec i ely) has o be applied [7]. Fig. 1.1 shows all he domains and 9 10 Chap e 1. Mul i Ca ie Modula ions γ( , τ)H( , ) Γ(ν, )h(ν, τ) F F−1 F F−1 F−1 F F−1 F Figu e 1.1: Channel domains. he ela i e Fou ie ans o ms. In he op igh posi ion o Fig. 1.1 we ind he ime- equency domain, wi h unc ion H( , ). Conside ing only di ec Fou ie ans o ms, i.e., F, we i s mo e o he Dopple - equency domain (Γ(ν, )), hen o he Dopple -delay domain (h(ν, τ)), and inally o he ime-delay do- main (γ( , τ)). I is in e es ing o no e ha he channel desc ip ion in he Dopple -delay domain elies in i s physical ep esen a ion, o geome y, which simpli ies he o e all ma hema ical analysis [4, 5, 7]. In ac , ypically, only a small numbe o e lec o s (o p opaga ion pa hs) akes pa o a channel, which is hus spa se and can be modeled wi h ew pa ame e s. Mo eo e , he geome y o he su ounding en i onmen slowly changes in ime (w. . . he ame du a ion), beha io which could be exploi ed du ing he algo i hmic design. The spa se ep esen a ion o he channel h(ν, τ) can be gi en as [7] h(ν, τ) = P−1 X p=0 hpδ(τ−τp)δ(ν−νp),(1.1) whe e Pis he numbe o p opaga ion pa hs, hp,τp, and νp ep esen he pa h gain, delay, and Dopple shi associa ed o he p- h pa h. The key poin is ha he channel beha io is disc e e in he numbe o pa hs. In o he wo ds, a single symbol ansmi ed o e he channel is shi ed in he delay domain, i.e., is ecei ed wi h a delay o τp, and i s equency is shi ed o νp(Dopple 1.2. O hogonal F equency Di ision Mul iplexing (OFDM) Modula ion 11 e ec ). An ex ension o he (1.1) aking in o accoun MIMO an enna sys ems can be ound in Chap e 3 and in [7]. 1.2 O hogonal F equency Di ision Mul iplexing (OFDM) Modula ion Be o e en e ing in o he de ails o OFDM signal p ocessing, o ada and communica ion pu poses, we will b ie ly desc ibe he basics o his modula- ion echnique, o be e unde s and hings o come. This p e ends o be an o e iew o mainly aspec s which a e ele an o ou analysis, while a mo e in-dep h desc ip ion can be ound in many di e en digi al communica ion books (see, e.g., [2, 3]) and wo ks in li e a u e [2, 57, 58]. As he name sugges s, OFDM is a mul iplexing scheme which modula es da a (in o ma ion symbols) on dis inc pa allel o hogonal equencies (see also he pionee ing wo k [58]). In gene al, OFDM uses a ce ain numbe o sub- ca ie s, o equally spli he a ailable bandwid h, and some ime slo s, which, oge he , iden i y an OFDM ame. The dimension o such ame depends on he pa icula applica ion, which could aim a low-la ency sys ems, i.e., smalle ames in ime, o necessi a es la ge dimensions o cope and es ima e unknown channel impai men s. As a no a ion, a se o modula ion symbols ansmi ed o e di e en subca ie s a he same ime is called OFDM sym- bol, while mo e OFDM symbols (in ime) o m he OFDM ame. The o hogonali y o he equency di ision is achie ed by choosing a con- s an subca ie spacing ∆ , gene ally de ined as he in e se o he symbol du a ion T, i.e., ∆ = 1/T, in o de o a oid da a loss du ing il e ing ope - a ions. Thus, by assuming a ec angula shaping pulse o du a ion T o mod- ula e cons ella ion symbols a he ansmi e (Tx) side, whose exp ession is 18 Chap e 1. Mul i Ca ie Modula ions c oss-ambigui y unc ion (CAF) be ween he wo pulses, use ul o suc- cessi e conside a ions, i.e., Cg x,g x ( , ),Zg∗ x( 0− )g x( 0)e−j2π 0d 0.(1.15) We adop ed he de ini ion o [13], while o he exp essions migh be ound in li e a u e (wi h no signi ican changes on he inal esul s and sys em beha io ).  Conside a ime- a ying channel whe e he maximum delay and Dopple shi o e all mul ipa h componen s a e gi en by τmax and νmax, espec- i ely. The pa ame e s Tand ∆ de e mine he maximum ole able delay and Dopple , espec i ely, such ha νmax <∆ and τmax < T. We will now look in o a de ailed de i a ion o he OTFS inpu -ou pu ela ion, which is he base o any signal p ocessing applied a e wo ds. 1.3.2 Modula ion and T ansmission o e he Channel The OTFS Tx i s maps symbols x[k, l] o samples X[n, m], om he Dopple - delay domain o he ime- equency domain, acco ding o g ids Γ and Λ, using he ISFFT, i.e., X[n, m] = 1 √NM N−1 X k=0 M−1 X l=0 x[k, l]ej2π(nk N−ml M),(1.16) o n= 0,...N −1, m = 0,...M −1.2Eq. (1.16) shows ha each in o ma- ion symbol x[k, l], belonging o any complex cons ella ion alphabe , is mod- ula ed by a wo-dimensional basis unc ion in he ime- equency domain, i.e., 2No e ha , since he ISFFT is a Fou ie ans o ma ion be ween wo-dimensional do- mains, a no maliza ion ac o has o be aken in o accoun . The no maliza ion ac o 1/(NM) could be a bo h di ec and in e se ans o ma ion, wi h a squa e oo , o jus a one side, wi hou he squa e oo . 1.3. O hogonal Time F equency Space (OTFS) Modula ion 19 exp j2πnk N−ml M. Nex , he ime- equency modula o con e s he sam- ples X[n, m] o a con inuous- ime wa e o m s( ), by he use o he ansmi (shaping) pulse g x( ), i.e., s( ) = N−1 X n=0 M−1 X m=0 X[n, m]g x( −nT)ej2πm∆ ( −nT ).(1.17) Eq. (1.17) can be seen as a disc e e Heisenbe g ans o m pa ame e ized by g x( ) [5, 4]. The pulse s( ) is he p oduc o he supe posi ion o delay-and- modula e ope a ions on he pulse wa e o m g x( ), shi ed in ime and in e- quency. No e ha i is use ul o exp ess s( ) h ough an Heisenbe g ans o m since he cascade o wo Heisenbe g ans o ms, one o he modula o and one o he channel, can be exp essed as a unique unc ion. No e ha he equali y ∆ T = 1 implies ej2πm∆ ( −nT )=ej2πm∆ ,(1.18) which simpli ies (1.17) leading o he equi alen signal model s( ) = N−1 X n=0 M−1 X m=0 X[n, m]g x( −nT)ej2πm∆ ,(1.19) which can be ound, e.g., in [26, Page 13, Equa ion (3.4)]. The signal s( ) is ansmi ed o e he ime- equency a ying channel wi h complex baseband CIR h(ν, τ) speci ied in (1.1). The ecei ed signal, neglec ing o simplici y he noise, is ( ) = ZZ h(ν, τ)s( −τ)ej2πν dτdν , (1.20) which is a con inuous Heisenbe g ans o m pa ame e ized in h(ν, τ). No e ha , by subs i u ing (1.1) in o (1.20), he double in eg a ion is alid only whe e he wo del as a e equal o one, simpli ying in a single summa ion o e p= 0, . . . , P −1. This subs i u ion is done in subsequen calculus. 20 Chap e 1. Mul i Ca ie Modula ions 1.3.3 Demodula ion A he Rx, a ma ched il e compu es he CAF (see (1.15)) in he ollowing way Y( , ) = Ag x, ( , ) = Zg∗ x  0−   0e−j2π 0d 0.(1.21) By subs i u ing (1.20) in (1.21), we ob ain Y( , ) = Zg∗ x  0− ZZ h(ν, τ)s 0−τej2πν 0dτdνe−j2π 0d 0, (1.22) and, by using (1.17) Y( , ) = Zg∗ x  0− "ZZ N−1 X n0=0 M−1 X m0=0 h(ν, τ)Xn0, m0g x  0−τ−n0T ej2πm0∆ ( 0−τ−n0T)ej2πν 0dτdν#e−j2π 0d 0,(1.23) while, by eo de ing e ms Y( , ) = N−1 X n0=0 M−1 X m0=0 Xn0, m0"ZZ h(ν, τ)(Zg∗ x  0− g x  0−τ−n0T ej2πm0∆ ( 0−τ−n0T)ej2πν 0e−j2π 0d 0)dτdν#.(1.24) The ma ched il e ou pu is ob ained by sampling Y( , ) as Y[n, m] = Y( , ) =nT, =m∆ .(1.25) Now, by ecalling he unc ion inside he squa e b acke s and sampling, we de ine Hn,m n0, m0=ZZ h(ν, τ)"Zg∗ x  0−nTg x  0−τ−n0T ej2πm0∆ ( 0−τ−n0T)ej2π(ν−m∆ ) 0d 0#dτdν . (1.26) 1.3. O hogonal Time F equency Space (OTFS) Modula ion 21 By subs i u ing 00 = 0−τ−n0T, we ge Hn,m n0, m0=ZZ h(ν, τ)"Zg∗ x  00 −n−n0T+τg x  00 ej2πm0∆ 00 ej2π(ν−m∆ )( 00+n0T+τ)d 00#dτdν =ZZ "Zg∗ x  00 −n−n0T+τg x  00 e−j2π((m−m0)∆ −ν) 00 d 00#h(ν, τ)ej2π(ν−m∆ )(n0T+τ)dτdν =ZZ h(ν, τ)Ag x,g x n−n0T−τ, m−m0∆ −ν ej2πνn0Tej2πντ e−j2πm∆ τ dτdν , (1.27) and, by conside ing he channel speci ied in (1.1), i becomes Hn,m n0, m0= P−1 X p=0 hpAg x,g x n−n0T−τp,m−m0∆ −νp ej2πνpn0Tej2πνpτpe−j2πm∆ τp.(1.28) I is s aigh o wa d o ob ain he inpu -ou pu ela ion o OTFS, gi en by Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0.(1.29) No e ha Eq.(1.29) can be spli o di ec ly and sepa a ely show he pa s 22 Chap e 1. Mul i Ca ie Modula ions in ol ed in (Dopple -delay) ISI and ICI, hus Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0 =Hn,m [n, m]X[n, m] + M−1 X m0=0 m06=m Hn,m n, m0Xn, m0 + N−1 X n0=0 n06=n M−1 X m0=0 Hn,m n0, m0Xn0, m0,(1.30) in which he i s e m indica es he cu en symbol X[n, m], wi h he associ- a ed channel esponse, he second e m is he ICI, i.e., he o al in e e ence a di e en equencies m06=mbu wi hin he same ime slo no he cu en symbol X[n, m], and, a las , he hi d e m is he ISI. No e ha , a his poin , he shape o he pulses is unknown, so he e a e possibly in ini e (pas and u u e) in e e ing e ms. P oceeding u he , s a ing om (1.30) and exploi ing (1.16), we ob ain Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0 = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0"N−1 X k0=0 M−1 X l0=0 x[k0, l0] √NM ej2πn0k0 N−m0l0 M#.(1.31) By applying he ISFFT ( om now on, o he sake o b e i y, we emo e he summa ion subsc ip s and supe sc ip s, which a e, howe e , in acco d o he 1.3. O hogonal Time F equency Space (OTFS) Modula ion 23 a o emen ioned ea men ) y[k, l] = X n,m X n0,m0X k0,l0 Hn,m n0, m0x[k0, l0] NM ej2πn0k0 N−m0l0 Me−j2π(nk N−ml M) =X k0,l0 x[k0, l0] NM  X n,m X n0,m0 Hn,m n0, m0ej2πn0k0 N−m0l0 Me−j2π(nk N−ml M)  =X k0X l0 x[k0, l0] NM hk,l k0, l0,(1.32) in which, by using he de ini ion in (1.28), including he CIR in (1.1), we ge hk,l k0, l0=X N,M X n0,M0 P−1 X p=0 hpAg x,g x n−n0T−τp,m−m0∆ −νp ej2πνpn0Tej2πνpτpe−j2πm∆ τpej2πn0k0 N−m0l0 Me−j2π(nk N−ml M).(1.33) No e ha Ag x,g x n−n0T−τ, m−m0∆ −ν= Zg∗ x  0−n−n0T+τg x  0e−j2π[(m−m0)∆ −νp] 0d 0,(1.34) and since he ecei ed signal ( ) is sampled a ime in e als 0= 1/(M∆ ) o equi alen ly 0=T/M, we ge Ag x,g x =T M i=∞ X i=−∞ g∗ x iT M−n−n0T+τpg x iT M e−j2π[(m−m0)∆ −νp]i M∆ .(1.35) 24 Chap e 1. Mul i Ca ie Modula ions So, by eo de ing e ms (no e ha ∆ T = 1) and de ining h0 i=hiej2πντ hk,l k0, l0=T M P−1 X p=0 h0 p(X n e−j2π(k N)n"X n0 ej2πk0+νpNT Nn0 i=∞ X i=−∞ g∗ x iT M−n−n0T+τpg x iT M X m ej2π−∆ i M∆ +τp+l MmX m0 e−j2πl0 M−i Mm0ej2πνpi M∆ #) =T M P−1 X p=0 h0 p(X n e−j2π(k N)n"X n0 ej2πk0+νpNT Nn0 i=∞ X i=−∞ g∗ x iT M−n−n0T+τpg x iT M (X m ej2π(−i−M∆ τp+l)m M)(X m0 e−j2π(−i+l0)m0 M)ej2πνpi M∆ #). (1.36) A his poin , i is use ul o de ine he Di ichle ke nel unc ion, ha is Di (φ, Z), Z−1 X z=0 ej2πφ z Z=ej2πφ −1 ej2πφ/Z −1=ejπφ ejπφ −e−jπφ ejπφ/Z ejπφ/Z −e−jπφ/Z =ejπφ(Z−1)/Z sin (πφ) sin (πφ/Z).(1.37) A plo o he unc ion is gi en in Fig. 1.3. The alue o he unc ion is equal o Z(o −Zdepending i Zis e en o odd) when φis a mul iple o Z, and is equal o ze o o all o he s in ege alues o φ. Mo eo e , no e ha he Di ichle ke nel has he ollowing p ope y sin (πφ) sin π Zφ=sin (π(φ+Z)) sin π Z(φ+Z).(1.38) which could be use ul o successi e analysis. 1.3. O hogonal Time F equency Space (OTFS) Modula ion 25 -20 -15 -10 -5 0 5 10 15 20 -1 0 1 2 3 4 5 (a) Z= 5. -15 -10 -5 0 5 10 15 -6 -4 -2 0 2 4 6 (b) Z= 6. Figu e 1.3: Example o Di ichle unc ions Di (φ, Z) = sin(πφ) sin(πφ/Z). Thus, he exp ession o y[k, l] becomes y[k, l] = X k0,l0 x[k0, l0] NM T M P−1 X p=0 h0 p(X n e−j2πk n NX n0"∞ X i=−∞ g x iT M g∗ x iT M−n−n0T+τpDi (l−i−τpM∆ , M) Di l0−i, Mej2πνpi M∆ #ej2π(k0+νpNT )n0 N) =X k0,l0 x[k0, l0] NM T P−1 X p=0 h0 p(X nX n0 g∗ x l0T M−n−n0T+τp g x l0T MDi l−l0−τpM∆ , Me−j2πk n Nej2πνpl0 M∆ ej2π(k0+νpNT )n0 N).(1.39) 1.3.4 Special Case: Rec angula Wa e o ms As a p ac ical special case, conside a ec angula wa e o m o du a ion Tand ampli ude 1/√T, i.e., g x( ) = g x( ) = ec ( ), as de ined in (1.2). I τmax < T, only he signal o he i s p eceding slo is in ol ed in he ISI calcula ion, i.e., 26 Chap e 1. Mul i Ca ie Modula ions n0=n−1. In his case g∗ x pT M−T+τpg x pT M=1 √T 1 √T=1 T,(1.40) o he alues o pwhe e he p oduc o he wo pulses is nonze o. Thus, o ec angula pulses, he sum Pn0 akes in o accoun only wo e ms, i.e., n0=n and n0=n−1. By using hese esul s, s a ing om (1.39), we ob ain y ec [k, l] = X k0,l0 x[k0, l0] NM P−1 X p=0 h0 p (X n e−j2π(k−k0−νpNT )n Nej2πνpl0 M∆ Di l−l0−τpM∆ , M)+(X n e−j2π(k−k0−νpNT )n N Di l−l0−τpM∆ , Mej2πνpl0 M∆ e−j2π(k0+νpNT ) N)! =X k0,l0 x[k0, l0] NM P−1 X p=0 h0 pDi νpNT −k+k0, Nej2πνpl0 M∆ Di l−l0−τpM∆ , M1 + e−j2πk0+νpNT N.(1.41) Howe e , no e ha , ha ing conside ed he mul iplica ion be ween he wo ec angula pulses g x( ) and g x( ), he l0in ol ed in he wo e ms inside he cu ly b acke s is di e en , since i conside s he pulses o e lap wi hin he in e al [0, M −1−dτp/(T/M)e] and [M−1−bτp/(T/M)c, M −1], whe e d·e and b·c indica es he nea es uppe and lowe in ege , espec i ely. We inally ge y[k, l] = P−1 X p=0 h0 pX k0 Di νpNT −k+k0, NX l0 Di l−l0−τpM∆ , M ej2πνpl0 M∆ x[k0, l0] NM ×   1 i l0∈l0 ISI e−j2πνpT+k0 Ni l0∈l0 ICI ,(1.42) 1.3. O hogonal Time F equency Space (OTFS) Modula ion 27 whe e we used    l0 ISI ,n0, M −1−d τp T/M eo l0 ICI ,nM−1−b τp T/M c, M −1o.(1.43) Thus, he channel ma ix exp ession o he p- h pa h becomes Ψp k,k0l, l0=1 NM Di νpNT −k+k0, NDi l−l0−τpM∆ , M ej2πνpl0 M∆ ×   1l0∈l0 ICI e−j2πνpT+k0 Nl0∈l0 ISI .(1.44) The inpu ou pu ela ion becomes y[k, l] = X k0,l0 P−1 X p=0 h0 pΨp k,k0l, l0xk0, l0,(1.45) which can be ep esen ed in ma ix o m as y=  P−1 X p=0 h0 pΨp x,(1.46) wi h yand x ec o s o dimension NM ×1 ob ained by s acking he ecei ed samples and in o ma ion symbols, espec i ely, and Ψpma ix o dimension MN ×MN, whose {k, k0, l, l0}elemen is de ined in (1.44). 1.3.5 Conside a ions on ma ix Ψ We now conside some special cases o ma ix Ψp, o any single pa h p.  Ze o delay — Ze o Dopple :τp=νp= 0. Ψp k,k0l, l0=   1 i (l=l0, k =k0) 0 else .(1.47) When bo h delay and Dopple a e ze o, Ψp esul s o be he iden i y ma- ix. The channel does no modi y he ansmi ed symbols, he ecei ed signal is simply y=x+ noise. 34 Chap e 1. Mul i Ca ie Modula ions −5 0 −4−2024 0 0.5 1 |A| Figu e 1.5: Two dimensions isual example o Di ichle unc ions in bo h do- mains, whe e |A|is he no malized ampli ude alue, in acco d o he hea map o Fig. 1.4. suppo . By s a ing om (1.39) and de ining q,(n−n0), we ge y[k, l] = T NM X k0,l0 xk0, l0P−1 X p=0 h0 p(X n e−j2πk n NX n0 ej2π(k0+νpNT )n0 Nej2πνpl0 M∆ g∗ x l0T M−n−n0T+τpg x l0T MDi l−l0−τpM∆ , M) =X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 p(X n e−j2πk n N ∞ X q=−∞ ej2π(k0+νpNT )n−q Nej2πνpl0 M∆ g∗ x l0T M−qT +τpg x l0T MDi l−l0−τpM∆ , M) =x[k0, l0]T NM X k0,l0 P−1 X p=0 h0 pg x l0T MDi l−l0−τpM∆ , Mej2πνpl0 M∆ X n ej2π(k0+νpNT −k)n N!∞ X q=−∞ e−j2π(k0+νpNT )q Ng∗ x l0T M−qT +τp. (1.60) 1.3. O hogonal Time F equency Space (OTFS) Modula ion 35 No e ha only he e m in b acke s depends on n, and i is a Di ichle unc ion X n ej2π(k0+νpNT −k)n N,Di νpNT −k+k0, N.(1.61) Hence y[k, l] = X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 pDi l−l0−τpM∆ , MDi νpNT −k+k0, N g x l0T M(∞ X q=−∞ e−j2π(k0+νpNT )q Ng∗ x l0T M−qT +τp) ej2πνpl0 M∆ .(1.62) The e m unde cu ly b acke s akes in o accoun in e e ing pulses shi ed o mul iples o Tw. . . he summa ion index q. Depending on he suppo o he pulses, i.e., when he ene gy is abo e a ce ain h eshold i he pulse suppo is in ini e, only a ixed numbe o in e e es appea s ( o le and o igh ), indica ed by Ip. The summa ion o e qcan be hus limi ed o [−Ip, Ip]. We inally ge y[k, l] = X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 pDi l−l0−τpM∆ , MDi νpNT −k+k0, N ej2πνpl0 M∆ g x l0T M(Ip X q=−Ip e−j2π(k0+νpNT )q Ng∗ x l0T M−qT +τp). (1.63) No e ha a pulse no sa is ying he Nyquis condi ion, i.e., la equency ep esen a ion, changes he noise p ope ies. The noise associa ed o he e- cei ed samples is no whi e anymo e, and changes wi hin he pulse de ini ion. This ac mus be conside ed in he ma hema ical and simula ion model. Wha is he ole o he shaping pulses in he cons uc ion o he channel ma ix Ψ? The s uc u e o Ψis domina ed by he Di ichle unc ion alues, w. . . he in ege / ac ional delay and Dopple shi s and he indices l, l0, k, k0. 36 Chap e 1. Mul i Ca ie Modula ions The ISI and ICI e ec s caused by he shaping pulses add o he Di ichle be- ha io , bu how? One can hink ha op imized “well-known” pulses ha ing limi ed CAF in ime- equency domain should be adop ed [60], bu he e ec in he dual Dopple -delay domain emains no clea . In ac , he Dopple -delay ISI and ICI weakly depend on he adop ed shaping pulse, and a e domina ed by he Di ichle unc ions, whose exp essions appea om he pa icula ans- o ma ions pe o med by modula o and demodula o , and no om he choice o he ansmi ed and ecei ed pulses (see Sec. 1.3.1). Fo hese easons, once common well-con ined ime- equency pulses a e adop ed [60], i is no gua - an eed o achie e good pe o mance also in he Dopple -delay domain. This ac is also con i med in [61]. The conclusion could be ha , wha e e he chosen pulse (also di e en be ween ansmi e and Rx), he e a e no pe o mance gua an ees. Fo com- ple eness, he nex sec ion will p esen some known pulses and he associa ed CAFs, oge he wi h inal conside a ions on he adop ed pulses. 1.3.8 The C oss-Ambigui y Func ion In ada scena ios, he CAF is a wo-dimensional unc ion o delay and Dopple showing he dis o ion o a e u ned pulse a he Rx ma ched il e due o delay and Dopple shi o he mo ing a ge (see, e.g., [13, 60]). The ambigui y unc- ion desc ip ion is only de e mined by he p ope ies o he ansmi ed pulse and he ma ched il e ( ecei ed pulse). By aking in o accoun he de ini ion o CAF in (1.15) [13], ecalled he e o con enience Cg x,g x ( , ),Zg∗ x( 0− )g x( 0)e−j2π 0d 0,(1.64) we show i s beha io when di e en pulses g x( ) and g x( ) a e used. Fig. 1.6 shows ha di e en pulses achie e dis inc pe o mance in e ms o sp eading in ime and in equency o he CAF. Fo ins ance, as shown in Fig. 1.6 (a), since ec angula pulses ha e in ini e suppo in he equency domain, he p ojec ion o he CAF on he equency plane slowly decays and 1.3. O hogonal Time F equency Space (OTFS) Modula ion 37 necessi a es some ime o each he ze o “ loo ”, while alues nea he peak poin ha e ema kable magni ude. A di e en beha io is shown when wo Gaussian pulses a e adop ed (Fig. 1.6 (b)). In ac , ha ing Gaussian pulses ini e suppo in he equency domain, hey exhibi a apid decay o ze o a ound peak poin . A simila conside a ion occu s in he ime domain, which is con i med by looking a he p ojec ion on he ime plane. A las , compa e Fig. 1.6 (b) and 1.6 (c) o ha e an idea on how Gaussian pulses wi h di e en a iances change he beha io o he p ojec ions o e he ime and equency planes. The de ini ion o good shaping pulses, wi h he co esponding CAF, is a p oblem o p ima y in e es in ypical ada sys ems. Fo mo e de ails, o ins ance, e e o he analysis ca ied on in [60], sugges ing he use o pulses well localized in he ime- equency domain. Rega ding OTFS, pionee ing wo ks [4, 5] a e based on he assump ion o ideal bi-o hogonal pulses sa is ying pe ec in e e ence p ope ies, i.e., e- sul ing o be del as in bo h ime and equency planes. Howe e , hese pulses canno be c ea ed in eal elec onic ci cui s, and di e en solu ions mus be adop ed. Thus, i is possible o ind in li e a u e many examples o OTFS mod- ula ion based on ec angula pulses [62, 46], whose CAF sp eads in equency, acco ding o Fig. 1.6, bu i easie o handle (ma hema ically speaking) wi hin he de i a ion o he OTFS inpu -ou pu ela ion. An in-dep h nume ical anal- ysis compa ing he pe o mance o OTFS wi h di e en pulses ( ec angula , oo - aised cosine, Gaussian) has been ca ied on, esul ing in simila pe o - mance, and hus no shown he e o he sake o b e i y. This ac has been also con i med in [61]. Thus, we ook ad an age o he simplici y in e ms o ma hema ical ea ing o ec angula pulses, as gene ally done in li e a u e, by keeping in mind ha a di e en ea men is possible, bu does no lead o ema kable pe o mance imp o emen , a leas in he conside ed scena ios. 38 Chap e 1. Mul i Ca ie Modula ions −1 0 1−10 −50510 0 0.5 1 |A| (a) Rec angula pulses, g x( ) = g x( ). −2 0 −10 −50510 0 0.5 |A| (b) Gaussian pulses, σ2 1,g x( ) = g x( ). −2 0 −10 −50510 0 0.2 0.4 |A| (c) Gaussian pulses, σ2 2> σ2 1,g x( )6= g x( ). Figu e 1.6: C oss-ambigui y unc ion o di e en pulses. Chap e 2 ML Me hods o Rada Pa ame e Es ima ion 2.1 Join S a e Sensing and Communica ion 2.1.1 OFDM 2.1.1.1 Maximum Likelihood Es ima o S a ing om he a o emen ioned analysis, by ocusing o simplici y on a single- a ge case (P= 1), we neglec he p-pa h subsc ip o he ollowing de i a ions based on OFDM. Since da a symbols a e known by he ada Rx (which could be coloca ed wi h he Tx (monos a ic ada ) o no (bis a ic ada )), and he noise is i.i.d. Gaussian ci cula ly symme ic, he ada Rx can shi he da a symbol phase wi hou changing he noise s a is ics. The e- o e, he ada obse a ion, including he noise and symbol-by-symbol phase o a ion, can be w i en as zn,m =An,mhej2πnToνe−j2πm∆ τ +wn,m ,(2.1) whe e An,m =|xn,m|deno es he ampli ude o he ansmi ed symbol and wn,m is he addi i e whi e Gaussian noise (AWGN) wi h ze o mean and uni 39 40 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion a iance. The maximum likelihood (ML) es ima o o channel gain, ange, and eloc- i y o he obse a ion model in (2.1) is ob ained by gene alizing he app oach in [26, Chap e 3.3.3] o he case o a bi a ily ampli ude Asymbols, wi h Abeing he ma ix collec ing he ime- equency en ies {An,m}. Fo he se o pa ame e s θ= (h, ν, τ), we wish o ind he es ima o minimizing he likelihood unc ion l(z|θ,A) = X nX mzn,m−hAn,mej2π(νnTo−m∆ τ) 2 .(2.2) Assuming ha he pai (ν, τ) is known, by le ing he de i a i e o l(z|θ,A) w. . . hequal o ze o, we ob ain he es ima ion ˆ ho he complex channel gain h, which is ˆ h=Z(ν, τ) Pn,m A2 n,m ,(2.3) whe e he DFT/in e se disc e e Fou ie ans o m (IDFT) ope a ion is de ined as Z(ν, τ), M−1 X m=0 N−1 X n=0 zn,mAn,me−j2πνnToej2πm∆ τ ,(2.4) which is a wo-dimensional pe iodog am. By plugging (2.3) in o (2.2) and ollowing simila s eps as [13, Chap e 7.2.2]), he es ima o o he emaining unknown pa ame e s in gi en by (ˆν, ˆτ) = a g max (ν,τ)∈Γ0|Z(ν, τ)|2,(2.5) whe e we conside ed a disc e ized se Γ0o delay and Dopple equency axes wi h s ep sizes 1/(M0∆ ) and 1/(N0To), espec i ely, wi h N0≥Nand M0≥ M. No e ha Γ0is in line wi h he de ini ion in (1.13), bu wi h g ea e g anula i y o achie e an highe accu acy wi hin he es ima ion p ocess. In summa y, o compu e he join ML es ima o o he se o unknown pa ame e s (h, τ, ν) he ollowing s eps a e done: 2.1. Join S a e Sensing and Communica ion 41 1. Compu e he DFT/IDFT ou pu Z(ν, τ). This s ep which can be e i- cien ly implemen ed by using as Fou ie ans o m (FFT)-based design. 2. Choose (ˆν, ˆτ) maximizing |Z(ν, τ)|2o e Γ0, o some N0and M0(de- pending o he a ge accu acy le el). 3. Le he channel gain be ˆ h=Z(ˆν, ˆτ)/Pn,m A2 n,m. Clea ly, s a ing om he delay and Dopple es ima ions, i is possible o de i e he co esponding ange and eloci y es ima ions, espec i ely gi en by ˆ = ˆτc/2 and ˆ = ˆνc/(2 c). 2.1.1.2 C ame´ -Rao lowe bound (CRLB) I is well known ha he C ame´ -Rao lowe bound (CRLB) p o ides a heo- e ical lowe bound on he a iance o any es ima o [3, 34]. The de i a ion o he CRLB depends on he pa icula sys em se ing, bu i is always based on a common denomina o , i.e., he cons uc ion o he Fishe in o ma ion ma ix. Suppose we ha e only one pa h, i.e., P= 1, o simpli y he no a ion. Fo he calcula ion o he CRLB conside a ec o o unknown pa ame e s θ o be es ima ed. Gi en (y|θ), which is he condi ional dis ibu ion o he channel ou pu ygi en he se o unknown pa ame e s θ, i he egula i y condi ion is sa is ied Ey∂ ∂θln (y|θ)=   Ey∂ ∂h ln (y|θ) Ey∂ ∂τ ln (y|θ) Ey∂ ∂ν ln (y|θ)  =   0 0 0   ,(2.6) hen, any unbiased es ima o p o iding ˆ θhas co a iance ma ix Cˆ θ=Ehˆ θ−Ehˆ θiˆ θ−Ehˆ θi∗i,(2.7) which sa is ies Cˆ θ−I(θ)−1≥0∀θ.(2.8) 42 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion The ma ix I(θ) is he Fishe in o ma ion ma ix, whose (i, j) elemen is [I(θ)]i,j =−Ey∂2ln (y|θ) ∂θi∂θj,(2.9) whe e indexes (i, j) selec he unknown pa ame e s wi hin he ec o θ. Mo e- o e , (2.8) implies ha diagonal elemen s o Cθdomina es hose o I(θ)−1, hence Va hˆ θii≥hI(θ)−1iii ≥1 [I(θ)]ii .(2.10) Conside he ec o o unknown pa ame e s θ= (α, ϕ, , ), whe e α=|h|, ϕ=∠(h), =Toν, and = ∆ τ, om (2.1) we ob ain zn,m =An,mαejϕej2πn e−j2πm +wn,m .(2.11) By le ing sn,m =An,mαejϕej2πn e−j2πm , we de i e he 4 ×4 Fishe in o ma- ion ma ix de ined as [I(θ,A)]i,j = 2Pa gRe (X n,m ∂sn,m ∂θi∗∂sn,m ∂θj),(2.12) whe e Pa g akes in o accoun any possible powe cons ain on ansmi ed symbols. We hus ha e ∂sn,m ∂α =An,mejϕe+j2πn e−j2πm (2.13a) ∂sn,m ∂ϕ =jαAn,mejϕe+j2πn e−j2πm (2.13b) ∂sn,m ∂ = (j2πn)αAn,mejϕe+j2πn e−j2πm (2.13c) ∂sn,m ∂ = (−j2πm)αAn,mejϕe+j2πn e−j2πm .(2.13d) Fo a model gi en in (2.11), he MMSE o and is lowe bounded by σ2 ˆ ≥N0 2|h|2 e (A) d(A)(2.14a) σ2 ˆ ≥N0 2|h|2 e (A) d(A),(2.14b) 2.1. Join S a e Sensing and Communica ion 43 whe e e (A), e (A), d(A) a e gi en by e (A) = (2π)2(X n,m (A2 n,m)·X n,m (m2A2 n,m)−hX n,m mA2 n,mi2),(2.15a) e (A) = (2π)2(X n,m (A2 n,m)·X n,m (n2A2 n,m)−hX n,m nA2 n,mi2),(2.15b) d(A) = (2π)4nX n,m (A2 n,m)hX n,m n2A2 n,mihX n,m m2A2 n,mi + 2hX n,m nA2 n,mihX n,m mA2 n,mihX n,m nmA2 n,mi−X n,m (A2 n,m)hX n,m nmA2 n,mi2 −hX n,m n2A2 n,mihX n,m mA2 n,mi2−hX n,m m2A2 n,mihX n,m nA2 n,mi2o.(2.15c) In he egime o la ge Mand N, he CRLB o and a e gi en by σ2 ˆ ≥6 |h|2Pa g(2π)2MN(N2−1) ,(2.16a) σ2 ˆ ≥6 |h|2Pa g(2π)2MN(M2−1) .(2.16b) Fo a special case o cons an en elope (An,m =pPa g o all n, m), he abo e exp essions coincide wi h hose in [26, Sec ion 3.3]. 2.1.2 OTFS 2.1.2.1 Maximum Likelihood Es ima o Based o he esul s o Chap e 1 and Sec. 1.3.1, he ec o ized inpu -ou pu ela ion is y= P−1 X p=0 hpΨp(τp, νp)x+w.(2.17) We wish o es ima e he se o unknown pa ame e s θ= (¯ h,¯ τ,¯ ν), wi h ¯ h= [h0, . . . , hP−1], ¯ τ= [τ0, . . . , τP−1], and ¯ ν= [ν0, . . . , νP−1], whe e he ba indica es he ue channel pa ame e alue. 50 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion maximiza ion based es ima o in his disse a ion): namely, when he noise domina es he use ul signal, he maxima o he likelihood unc ions end o be andomly placed anywhe e on he sea ch g id. In he ollowing, we analyze he wa e all e ec , o ansi ion, o he single pa h case P= 1, i.e., we sea ch and s udy he egion a ound he h eshold SNR alue whe e he apid de e io a ion occu s. Mo eo e , simula ions esul s show ha he p o ided wa e all p edic ion o P= 1 is also e y accu a e o he mul ipa h case P > 1.1This also con i ms he e idence ha he p oposed app oxima ed i e a i e ML es ima ion desc ibed in Algo i hm 1 is e ec i ely e y good, and pe o ms e y close o he ue ML.2 Following he easoning o [64, 65], we de ine as “ou lie ” he e en ha a maximum o he likelihood unc ion is andomly placed on he g id Γ, a he han wi hin he clus e o poin s su ounding he ue alue (¯τ0,¯ν0). Le α∈ {τ, ν}be he unknown pa ame e o be es ima ed, i.e., τo νindi e en ly. By he law o o al p obabili y o e he disc e ized g id Γ (whe e we calcula e he ML es ima o ), we can exp ess he es ima ion MSE as MSE = Eh(ˆα−¯α)2i=X i∈Γ P (ε(i)) (ˆαi−¯α)2 ≤X i∈Γ P (˜ε(i)) (ˆαi−¯α)2.(2.36) whe e ¯αis he ue alue o he pa ame e , ˆαis an es ima ion o such pa am- e e , and P (ε(i)) deno es he p obabili y o e o o choosing αi a he han ¯α. While e alua ing P (ε(i)) may be ex emely di icul , also because he ue pa ame e ¯α∈R, we ob ain an uppe bound by conside ing pai wise e o p obabili ies, i.e., eplacing P (ε(i)) wi h he p obabili y ha he de ec o chooses αi a he han ¯αwhen hese a e he only wo al e na i es. Since he 1This is also ue because he MSE pe o mance be ween he single pa h and he mul ipa h case a e e y simila . 2Ob iously, he CRLB o P= 1 yields also a lowe bound o he case P > 1, in he case whe e we simply add mo e mul ipa h componen s wi h independen s a is ics, and he p esence o mo e unknown pa ame e s does no gene ally help he es ima ion o each single a ge pa ame e s. 2.1. Join S a e Sensing and Communica ion 51 pai wise e o e en ˜ε(i) con ains he ue e o e en ε(i), i ollows ha he inequali y o (2.36) p o ides an uppe bound. Thus, he de ini ion o he pai wise e o p obabili y is P (˜ε(i)) ,P nl(y|θi,x)> l y|¯ θ,xo,(2.37) whe e l(y|θi,x) and ly|¯ θ,xa e he alues ob ained om he e alua ion o he likelihood unc ion a g id poin i, wi h pa ame e θi, and when he se o ue pa ame e s ¯ θis aken in o accoun , espec i ely. A his poin , he p oblem is educed o he compu a ion o he pai wise e o p obabili ies P (ε(i)), o i∈Γ, which can be de i ed as ollows. We no ice ha , aking in o accoun he use ul signal appea ing in (2.28) P (˜ε(i)) ≈P |xHΨH iy|2>|xH¯ ΨHy|2,(2.38) wi h ¯ Ψ he channel ma ix compu ed wi h he ue se o pa ame e s, and whe e we exploi ΨH iΨi=I,∀i, o P= 1. De ining he join ly condi ionally Gaussian andom a iables zi,xHΨH iy=xHΨH i¯ Ψx+xHΨH iw,(2.39) ¯z,xH¯ ΨHy=xHx+xH¯ ΨHw,(2.40) wi h i s and second o de momen s    E [¯z] = kxk2,Va [¯z] = σ2 wkxk2 E [zi] = xHΨH i¯ Ψx,Va [zi] = σ2 wkxk2,(2.41) and Co [¯z, zi] = σ2 wxH¯ ΨHΨix,(2.42) a good app oxima ion o P (˜ε(i)) is gi en by [64] P (˜ε(i)) ≈1 2exp (−kxk4 2σ2 wNM )I0xHΨH i¯ Ψx·kxk2 2σ2 wNM ,(2.43) 52 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion −30 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10 10−1 100 101 102 103 104 SNR RMSE CRLB Wa e all Es ima ion Random Es ima ion Figu e 2.1: Wa e all beha io example w. . . o he bounds. whe e I0is he modi ied Bessel unc ion o he i s kind o o de 0. The MSE es ima ion can be hus compu ed by subs i u ing (2.43) in o (2.36), using he abo e de ini ions. The esul ing (app oxima ed) uppe bound ends o be loose a low SNR, whe e he ou lie s a e likely o occu and he associa ed e o p obabili ies a e high, bu becomes mo e accu a e mo ing owa ds he ML wa e all egion, hus, when he ou lie p obabili y dec eases. Fu he mo e, he MSE s ic ly depends on he g id esolu ion and may o may no each he CRLB o inc eas- ing SNR depending on he sys ema ic e o incu ed by he g id disc e iza ion. As a ma e o ac , in ou nume ical esul s we ook ca e o using a sea ch g id ine enough such ha he disc e iza ion sys ema ic e o is no isible in he explo ed ange o SNRs, and, pa icula ly, when he wa e all alls o he CRLB. Mo eo e , o con as he looseness o he MSE uppe bound ob ained in (2.36) a low SNR, we de ine he i ial uppe bound MSE ≤1 |Γ|X i∈Γ (ˆαi−¯α)2,(2.44) 2.2. Rada Resolu ion and Mul i-Ta ge De ec ion 53 ha co esponds o es ima e by choosing andom g id poin s, i.e., comple ely dis ega ding he ecei ed signal. Then, pu ing oge he (2.36) (wi h he pai wise e o p obabili y app ox- ima ion in (2.43)) and he abo e “ andom es ima ion” bound in (2.44), we inally ob ain he app oxima ed MSE uppe bound as MSE .min (X i∈Γ P (˜ε(i)) (ˆαi−¯α)2,X i∈Γ (ˆαi−¯α)2 |Γ|),(2.45) whose beha io is shown in Fig. 2.1, oge he wi h he CRLB in oduced he e- a e . Thus, he ML pe o mance o he es ima o s ands on he andom es- ima ion s aigh line o low SNR, ollows he CRLB o high SNR, and he ansi ion be ween hese wo ex emes occu s a ound he wa e all es ima ion. Simula ion esul s show ha he a o emen ioned analysis is able o accu- a ely p edic he wa e all beha io o he ML es ima o . 2.2 Rada Resolu ion and Mul i-Ta ge De ec ion Unde he assump ion o he poin a ge model [17, 66], un il now we ha e an- alyzed a single a ge scena io. Rada ope a ions can be essen ially pe o med wi h wo di e en app oaches. The i s one, conside s a ada pe iodically scanning angula sec o s, as ypical na al o ai plane ada s. In his con ex , by making he beam as di ec i e as possible, i.e., assuming a e y na ow beam and angula co e age, i is easonable o assume he p esence o jus one a ge in any di ec ion (o mul iple a ge s sha ing he same di ec ion and assuming no blockage o he signal p opaga ion). As an al e na i e, he second app oach is based on a wide angula sec o co e age. In such a case, since he a ge loca ion is no a p io i known, a join es ima ion o Dopple , delay, and angle o a i al (AoA) should be pe o med a he ada Rx. In bo h app oaches, he ada ange and eloci y esolu ions acqui e a cen al ole, p o iding he min- imum dis ance and eloci y a which wo di e en a ge s can be sepa a ely de ec ed, i.e., such ha hey a e no seen as a single en i y. 54 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion Conside he ansmi ed symbols o be a anged wi hin he Dopple -delay g id in (1.13). The de ini ion o N,M,B(bandwid h), T, and c(ca ie e- quency) is e y impo an . By ollowing he con inuous ime signal exp ession in (1.17), symbols a e spaced by T(seconds) in ime and ∆ (He z) in e- quency. The ada esolu ions a e compu ed in he ollowing. S a ing om he de ini ion o Dopple shi we ge ν,2 c c⇒ ,νc 2 c ,(2.46) and since he minimum Dopple s ep is ∆ /N, he eloci y esolu ion esul s o be νs ep =∆ N⇒ es ,νs epc 2N c =∆ c 2N c .(2.47) Equi alen ly, s a ing om he de ini ion o delay we ge τ,2 c⇒ ,τc 2,(2.48) and since he minimum delay s ep is T/M, he ange esolu ion is τs ep =T M⇒ es ,τs epc 2M=Tc 2M.(2.49) I we impose, as gene ally done in mul i-ca ie sys ems, he equali y ∆ , 1/T, oge he wi h B=M∆ , p e ious o mulas become    es =Tc/2 = c/(2B) es = ∆ c/(2 ) = Bc/(2NM c) .(2.50) This analysis p o ides he limi s o a ada sys em based on a mul i-ca ie communica ion wa e o m. E en i he ange esolu ion s ic ly depends on he bandwid h B, and his is alid o any ada sys em, he only way o educe he eloci y esolu ion, ha ing imposed ∆ = 1/T, is o inc ease he dimension o he ansmi ed block o da a, h ough Nand M. Howe e , his ansla es in inc easing he o al signal du a ion and dec easing he subca ie s spacing, which could lead o an inc ease o he compu a ional complexi y (i 2.3. Simula ion Resul s 55 blockwise ope a ions ha e o be compu ed), o inc eased in e e ence e ec s ( o ins ance o OFDM). This is he cos o he p oposed join ada pa ame e es ima ion and communica ion se up using OFDM and/o OTFS, which allows he simul aneous ansmission o use ul in o ma ion and ada sensing wi hou any adeo , bu equi es complex signal p ocessing ope a ions ( o con as he block dimension o he in e e ence). On he o he hand, one can bypass he a o emen ioned p oblem by con- side ing he addi ional spa ial dimension, i.e, wi h a mul i an enna sys em, such ha dis inc a ge s can be iden i ied in h ee di e en domains: ange (delay), eloci y (Dopple ), and angula posi ion. As we will see, ou p oposed me hod is able o co ec ly de ec a ge s which a e sepa able in a leas one domain ou o h ee (delay, Dopple , and angle). 2.3 Simula ion Resul s 2.3.1 Join S a e Sensing and Communica ion The ada (backsca e ed) and o wa d communica ion SNRs a e de ined as SNR ad =λ2σ csG2 (4π)3 4 Pa g σ2 w ,(2.51) SNRcom =λ2G2 (4π)2 2 Pa g σ2 w ,(2.52) espec i ely, whe e λ=c/ cis he wa eleng h, σ cs is he ada c oss-sec ion in m2,Gis he an enna gain, and is he dis ance be ween Tx and Rx. In he case o mul ipa h, we ix SNRcom o be he SNR o he line o sigh (LoS) componen , and we add mul ipa h componen s wi h p og essi ely lowe SNRs, such ha he sum SNR o he channel inc eases wi h he numbe o pa hs P. This co esponds o he physically meaning ul case ha a iche p opaga ion en i onmen con eys mo e signal powe . Table 2.1 summa izes he ele an simula ion pa ame e s inspi ed by he au omo i e communica ion s anda d 56 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion Table 2.1: Simula ion pa ame e s c= 5.89 GHz M= 64 B= 10 MHz N= 50 ∆ =B/M = 156.25 kHz T= 1/∆ = 6.4µs σ cs = 1 m2G= 100 = 20 m = 80 km/h IEEE 802.11p [21], whe e and deno e he a ge ange and eloci y aken in o accoun .3 By b ie ly e iewing OFDM modula ion, we in oduce a widely used ada wa e o m known as equency modula ed con inuous wa e (FMCW) [13, Chap- e 4.6], in o de o unde s and he gain o he p oposed me hods agains well-known and cu en ly used app oaches o ada ehicula applica ions. Fo bo h OFDM and FMCW we conside a symbol leng h o To=TGI +T, i.e., including he possible p esence o a gua d in e al deno ed by TGI, gene - ally longe han he maximum pa h delay τmax. In OFDM he gua d in e al esul s o be he CP, ex ensi ely discussed in Sec. 1.2. In FMCW, he ada ansmi e sends a sequence o iden ical “chi p” pulses o du a ion T, each ollowed by a gua d in e al o leng h TGI o a oid in e -pulse in e e ence. By deno ing wi h φ( )=( c+B 2T ) he phase a ime , he ansmi signal is gi en by s( ) = N−1 X i=0 ej2πφ( −iT0) ec  −iT0 T,(2.53) whe e Nis he numbe o consecu i e pulses. T ansmi ing (2.53) o e he channel in (1.1), we ge he ecei ed signal ( ), whose exp ession, by neglec ing 3No e ha , i he condi ions τmax < T and νmax <∆ a e sa is ied, he nume ical esul s a e independen o he choice o and . 2.3. Simula ion Resul s 57 he noise, is gi en in (1.20), bu wi h he cu en s( ). A e some algeb a, i is easy o show ha he p oduc o he ecei ed and he ansmi signals gi es y( ) = ( )s∗( ) = P−1 X p=0 hpej2πνp e−j2π cτp N−1 X i=0 e−j2π b,p( −iT0−τp/2) ,(2.54) whe e b,p =B Tτpdeno es he so-called bea ing equency o pa h p. The ecei e samples y( ) e e y T/M o each pulse, i.e., o =iTo+lT Mwhe e i deno es he pulse index and ldeno es he sample index. By le ing L=M+C deno es he numbe o samples pe pulse, he sampled ecei ed signal can be ew i en as y[i, l] = yiT0+l s=X p hpej2π( b,p+νp)l sej2πνpiT0,(2.55) o i= 0, . . . , N −1 and l= 0, . . . , L −1, whe e hpabso bs a cons an phase e m independen o he indices i, l. The es ima ion o he 2Punknown pa ame e s {τp, νp}is hus ob ained by selec ing he peak o he ange-Dopple map ound by applying a wo- dimensional DFT o he noisy samples in (2.55) as p oposed in [26, 20]. The ange and eloci y o he a ge a e ob ained by he es ima es o {τ0, ν0}. 2.3.2 Join Rada and Communica ion Pe o mance The i s wo sub igu es o Fig. 2.2 show he eloci y and ange es ima ion oo MSE (RMSE) e sus SNR ad o a pu e LoS channel (P= 1) and o OTFS, OFDM, and FMCW. We no ice ha bo h digi al modula ion o ma s p o ide as accu a e ada pe o mance as FMCW, while ansmi ing use ul in o ma ion o any possible Rx. In addi ion, he hi d sub igu e o Fig. 2.2 shows he achie able a e o OFDM and OTFS, o simplici y o de i a ion, when Gaussian independen and independen and iden ically dis ibu ed (i.i.d.) symbols CGauss a e sen h ough he channel. This p o ides an achie able a e in he case o join de- ec ion and decoding, wi h uncons ained complexi y, as a unc ion o SNRcom, 58 Chap e 2. ML Me hods o Rada Pa ame e Es ima ion −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 10−2 10−1 100 101 102 103 SNR ad [dB] RMSE[ˆ ] / [m/s] OTFS OTFS CRLB OFDM FMCW OTFSWa e all Es ima ion OTFSRandom Es ima ion −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 10−2 10−1 100 101 102 103 SNR ad [dB] RMSE[ˆ ] / [m] OTFS OTFS CRLB OFDM FMCW OTFSWa e all Es ima ion OTFSRandom Es ima ion 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 SNRcom [dB] CGauss / [bi s/s/Hz] COTFS Gauss COFDM Gauss Figu e 2.2: F om op o bo om: he RMSE o he a ge eloci y es ima ion ˆ s SNR ad, he RMSE o he a ge ange es ima ion ˆ s SNR ad, and he Gaussian capaci y CGauss s. SNRcom, o he cu es indica ed in he di e en legends. 2.3. Simula ion Resul s 59 gi ing a quali a i e idea o pe o mance cu es. A mo e accu a e model should conside only de ec ion, i.e., sepa a ed om decoding, calcula ing he achie - able communica ion a e in e ms o p agma ic capaci y, as ex ensi ely done in Chap e 4. Gi en he ac ha we can model he block inpu -ou pu ela ion o OTFS as a MIMO channel, he mu ual in o ma ion wi h Gaussian inpu s and pe ec channel s a e in o ma ion (CSI) a he ecei e is gi en by [67] COTFS Gauss =NT NT +TGI 1 NM log2de I+ SNRcomΨΨH.(2.56) A simila exp ession o OFDM, owing o he ac ha he channel ma ix in OFDM is diagonal, wi h consequen symbol-by-symbol de ec ion, yields COFDM Gauss =T T+TGI log2(1 + SNRcom).(2.57) The ac o s NT/(NT +TGI) and T/(T+TGI) o OTFS and OFDM, espec- i ely, a e in oduced in o de o ake in o accoun he (possible) inse ion o a gua d in e al. In OTFS, a gua d in e al o du a ion TGI is inse ed a he end o each ame, whose du a ion is TOTFS =NT, comp ising Nconsecu i e symbols in he ime domain (see Chap e 1). In con as , in OFDM, he gua d in e al akes he o m o a CP, inse ed o each OFDM symbol o du a ion T. The ame size is hus TOFDM =NTo=N(T+Tcp). I is clea ha o p ac ical alues o he OTFS ame leng h N, he o e - head paid by OTFS is much less han he CP o e head paid by OFDM (in hese esul s we used TGI =T/4, which is ypical in he IEEE 802.11 amily o s anda ds). On he o he hand, he la ge o e head incu ed by OFDM yields a pa icula ly simple ecei e s uc u e, allowing symbol-by-symbol de ec ion hanks o he diagonaliza ion o he channel. In con as , OTFS equi es block- wise de ec ion o e he whole ame, which can be e y compu a ionally in- ensi e, especially o la ge Nand M. This is why he p oposed so -ou pu symbol de ec o p esen ed in Sec ion 4.2.1 is o pa icula in e es . Nex , we p esen a second se o esul s whe e we conside only OTFS bu in he p esence o a mul ipa h channels (P > 1, up o P= 4), showing 66 Chap e 3. ML Rada Me hods in MIMO Con igu a ions on a de ec ed a ge , maximizing he BF gain and he ecei ed SNR, shall be used o minimize “mul i- a ge ” in e e ence du ing a possible acking and communica ion phase [49, 13, 70]. Clea ly, wi hin he a ge de ec ion phase, a non- i ial adeo appea s. On one hand, wide angula sec o s co e age enable he de ec ion o mul iple a ge s i he ecei ed backsca e ed powe is high enough. On he o he hand, a mo e di ec ional BF g an s a highe ecei ed SNR, and hus longe de ec able a ge dis ance, a he cos o a ime-consuming sea ch o e na owe angula sec o s (as classical ada suc- cessi ely swapping adjacen egions, see, e.g., [13]). Di e en solu ions o he a o emen ioned p oblem can be ound in li e a u e (see, e.g., [52, 53, 15, 54]). As an ex ension o [71] and o he concep s in oduced in he p e ious pa s, his chap e s udies he join a ge de ec ion and pa ame e es ima- ion p oblem wi h a mono-s a ic MIMO ada adop ing OTFS, and ex ends he li e a u e esul s on MIMO con igu a ions (see, e.g., [44, 72]). The use o communica ion wa e o ms o MIMO ada has been mo i a ed by he join ada and communica ion pa adigm (as showed in Sec. 2.1), whe e wo unc- ions a e implemen ed by sha ing he same esou ces and he same wa e o m (see e.g. [24, 27, 25] and e e ences he ein). Con a y o he exis ing wo ks on ada sensing using OTFS [46, 71, 28], his use case conside s a MIMO ada unde a p ac ical mmWa e sys em a chi ec u e, such ha he numbe o adio equency (RF) chains (N ) is much smalle han he numbe o an- ennas (Na). In ac , he di icul ies ela ed o he implemen a ion o a ully digi al BF, o , equi alen ly, associa e one RF chain pe an enna (including A/D con e sion, modula ion, and ampli ica ion) in a small o m ac o and highly in eg a ed echnology o e a la ge signal bandwid h, a e well known. The e o e, ocusing on mmWa e au omo i e applica ions, we conside hyb id digi al-analog (HDA) BF schemes as ypically conside ed in li e a u e (see, e.g., [55, 56] and e e ences he ein). We will s udy wo di e en scena ios, explo ing he a o emen ioned BF adeo . The i s scena io conside s a Tx BF design wi h beam co e ing a wide angula sec o , o join ly pe o m a ge de ec ion, pa ame e es ima ion, and mul icas ing o a common message o 3.1. In oduc ion 67 all possible ac i e use s (see Fig. 3.1a). A possible applica ion o his model is, o ins ance, a base s a ion moun ed nea a highway able o collec a - ic in o ma ion h ough ada capabili ies while communica ing i s knowledge o ac i e Rx.1Assuming he communica ion phase es ablished, i.e., de ec ion al eady pe o med, he second scena io conside s a Tx BF wi h di ec ed na - ow beams, such ha indi idual in o ma ion s eams a e sen o he de ec ed use s, o g oups o use s, as depic ed in Fig. 3.1b. I is impo an o s ess he ac ha he ada Rx uses a wide beam pa e n consis ing o N beams, as illus a ed in Fig. 3.2, in o de o ob ain a meaning ul ec o obse a ion, necessa y o AoA es ima ion, ega dless o he ope a ing phase. This is in a sha p con as o he hyb id beam alignmen conside ed in a ypical communi- ca ion sys em, whe e he Rx also applies BF and ob ains a scala obse a ion p ecluding he es ima ion o he AoA (see, e.g., [55, 50] and e e ences he ein). Unde his se up, we p opose an e icien ML-based scheme combined wi h HDA BF o join ly pe o m a ge de ec ion and pa ame e es ima ion. Mo e p ecisely, ou scheme i s pe o ms a ge de ec ion and supe - esolu ion es- ima ion o delay, Dopple shi , and AoA using a wide angula beam along which a single da a s eam is sen . Then, once he a ge s a e de ec ed, he subsequen acking phase pe o ms he pa ame e es ima ion using mul iple na ow beams along which mul iple da a s eams can be sen . Ou nume i- cal esul s demons a e ha he p oposed scheme is able o eliably de ec mul iple a ge s while essen ially achie ing he CRLB o ada pa ame e es- ima ion. Fu he mo e we in es iga e a ious scena ios o nea - a e ec s o a ge s, showing ha a successi e in e e ence cancella ion (SIC) mechanism is able o e icien ly emo e he masking e ec be ween a ge s loca ed a di - e en anges om he ada , and we p o ide an in-dep h analysis o he wo scena ios o in e es , showing hei limi s and ad an ages. 1I is clea ha , ge ing id o he message sen , ada asks can be pe o med by using any well known ada wa e o m [13], and he use o a digi al communica ion o ma is poin less ( he ansmission o in o ma ion could s a in a second communica ion phase, e.g., wi hin a ime di ision p o ocol). 68 Chap e 3. ML Rada Me hods in MIMO Con igu a ions Tx / Rada Rx Communica ion Rx / Rada Ta ge Tx / Rada Rx Wide Angula Sec o / GTX Wide Angula Sec o / GTX (a) De ec ion phase Tx / Rada Rx Communica ion Rx / Rada Ta ge Ta ge BF / GTX Ta ge BF / GTX Tx / Rada Rx Ta ge BF / GTX Ta ge BF / GTX (b) T acking phase Figu e 3.1: Two scena ios wi h wo di e en Tx beam pa e ns. In (a), a Tx (a base s a ion o a ca ) b oadcas s a common message explo ing a wide angula sec o . In (b), we conside di ec ional BF owa ds he de ec ed a ge s. The Rx always makes use o a wide beam wi hin he angula sec o o in e es . We ema k he e he used no a ion. (·)Tdeno es he anspose ope a ion. (·)Hdeno es he He mi ian (conjuga e and anspose) ope a ion. Ope a o |·| deno es he absolu e alue |x|i x∈R, o he ca dinali y (numbe o elemen s) o a disc e e se , i.e., |F|, i Fis a disc e e se . 3.2 Physical model We conside a join ada de ec ion and pa ame e es ima ion sys em ope - a ing o e a channel bandwid h Band a he ca ie equency c. The Tx is equipped wi h a mono-s a ic MIMO ada wi h Naan ennas and N RF 3.2. Physical model 69 Rada Rx N Beams Wide Angula Sec o Figu e 3.2: Beam con igu a ion a he ada Rx o he wo scena ios depic ed in Fig. 3.1. N beams co e a wide angula sec o . chains, ope a ing in ull-duplex mode.2The ada Rx (coloca ed wi h he Tx, i.e., mono-s a ic assump ion) p ocesses he backsca e ed signal o iden i y he p esence o a ge s wi hin he beam, while es ima ing pa ame e s o in e es such as ange, eloci y, and AoA. A poin a ge model is aken in o accoun , such ha each a ge can be ep esen ed h ough i s LoS pa h only [17, 21, 15]. By le ing he s ee ing angle φ∈[−π 2,π 2], by conside ing an an enna a ay wi h λ/2 spacing (whe e λis he wa eleng h), he Tx and Rx a ays a e gi en by a(φ) and b(φ) espec i ely, whe e a(φ)=(a1(φ), . . . , aNa(φ))T∈CNa, deno es he uni o m linea a ay esponse ec o o he ada Rx wi h an(φ) = ej(n−1)πsin(φ), n = 1, . . . , Na,(3.1) and bn(φ) = an(φ). In ac , gi en he mono-s a ic ada , he same s ee ing angle φis bo h a ada Tx and Rx, hus, ec o s aand b esul o be equal. The channel is modeled as an ex ension o he P- ap ime- equency selec i e 2Full-duplex ope a ions can be achie ed wi h su icien isola ion be ween he Tx and he ( ada ) de ec o and possibly in e e ence analog p e-cancella ion in o de o p e en he ( ada ) de ec o sa u a ion [31, 73, 32]. 70 Chap e 3. ML Rada Me hods in MIMO Con igu a ions channel in (1.1) including he addi ion spa ial dimension, which is gi en by [7] H( , τ) = P−1 X p=0 hpb(φp)aH(φp)δ(τ−τp)ej2πνp ,(3.2) whose dimension is Na×Na, whe e Pis he numbe o a ge s, hpis a complex channel gain including he pa hloss, νp=2 p c cand τp=2 p ca e he ound- ip Dopple shi and delay, while φpdeno es he AoA, each co esponding o he p- h a ge , espec i ely. 3.2.1 OTFS Inpu Ou pu Rela ion We conside OTFS wi h Msubca ie s o bandwid h ∆ each, such ha he o al equency band is gi en by B=M∆ . As al eady w i en in many p e ious chap e s, Tdeno es he symbol ime, and he OTFS ame du a ion is NT. The Dopple -delay dimensions, acco ding o (1.13), a e Nand M, while he equa ion T∆ = 1 is always alid (see Chap e 1 and Sec. 1.3 o mo e de ails). In o de o conside he a o emen ioned di e en ope a ional modes, i.e., de ec ion and acking phases, we le Nsdeno e he numbe o da a s eams o be sen in each ime- equency slo , such ha Ns= 1 co esponds o he mul icas ing o a single da a s eam (in a possible de ec ion phase whe e a base s a ion wan s o sha e a common in o ma ion while de ec ing ac i e a ge s) and Ns≤N co esponds o he b oadcas ing o indi idual da a s eams ( owa ds he ac i e a ge s al eady de ec ed in he p e ious phase). Following he s anda d de i a ion o he inpu -ou pu ela ion o OTFS (see Sec. 1.3), he e we ex end he equa ions o also conside he addi ional spa ial dimension. Le us deal wi h Ns-dimensional da a symbols {xk,l}, o k= 0, . . . , N−1, l= 0, . . . , M−1, belonging o any cons ella ion, and a anged in he N×M wo-dimensional Dopple -delay g id Γ = {(k/NT, l/M∆ )}in (1.13). The Tx i s applies he ISFFT o con e da a symbols {xk,l}in o a Ns×1 block {X[n, m]}in he ime- equency domain X[n, m] = N−1 X k=0 M−1 X l=0 xk,lej2π(nk N−ml M),(3.3) 3.2. Physical model 71 o n= 0, . . . , N −1, m= 0, . . . , M −1, sa is ying he a e age powe cons ain E[X[n, m]HX[n, m]] = Pa g/(NMNa)INa, whe e INsdeno es an iden i y ma ix o dimension Nsand Pa g some o al maximum powe allowed by he sys em. A e assigning Nss eams o N RF chains h ough a mapping ma ix V∈ CN ×Ns(since gene ally Ns≤N , hus a mapping is necessa y), he Tx gene a es he N -dimensional con inuous- ime signal s( ) = V N−1 X n=0 M−1 X m=0 X[n, m]g x( −nT)ej2πm∆ ( −nT ),(3.4) in which g x( ) is he shaping pulse applied in ansmission (see Chap e 1 o mo e de ails and Sec. 1.3.8). Since he numbe o RF chains is ypically much smalle han he numbe o an ennas, di e en ypes o HDA a chi ec u es be ween RF chains and an ennas ha e been conside ed in he li e a u e (see e.g. [55]). In his pape , we ocus on he ully-connec ed HDA scheme o [55]. Fo any HDA a chi ec u e, he Tx applies he hyb id BF ma ix deno ed by F∈CNa×N ha cap u es bo h baseband and RF analog BF (see [51, 54]), while he Rx sees he ecei ed signal o a educed dimension h ough a p ojec ion ma ix deno ed by U∈CN ×Na. By imposing (FVVHFH) = Na, he o al powe cons ain Pa g is sa is ied. In o he wo ds, he Rx canno access o each an enna elemen , bu ob ains only a p ojec ion o he ecei ed signal. By ansmi ing he signal (3.4) o e he channel (3.2), he N -dimensional con inuous- ime ecei ed signal is gi en by ( ) = P−1 X p=0 hpUb(φp)aH(φp)Fs( −τp)ej2πνp ,(3.5) whe e we omi ed he noise o simplici y. I is in e es ing he e o compa e hese equa ions wi h he single spa ial dimensional ones p esen ed in Sec. 1.3. The ou pu o he Rx il e -bank adop ing a gene ic ecei e shaping pulse 72 Chap e 3. ML Rada Me hods in MIMO Con igu a ions g x( ) is y( , ) = Z ( 0)g∗ x( 0− )e−j2π 0d 0 =Z 0 g∗ x( 0− ) P−1 X p=0 hpUb(φp)aH(φp)Fs( 0−τp)ej2πνp 0e−j2π 0d 0 =X p,n0,m0 hpUb(φp)aH(φp)FVX[n0, m0]Z 0 g∗ x( 0− )g x( 0−τp−n0T) ej2πm0∆ ( 0−τp−n0T)ej2π(νp− ) 0d 0.(3.6) By sampling a =nT and =m∆ , we ob ain y[n, m] = y( , )| =nT, =m∆ = N−1 X n0=0 M−1 X m0=0 hn,m[n0, m0],(3.7) whe e he ime- equency domain inpu -ou pu ela ion hn,m[n0, m0] is hn,m[n0, m0] = P−1 X p=0 h0 pUb(φp)aH(φp)FVX[n0, m0]ej2πn0T νp Cg x,g x ((n−n0)T−τp,(m−m0)∆ −νp)e−j2πm∆ τp,(3.8) whe e Cu, (τ, ν) deno es he CAF, we le h0 p=hpej2πνpτp, and imposed he e m e−j2πmn0∆ T = 1, ∀n0, m, unde he hypo hesis T∆ = 1. Since each Xi[n, m] is gene a ed ia ISFFT, he ecei ed signal in he delay-Dopple do- main is ob ained by he applica ion o he SFFT y[k, l] = X n,m y[n, m] NM ej2π(ml M−nk N)=X k0,l0 gk,k0l, l0,(3.9) whe e he ISI coe icien o he Dopple -delay pai [k0, l0] seen by sample [k, l] is gi en by gk,k0l, l0=X p h0 pUb(φp)aH(φp)FVxk0,l0Ψp k,k0[l, l0],(3.10) 3.2. Physical model 73 −60−50−40−30−20−10 0 10 20 30 40 50 60 −50 −40 −30 −20 −10 0 10 20 Angle BF Gain [dB] Co e age 10◦ Co e age 20◦ Co e age 30◦ Figu e 3.3: BF design o di e en angula co e age. Clea ly, wide he beam, less he BF gain. wi h Ψp k,k0[l, l0] de ined as Ψp k,k0[l, l0] = X n,n0,m,m0 Cg x,g x ((n−n0)T−τp,(m−m0)∆ −νp) NM ej2πn0T νp e−j2πm∆ τpej2πn0k0 N−m0l0 Me−j2π(nk N−ml M),(3.11) while simpli ied e sion o Ψp k,k0[l, l0] ob ained by app oxima ing he CAF can be ound in Sec. 1.3. 3.2.2 Beam o ming ma ices The design o he BF ma ix Fa he ada Tx depends on he ope a ing phase, and anyway Fis chosen such ha Tx and Rx a e aligned owa ds he same wide angula sec o . Following [51, Sec ion III.C], we cons uc F∈CNa×N o co e a wide angula sec o [−θ, θ] as ollows. By ep esen ing his angula sec o by a disc e e se o N angles, deno ed by Θ = {±(θ/(2N ) + kθ/N )}, 74 Chap e 3. ML Rada Me hods in MIMO Con igu a ions o k= 0, . . . , N /2−1, each column o F= [ 1,..., N ] akes he o m i=a(θi) |a(θi)|, i = 1, . . . , N ,(3.12) whe e a(θi) is de ined in (3.1) and aking in o accoun a sui able no maliza ion. An example on how he BF beams, w. . . he associa ed gain, look like is shown in Fig. 3.3, o di e en angula co e age. Du ing he a ge acking phase, we o m mul iple na ow beams co e- sponding o he es ima ed AoA o he de ec ed a ge s. This is illus a ed wi h ed and blue beams, co esponding o wo di e en AoA, in Fig. 3.1b. Assum- ing ha P a ge s a e de ec ed and hei espec i e AoA a e es ima ed, we cons uc Fby eplacing θiby ˆ φp o he i s Pcolumns in (3.12) [51, Sec ion III.B]. In such a case, he BF gain owa ds a single a ge is app oxima ely ≃Na, while dec eases o each o he conside ed a ge , i.e., ≃Na/P. Con a y o he ansmi beam o ming ma ix, he educ ion ma ix Ua he ada Rx emains he same o bo h de ec ion and acking phases. Namely, we se U=FH, whe e each column is gi en in (3.12). This is illus a ed in Fig. 3.2. This choice o an iso opic ecei e beam enables o ob ain a mul i- dimensional signal o AoA es ima ion in bo h de ec ion and acking phases. 3.3 Join De ec ion and Pa ame e s Es ima ion We wish o es ima e he se o ou pa ame e s θ={h0 p, φp, τp, νp}∈TP, wi h T=C×R×R×R. By de ining Gp(τp, νp, φp),(Ub (φp)aH(φp)FV)⊗Ψp,(3.13) whe e ⊗is he K onecke p oduc ,3as he N NM ×NsNM ma ix ob ained by mul iplying Ψpby a di e en coe icien o (Ub (φp)aH(φp)FV). Thus, by s acking Xin o a NsNM-dimensional ec o xand de ining an ou pu ec o 3No e ha AX×Y⊗BZ×K=CXZ×Y K . 3.3. Join De ec ion and Pa ame e s Es ima ion 75 yo dimension NMN ×1, he ecei ed signal in he p esence o noise is y= P−1 X p=0 h0 pGp(τp, νp, φp)x+w,(3.14) whe e wdeno es he AWGN ec o wi h independen and iden ically dis- ibu ed en ies o ze o mean and a iance σ2 w. The p oblem consis s, i s , o he de ec ion o he P a ge s (in case o a i s acquisi ion phase), oge he wi h he es ima ion o he associa ed 4Ppa ame e s (complex channel coe i- cien , Dopple , delay, and angle) om he N MN-dimensional ecei ed signal. To his end, we de ine he ML unc ion as l(y|θ,x) =  y−X p h0 pGpx 2 , =yHy−yHX p h0 pGx −X p h0∗ pxHGHy +xH X p h0 pG!H X q h0 qGq!x,(3.15) whe e we use he sho hand no a ion Gp,G(τp, νp, φp). No e he simila i ies and di e ences w. . . (2.27). The ML solu ion is gi en by ˆ θ= a g min θ∈T P l(y|θ,x).(3.16) Fo a ixed se o {φp, τp, νp}, he ML es ima o o {h0 p}is gi en by sol ing he ollowing se o equa ions xHGH p  P−1 X q=0 h0 qGq x=xHGH py, p = 0, . . . , P −1.(3.17) By plugging (5.34) in o (5.32), i eadily ollows ha minimizing l(y|θ,x) educes o maximize he ollowing unc ion l2(y|θ,x) = X p h0 pyHGpx=X p Sp(τp, νp, φp)−Ip({h0 q}q6=p,θ),(3.18) 82 Chap e 3. ML Rada Me hods in MIMO Con igu a ions While wo dis inc a ge s in he angle domain can be iden i ied i he angula esolu ion mee s some condi ions (depending on he numbe o an- ennas, he angula dis ance be ween he wo a ge s, and he an enna a ay p ope ies) [13]. The eloci y and he ange esolu ion is de e mined by he sys em pa ame e s in Table 3.1 and gi en by es =Bc 2NM c [m/s] , es =c 2B[m] .(3.29) In o de o ge a easonable ange esolu ion, e.g., <1 [m], a la ge bandwid h has o be conside ed.7Since he eloci y esolu ion is di ec ly p opo ional o B, o a ixed c, he only way o ob ain lowe alues is o inc ease he block size NM, leading o a ema kable inc ease in compu a ional complexi y, which could be no a o dable. Fo his eason, we se he sys em pa ame e s by ocus- ing on a easonable ange esolu ion (and heo e ical maximum ange) unde a easible compu a ional complexi y. No e ha he maximum ange could no be achie ed i he backsca e ed powe is below he noise loo . Howe e , he chosen sys em se up leads o an una oidable e y la ge eloci y esolu ion. Unde he a o emen ioned assump ions, aken a he beginning o Sec ion 5.4, he p oblem o a ge s iden i iabili y appea s only in he angula domain. How- e e , his only happens a mmWa e, hus, ange and eloci y esolu ions a e epo ed he e o comple eness, since he p oposed scheme could a ge lowe equencies, whe e he a o emen ioned assump ion migh no be sa is ied. Rema k: The pa ame e es ima ion pe o mance o he p oposed ML- based algo i hm, in pa icula ange and eloci y es ima ion, s ic ly depends on he dimension o he block o da a sen , i.e., he p oduc N·M. Thus, he sys em pa ame e s o Tab. 3.1 can be easily uned o achie e he desi ed le els o ada esolu ions (modi ying he bandwid h), acquisi ion ime (based on he leng h o he OTFS ame in ime), maximum ange, e c. Clea ly, he CRLB changes acco dingly. Mo eo e , no e ha his is also possible hanks o 7No e ha a adeo appea s. La ge bandwid hs mean mo e p ecise esolu ion, bu lowe heo e ical maximum ange (wi h he same N×Mg id). We ema k ha ou algo i hm is comple ely independen o hese choices. 3.4. Nume ical Resul s 83 OTFS modula ion, which is no sensi i e o Dopple and delay e ec s. Rema k: The ( ada ) ange and eloci y esolu ion in (3.29) indica es he minimum necessa y a ge s spacing, in one o he wo domains, such ha bo h o hem a e dis inguishable a he ada Rx. This is no linked o he pe o mance o ou ML-based de ec o , which is able o accu a ely es ima e he pa ame e s a beyond he esolu ion in (3.29). Thus, he e is a huge di e ence be ween a ge s iden i iabili y and es ima ion pe o mance. 3.4.1 Simula ion Resul s Fig. 3.4 shows he ada pe o mance in e ms o p obabili y o de ec ion (PD), ange/ eloci y/AoA es ima ion du ing he de ec ion phase (Fig. 3.1a). When mo e han one a ge is conside ed wi hin he simula ion scena io, he PD Pd is a e aged w. . . all P a ge s, i.e., Pd=PP−1 p=0 Pd(p) P,(3.30) whe e Pd(p) deno ed he PD o he p- h a ge . Fi s , no e ha , by conside ing an angula co e age o 10 deg ees (blue line), he maximum ange o co ec ly iden i y a a ge , limi ed by he pa hloss and hus di e en om he heo e ical limi indica ed in Table 3.1, is abou 110 m. Fo any dis ance be ween Tx and a ge , he es ima ion pe o mance o ada pa ame e s o in e es ( ange, eloci y, and AoA) ollows he co espond- ing CRLB. Mo e in de ails, no e ha a he limi ange o 110 m, he RMSE o ange, eloci y, and angle a e espec i ely, ≃4·10−2[m], ≃1.6·101[m/s], ≃4·10−2[deg ee]. As expec ed, gi en he sys em pa ame e s, he eloci y RMSE is qui e poo , while he o he es ima ion pe o mances a e sa is ac o y. Howe e , a p ope BF design owa ds a ge s, in a subsequen acking phase, could imp o e he es ima ion pe o mance maximizing he ecei ed SNR, as showed in nex esul s. As seen om Fig. 3.4, by inc easing he angle sec o om 10◦ o 30◦, he backsca e ed powe ge s smalle (less BF gain), hence he maximum ange signi ican ly dec eases. The e exis s a non- i ial adeo 84 Chap e 3. ML Rada Me hods in MIMO Con igu a ions 10 20 30 40 50 60 70 80 90 100110120130 140 150 0 0.2 0.4 0.6 0.8 1 Range [m] Pd Sec o 10◦ Sec o 20◦ Sec o 30◦ 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−4 10−3 10−2 10−1 100 Sec o 10◦20◦30◦ [m] CRLB Range [m] RMSE[ˆ ] / [m] 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−1 100 101 102 Sec o 10◦20◦30◦ [m/s] CRLB Range [m] RMSE[ˆ ] / [m/s] 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−4 10−3 10−2 10−1 100 Sec o 10◦20◦30◦ ϕ[deg] CRLB Range [m] RMSE[ˆ ϕ] / [deg] Figu e 3.4: De ec ion phase. Single a ge a a di e en dis ance wi hin he illumina ed angula sec o o speci ied co e age. RMSE pe o mance wi h as- socia ed CRLB and de ec ion pe o mance. Na= 128. 3.4. Nume ical Resul s 85 be ween he wid h o beams and ada pe o mance. Wide angula sec o s al- low o explo e he en i onmen in less ime, bu wi h limi ed maximum ange, while na owe sec o s maximize he ecei ed powe and he maximum a ge ange, a he cos o a ime consuming beam sweeping sea ch. Clea ly, RMSE pe o mance can no be compu ed i he PD is equal o 0, i.e., he a ge is no de ec ed, hus RMSE cu es may s op a ce ain anges, as isible in Fig. 3.4. Fig. 3.5 shows he pe o mance o he SIC echnique p esen ed in Algo i hm 2 du ing he de ec ion phase in he scena io depic ed in Fig. 3.6. Namely, he Tx wishes o de ec wo a ge s, one a ixed dis ance o 10 [m], he o he mo ing w. . . he x-axis, i.e., om 20 o 150 [m] (see Fig. 3.6). SIC is necessa y because he close a ge (black ca ) will mask he u he a ge (blue ca ) so ha he la e canno be de ec ed. Fi s , he i s plo o Fig. 3.5, e e ed o he PD, shows ha , when he mo ing a ge is loca ed a anges g ea e han 90 [m], co esponding o he ela i e ange beyond 80 [m], he masking e ec is no emo ed e icien ly by he SIC echnique ( he esidual in e e ence is ema kable), and he a ge a longe dis ance is no de ec ed co ec ly. In ac , a he ex eme poin , he cu e la s o Pd= 0.5, because only one a ge ou o wo (clea ly, he closes o he ada Rx, i.e., he one ixed a 10 [m]) is co ec ly de ec ed. As clea ly isible, he pe o mance in e ms o RMSE, which conside s in his case he es ima ion pe o mance a e aged w. . . he de ec ed a ge s (no e ha he a ge loca ed a 10 [m] is always de ec ed co ec ly), sligh ly changes while conside ing one o wo a ge s, as a con i ma ion o he e ec i eness o he p oposed algo i hm. No e ha he blue cu es o Fig. 3.5 co espond o he blue ones o Fig. 3.4. Now we conside he acking phase co esponding o Fig. 3.1b. The sce- na io akes in o accoun one Tx and h ee a ge s wi hin an angula sec o o 10◦, as shown in Fig. 3.7. Fig. 3.8 shows he RMSE pe o mance o he e e - ence a ge (black ca ), in he p esence o o he wo a ge s (blue ca s), No e ha dis ance, eloci y, and angula posi ion o all h ee a ge s a e andomly chosen a e e y Mon e Ca lo i e a ion, in such a way he comple e masking 86 Chap e 3. ML Rada Me hods in MIMO Con igu a ions 20 30 40 50 60 70 80 90 100 110 120 130 140 150 0 0.2 0.4 0.6 0.8 1 Range [m] Pd 1 Ta ge 2 Ta ge s 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−3 10−2 10−1 N. Ta ge s 1 2 [m] CRLB Range [m] RMSE[ˆ ] / [m] 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−1 100 101 102 N. Ta ge s 1 2 [m/s] CRLB Range [m] RMSE[ˆ ] / [m/s] 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−3 10−2 10−1 N. Ta ge s 1 2 ϕ[deg] CRLB Range [m] RMSE[ˆ ϕ] / [deg] Figu e 3.5: De ec ion phase. Two a ge , one loca ed a 10 [m] and he o he mo ing a a di e en dis ance (x-axis) wi hin he illumina ed 10◦angula sec- o as shown in Fig. 3.6. RMSE pe o mance wi h associa ed CRLB. De ec ion pe o mance. Na= 128. 3.4. Nume ical Resul s 87 Fixed Ta ge Mo ing Ta ge Tx / Rada Rx Figu e 3.6: Example o scena io depic ed in Fig. 3.5. The ixed a ge (in black) masks he mo ing a ge , in blue, which changes i s loca ion wi hin he illumina ed angula sec o . Re e ence Ta ge Tx / Rada Rx In e e ence Ta ge In e e ence Ta ge Rx BF Pa e n Figu e 3.7: Example o scena io depic ed in Fig. 3.8. The goal is o co ec ly es ima e he pa ame e s o he e e ence a ge (in black), while in e e ence a ge s (in blue) lay wi hin he same Rx BF pa e n (depic ed in Fig. 3.2 and wi h he black shape in his igu e). 88 Chap e 3. ML Rada Me hods in MIMO Con igu a ions 50 100 150 200 250 300 350 400 10−4 10−3 10−2 10−1 100 101 102 NaSim CRLB 16 32 64 128 Range [m] RMSE[ˆ ] / [m] 50 100 150 200 250 300 350 400 10−1 100 101 102 103 NaSim CRLB 16 32 64 128 Range [m] RMSE[ˆ ] / [m/s] 50 100 150 200 250 300 350 400 10−4 10−3 10−2 10−1 100 101 NaSim CRLB εBW 16 32 64 128 Range [m] RMSE[ˆ φ] / [deg ee◦] Figu e 3.8: T acking scena io. Tx BF dis inc beams owa ds h ee di e en a ge s. 3.4. Nume ical Resul s 89 e ec p esen ed in Fig. 3.5 does no occu , and an a e age o RMSE esul s is inally compu ed. F om Fig. 3.8, we obse e ha he RMSE c i ically depends on he numbe o an ennas. This is because he BF gain g ows p opo ionally wi h he numbe o an ennas and inc eases he backsca e ed signal powe . Mo eo e , no e ha a ( e e sed) “wa e all” beha io is shown o ange and eloci y es ima ion. This is because, e en i he p esence o he a ge is gi en o g an ed, low SNR alues migh s ill lead o a la ge e o du ing he Dopple - delay ML maximiza ion (see Algo i hm 2). The wa e all beha io is ypical o ML es ima o s and has been ex ensi ely analyzed in Sec. 2.1.2.3. Also no e ha he AoA RMSE pe o mance is uppe limi ed by he 3-dB beamwid h o he beam pa e n (see, e.g., [51, 76]). In ac , supposing ha he a ge posi ion lies wi hin he 3-dB beamwid h, also he ini ial AoA es ima ion ( he uppe and lowe limi o ma ix Ω in (3.21)) is limi ed o ha wid h. As a consequence, he RMSE does no exceed a sys ema ic e o , indica ed as εBW, calcula ed by a e aging o e andom AoA es ima ion ealiza ions wi hin he ange o possibili ies, i.e., be ween he uppe and lowe limi se by he 3-dB beamwid h o he beam pa e n, equi alen ly o he analysis ela ed o (2.44), which can be also applied o ange and eloci y es ima ions. Chap e 4 OTFS De ec ion 4.1 In oduc ion We will now ocus on he OTFS da a de ec ion a he Rx side. By conside ing he communica ion-o ien ed channel model in (1.1), i.e., aking in o accoun one-way Dopple and delay shi (modeling a o wa d communica ion channel, agains he ada wo-way scena io), in line wi h mos o he cu en li e a u e on OTFS de ec ion (e.g., [62, 4]) we assume pe ec CSI a he Rx, i.e., he knowledge o channel ma ix Ψ. The acquisi ion o such CSI is a dis inc p oblem o independen in e es , ha will be analyzed in 5 and has been s udied in li e a u e in ecen yea s (see, e.g., [37, 44, 45]). In his chap e , we conside sepa a e de ec ion and decoding, whe e he Rx consis s o he conca ena ion o a so -ou pu symbol de ec o , p oducing so -es ima es o he coded symbols x, and a (sepa a e) decode ha akes such es ima es as he ou pu o a i ual channel ha inco po a es also he de ec o . Thus, we do no conside “ u bo equaliza ion” schemes, in which de ec ion and decoding a e join ly pe o med h ough successi e i e a ions in- ol ing eedback loops (as, e.g., in [77]). This choice is jus i ied by he ac ha he p esence o a o wa d e o co ec ion scheme, oge he wi h he speci ic de ini ion o he channel model, could obscu e he eal pe o mance o he de- 91 98 Chap e 4. OTFS De ec ion disc e e a iables aking alues in C. The e o e, i has a complexi y equal o |C|di−1, whe e |·|indica es he ca dinali y o he se , which may be p ohibi i ely la ge o la ge cons ella ions and, abo e all, i depends on he spa si y o he channel. The e o e, he exac applica ion o he SPA compu a ion ules o he FG ob ained di ec ly om he ma ix Ψis highly imp ac ical. Fo his eason, he au ho s o [62] p opose o use a Gaussian app oxima ion o he in e e ing symbols in he compu a ion a nodes Qi(x), which e ec i ely elies o a so in e e ence cancella ion app oach, as al eady widely used in u bo equaliza ion and in he con ex o mul iuse de ec ion in [39, 85]. The de ails o he esul ing MP algo i hm can be ound in [62]. I should be also men ioned ha , o he sake o simplici y and in o de o inc ease he spa si y o he FG in Fig. 4.2, he de ec o p oposed in [62] cons uc s he nominal ma ix Ψby ounding he delay shi s o in ege s on ecei e sampling g id. Unde his condi ion, he channel ma ix is e y spa se since many coe icien s co esponding o sampling a non-in ege delay shi s a e iden ically o ze o, and he numbe o connec ions o each node is e- duced, while p ese ing he leng h-4 cycles p oblem, una oidable wi h his app oach. Ne e heless, since he assump ion is gene ally no sa is ied by eal- wo ld channels, such app oxima ion o he channel ma ix esul s in neglec ing a signi ican componen o he ISI. We shall e i y ha when such in ege de- lay shi ounding is applied o he cons uc ion o he nominal ma ix Ψused by he de ec o bu he ac ual delay shi s ha e a ac ional componen (as i is always he case in p ac ice), he misma ch yields a signi ican pe o - mance deg ada ion. This shows ha neglec ing he ac ional pa o delays and Dopple shi s, as ou inely done in he li e a u e o OTFS, may be indeed qui e misleading. 4.2.3 Linea block-wise MMSE equaliza ion As a u he e m o compa ison we conside also he s anda d linea MMSE block equalize , applied o he channel model (2.17). In his case, he so - ou pu is simply he linea MMSE es ima e o symbols x om he obse a ion 4.3. Pe o mance o Sepa a ed De ec ion and Decoding 99 y, gi en by ˆ xLMMSE =ΨHΨΨH+σ2 wI−1y.(4.13) The complexi y o his app oach is p opo ional o O(NM)3, so i becomes quickly un easible o ypical alues o Nand M. A low-complexi y (mis- ma ched) linea minimum mean squa e e o (LMMSE) app oach was ecen ly p oposed in [47], elying on he cyclic p ope ies o channel ma ix Ψunde pe ec bi-o hogonali y o shaping pulses g x( ) and g x( ), oge he wi h he assump ion o on-g id Dopple and delay shi s (wi h Dopple -delay g id Γ). In eal channel condi ions, i.e., in he p esence o p ac ical ec angula pulses and non-in ege delay and Dopple shi s, he pe o mance o his app oach isibly deg ades. 4.3 Pe o mance o Sepa a ed De ec ion and Decoding As al eady men ioned, we cha ac e ize he pe o mance o sepa a ed de ec ion and decoding schemes in e ms o p agma ic capaci y, i.e., he mu ual in o ma- ion o he i ual channel wi h inpu he cons ella ion symbols, assumed wi h uni o m p obabili y, and ou pu p o ided by he so -ou pu o he de ec o . This mu ual in o ma ion p o ides an achie able a e o sepa a ed de ec ion and decoding, o any gi en de ec ion scheme [78, 79]. Le us conside a sequence o symbols {xk}belonging o he signal con- s ella ion C, and le Vk(xk) deno es he de ec o so -ou pu . In he case o MP-based de ec o s, Vk(xk) is gi en in he o m o a pos e io p obabili y dis ibu ion on xk∈ C, while in he case o linea equalize s (e.g., he linea MMSE es ima o in (4.13)), his is gi en as he noisy es ima e ˆxkwhich is ea ed as he ou pu o a ( i ual) AWGN channel. In any case, he p ag- ma ic capaci y is simply de ined as he symbol-by-symbol mu ual in o ma ion IPC (xk;V(xk)). When V(xk) akes on he o m o a pos e io p obabili y dis- 100 Chap e 4. OTFS De ec ion ibu ion, his can be easily calcula ed as IPC(xk;V(xk)) ,H(xk)−H(xk|V(xk)) = log2C − X xk∈C V(xk) log2 1 V(xk),(4.14) and, by conside ing a Mon e Ca lo simula ion o e Kdi e en ealiza ions IPC = K−1 X k=0 H(xk)−H(xk|V(xk)) = log2C − 1 K K−1 X k=0   X xk∈C V(xk) log2 1 V(xk)   .(4.15) In he case o linea equaliza ion (i.e., V(xk) = ˆxk), he p agma ic capaci y is simply gi en by he symme ic capaci y (i.e., wi h symbols used wi h uni o m p obabili y) o he signal cons ella ion C, o an AWGN channel wi h SNR equal o he ou pu signal- o-in e e ence noise a io (SINR) o he equalize [67]. Fo he sake o compa ison, we also show he symme ic capaci y o he signal cons ella ion in an AWGN channel wi h SNR equal o SNRcom (i.e., he SNR o he LoS pa h), deno ed by Csym AWGN, and he mu ual in o ma ion wi h Gaussian inpu s COTFS Gauss , o he case P= 1. Fig. 4.3 and Fig. 4.4 show he pe o mance o he a ious me hods o OTFS so -ou pu de ec ion o a 16-quad a u e ampli ude modula ion (QAM) and sys em pa ame e s lis ed in Tab. 2.1. Fig. 4.3 shows he esul s o he (un ealis ic) case whe e he channel Dopple shi s and delays a e exac ly on he disc e e Dopple -delay g id Γ used by he ecei e sampling. In con as , Fig. 4.4 shows he esul s when he ac ual channel has a bi a y Dopple and delay shi s (wi h a andom uni o mly dis ibu ed ac ional pa ), while ce - ain algo i hms s ill assume such in ege g id when cons uc ing he nominal channel ma ix Ψused by he de ec o (as ad oca ed o example in [62, 47]). We no ice ha he p oposed MP-based app oach based on he de ini ion o ma ix Gou pe o ms he one in [62] in bo h scena ios. In pa icula , i 4.3. Pe o mance o Sepa a ed De ec ion and Decoding 101 −4−2 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 SNRcom [dB] P agma ic Capaci y — IPC hbi s symboli 16-QAM P= 1 P= 2 P= 3 P= 4 MPG MPΨ LMMSE LMMSELC Csym AWGN COTFS Gauss Figu e 4.3: Symbols de ec ion pe o mance in e ms o p agma ic capaci y o 16-QAM modula ion. The cu es show he beha io o ma ix Gbased MP algo i hm and MP algo i hm o [62] unde app oxima ed channel condi ions, i.e., wi h delay and Dopple shi s on he Dopple -delay g id, o a mul ipa h channel wi h di e en numbe o componen s P. 102 Chap e 4. OTFS De ec ion −4−2 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 SNRcom [dB] P agma ic Capaci y — IPC hbi s symboli 16-QAM P= 1 P= 2 P= 3 P= 4 MPG MPΨ LMMSE LMMSELC Csym AWGN COTFS Gauss Figu e 4.4: Symbols de ec ion pe o mance in e ms o p agma ic capaci y o 16-QAM modula ion. The cu es show he beha io o ma ix Gbased MP algo i hm and MP algo i hm o [62] unde eal channel condi ions, i.e., wi h delay and Dopple shi s no on he Dopple -delay g id, o a mul ipa h channel wi h di e en numbe o componen s P. 4.3. Pe o mance o Sepa a ed De ec ion and Decoding 103 su e s om almos no deg ada ion due o he non-in ege Dopple and de- lay shi s, unlike he me hod o [62], which has been de i ed based on hese s ong assump ions. The LMMSE equalize (4.13) wi h ull complexi y yields e y good pe o mance, paying only a small SNR penal y w. . . he p oposed MP-based scheme o P= 1,2, and ou pe o ming he MP-based scheme o iche sca e ing P= 3,4. Howe e , as said be o e, i s complexi y is cubic in he ame dimension NM, which is una o dable in p ac ical implemen a- ions. Un o una ely, he low-complexi y LMMSE es ima o (cu es indica ed by LMMSELC in he igu es), p oposed in [47] and exploi ing a speci ic s uc- u e o he channel ma ix Ψ, which equi es doubly block ci culan ea u e o he OTFS channel ma ix, is sa is ied only when g x( ) and g x( ) a e bi- o hogonal. Unde such condi ion, as demons a ed in [47], his scheme coin- cides wi h he MMSE block equalize , bu wi hou he need o a la ge ma ix in e sion. Howe e , by adop ing physically ealizable and ealis ic ec angula pulses, which clea ly do no sa is y he bi-o hogonal condi ion he doubly block ci culan ea u e is los . Thus, ou simula ions show ha he app oach [47] is no compe i i e when applied o a channel model using ec angula pulses, as in ui i ely expec ed. I should be no iced ha bi-o hogonali y o pulses wi h ime- equency p oduc equal o 1 is ma hema ically impossible [84], and elaying on such assump ion may be e y misleading. No e ha we only ea a 16-QAM modula ion. Howe e , he esul s a e e y clea and plo s wi h di e en modula ion o ma s would ha e only been a con i ma ion o wha we s a ed abo e. Chap e 5 Channel Es ima ion 5.1 In oduc ion In a gi en communica ion scena io, he CSI, i.e., he knowledge o he commu- nica ion channel, is equi ed a he Rx o pe o m cohe en de ec ion [34], i.e., o co ec ly demodula e he ansmi ed symbols based on he (known) chan- nel ans o ma ion. The mos common app oach o acqui e he CSI is h ough he ansmission o known symbols, usually called pilo s [34]. Gene ally, hese pilo s a e a anged wi hin he block o in o ma ion symbols, ollowing a chosen ixed pa e n known o bo h Tx and Rx (see e.g, [35, 36, 86]), in such a way when a comple e block is ecei ed, he Rx is able o ins an ly pe o m cohe en de ec ion, o ins ance, wi hou wai ing o some o he side in o ma ion om ano he sou ce.1I is also well-known ha , subjec o he meaning ul and widely used assump ion o block ading (i.e., he p opaga ion channel emains cons an o e blocks o consecu i e ime-domain symbols, while i may change 1In some communica ion scena ios (o s anda d) he Tx could send an en i e block o pilo symbols, i.e., wi hou in o ma ion da a, ollowed by blocks wi h no pilo s. Unde he assump ion o slowly a ying channel s a is ics, he symbol de ec ion is based on he es i- ma ion made om he i s block. This case has no bee conside ed, by ocusing on a pe block channel es ima ion, wi h associa ed bene i s and losses. 105 106 Chap e 5. Channel Es ima ion independen ly om block o block), pilo -aided schemes a e indeed nea ly in o ma ion- heo e ically op imal in e ms o he capaci y scaling in he high spec al e iciency / high SNR egime (see, e.g., [87, 88, 89, 90]). Clea ly, de- pending on he pa icula applica ion and communica ion scena io, bo h he pilo pa e n and he channel es ima ion algo i hm should be op imized. As ypical elec onic sys ems a e a ec ed by he mal noise, he es ima ed CSI is no pe ec bu a lic ed by an es ima ion e o , whose magni ude depends on di e en pa ame e s, e.g., he channel SNR, he numbe o pilo s pe block, he es ima ion algo i hm (wi h ela ed con e gence speed, complexi y, accu acy). The es ima ed communica ion channel exp ession is hus used o pe o m co- he en de ec ion a he Rx side. By aking he achie able communica ion a e as he mos ele an and meaning ul pe o mance me ic, which is a measu e o he amoun o use ul in o ma ion sen in a block o symbols, a adeo appea s be ween he numbe o pilo s pe block, dedica ed o he CSI es ima ion, and he numbe o in o ma ion-bea ing symbols. The op imiza ion o his adeo is gene ally no i ial, depending on he modula ion o ma and on he chan- nel p opaga ion cha ac e is ics, and i s op imiza ion is usually pe o med by ials, gi en he di icul y o ind closed- o m op imiza ion c i e ia, as done in ou successi e analysis. Gi en he a o emen ioned sys em se up and ela ed p oblems and chal- lenges, we analyse he symbol de ec ion pe o mance in e ms o p agma ic capaci y, i.e., he achie able a e o he channel induced by he signal con- s ella ion and he de ec o so -ou pu [78, 79], o OTFS and OFDM, bo h designed o handle ime- equency selec i e channels as (1.1). This is equi a- len o wha is done in Chap e 4, ecalled he e o he sake o comple eness. In gene al, a so -ou pu de ec o a he Rx side p oduces an es ima e o he pos- e io p obabili y o he ansmi ed symbols gi en he ecei ed signal block (pilo s and da a). This es ima ed pos e io p obabili y (e.g., in he o m o log-likelihood a ios) is hen passed o a decode , ha ea s he sequence o so -ou pu symbols as he ou pu o a i ual channel. The p agma ic capaci y is he capaci y o such i ual channel, wi h disc e e inpu ep esen ed by he 5.1. In oduc ion 107 modula ion symbols, and so -ou pu gene a ed by he de ec o . Hence, he p agma ic capaci y is ep esen a i e o he achie able a e unde he assump- ion o sepa a ed de ec ion and decoding, i.e., wi hou “ u bo” ep ocessing o he channel ou pu once he decode ou pu is a ailable (see, e.g., [91, 92] and e e ences he ein). In p ac ice, i e a i e “ u bo” de ec ion is e y ha d o implemen since o en he de ec o is implemen ed in ha dwa e (e.g., in an in eg a ed ci cui ) and he decode is implemen ed in so wa e, and maybe e en in a di e en loca ion (as o example in he so-called 7.2 spli be ween ha dwa e and so wa e, enabling cloud-based p ocessing o he signals om emo e adio heads [93]). Fo his eason, we belie e ha p agma ic capaci y o sepa a ed de ec ion and decoding is a e y meaning ul pe o mance me ic o compa e di e en modula ion o ma s and he associa ed pilo schemes and so -ou pu de ec o s. In o de o make a ai compa ison be ween he wo modula ion o ma s in e ms o achie able communica ion a e, he pilo o e head necessa y o achie e a sa is ac o y es ima ion o he CSI has been aken in o accoun . Since he loss (no use ul in o ma ion is ansmi ed) associa ed o he p esence o pilo s canno be neglec ed, he achie able communica ion a e ine i ably de i- a es om he co esponding uppe bound, depic ed by he AWGN symme ic capaci y. Mo eo e , oge he wi h he pilo o e head, we also conside he (likely) p esence o a gua d in e al (GI) o CP, which addi ionally educes he achie able a e. A pe -block GI is used in OTFS o a oid in e -block in- e e ence (IBI) [61], while a CP is used o e e y OFDM symbol o a oid ISI and o make symbols o hogonal, hanks o he diagonaliza ion o he channel ma ix, unde he assump ion o low ICI, condi ion which is no ha d o be sa is ied. He e a i s signi ican di e ence be ween he wo modula ion o ma s e iden ly appea s. In ac , especially when he channel delay is signi ican , he CP leng h ends o be a la ge ac ion (e.g., 25%) o he symbol ime, leading o a ema kable loss in e ms o capaci y. On he o he hand, OTFS does no need a pe symbol sepa a ion, bu his comes a he cos o a non-negligible in- c ease in signal p ocessing complexi y caused by he ema kable Dopple -delay 114 Chap e 5. Channel Es ima ion We used as s opping c i e ion he maximum numbe o i e a ions. No e ha he sh inking ope a ion is allowed because ze o en ies o ec o ˆ ha i e a ion icanno assume a alue 6= 0 a i e a ion i0> i [99]. F om a complexi y poin o iew, he i s i e a ions a e he mos cos ly, while he algo i hm can un >106 imes keeping he complexi y almos cons an and he compu a ional ime linea (when he numbe o i e a ions is la ge enough, i.e., a away om s a ing cos ly ones). 5.2.1.1 Complexi y o he LASSO Sol e and S ep Size Re inemen The sensing ma ix Dis composed o Gcolumns o leng h NM. While he dimensions Nand Mdepends on sys em se ings and can be somehow con- olled o uned, he dimension G akes in o accoun he es ima ion p ecision, o g anula i y, o he sea ching g id Γ. Hence, he la ge he dimension G, he mo e eliable he esul . Using some examples:  I Γ is equi alen o he Dopple -delay g id (delay and Dopple shi s in ege mul iple o he g id), G=NM and Dis a NM ×NM ma ix. Blockwise ope a ions adop ed by any LASSO sol e a e easible in his amewo k.  I he s ep size o bo h Dopple and delay axis is a ac ion 1/ρ o he Dopple -delay g id s ep, G≃ONM ·ρ2and Dis app oxima ely a NM ×NM ·ρ2ma ix. Clea ly, inc easing he g anula i y o he g id quickly inc eases he complexi y. In o de o o e come he complexi y induced by sea ching g id wi h ine g anula i y, i is possible o e ine he s ep size in successi e phases, a he han di ec ly de ining a low ac ional alue o he en i e g id. The p oposed e inemen scheme is illus a ed in Algo i hm 3. Du ing he Peak Selec ion s ep, i he numbe o mul ipa h componen s Pis no a ailable a he ecei e , ins ead selec all local maxima o peaks (o he g oups o es ima es) whose magni ude is abo e a ce ain h eshold ( o be de ined). 5.2. OFDM Modula ion and he CS Algo i hm 115 Algo i hm 3: Re inemen o he G anula i y Resul : Fine es ima ion ˆ h o LASSO p oblem (5.14). Coa se Es ima ion: Fo any LASSO sol e , ge a i s coa se es ima ion ˆ hsuch ha he sea ching g id Γ is equi alen o he Dopple -delay g id (i.e., delay and Dopple shi s in ege mul iple o he g id). In his case G=NM and Dis a NM ×NM ma ix; Fo I e a ion i= 1,2, . . . do •Peak Selec ion: Selec he i s Plocal maxima o ˆ h; •S ep Re inemen A ound Maxima: Build a new sensing ma ix based on an ex ension o ma ix Dsuch ha he s ep size a ound he peaks is dec eased (i.e., he g anula i y and he p ecision a e inc eased); •Fine Es ima ion: Fo any LASSO sol e , ge a ine es ima ion ˆ h. End No e ha du ing he Coa se Es ima ion s ep, i.e., when Dis an NM × NM ma ix, i is possible o adop he app oach p oposed in [100] o sol e he LASSO minimiza ion in (5.14). The algo i hm o [100], bene i ing o he hie a chical s uc u e o ec o h, is able o p o ide a i s coa se and eliable es ima ion op imizing he compu a ional complexi y. Howe e , i h akes o - g id alues, he app oach o [100] becomes inapp op ia e, as con i med by he p esen ed simula ion esul s. Fo his eason, a e a Coa se Es ima ion, i.e., wi hin he I e a ion s ep, ano he LASSO sol e mus be chosen o ob ain he bes pe o mance in e ms o channel es ima ion. 5.2.1.2 So -Th esholding Ope a o In o de o sol e he LASSO minimiza ion p oblem, a so h esholding ope - a o is used in (5.17). The choice o his unc ion is jus i ied in he ollowing. 116 Chap e 5. Channel Es ima ion By s a ing om he inpu -ou pu ela ion in ma ix o m y=xSDh +z,Ah +z,(5.19) he esidual sum o squa es (RSS) o he LASSO sol e is gi en by RSS (h) = 1 2 NM−1 X i=0  yi− G−1 X j=0 hjAi,j  2 +λ G−1 X j=0 |hj|.(5.20) By aking he de i a i e o he i s and second e ms o he RSS w. . . θkwe ge ∂ ∂hk   1 2 NM−1 X i=0  yi− G−1 X j=0 hjAi,j  2   =− NM−1 X i=0 Ai,k    yi− G−1 X j=0 j6=k hjAi,j   +hk NM−1 X i=0 Ai,k ,−ρk+hkηk,(5.21) ∂ ∂hk   λ G−1 X j=0 |hj|   =         −λi hk<0 [−λ, λ] i hk= 0 λi hk>0 ,(5.22) in which he de i a i e a hk= 0 can assume wo di e en alues. By pu ing hings oge he ∂RSS (h) ∂hk =−ρk+hkηk+∂ ∂hk λ|hk|,(5.23) and, by se ing he de i a i e equal o ze o (since we a e ying o minimize he LASSO cos unc ion) ∂RSS (h) ∂hk =0=         −ρk+hkηk−λi hk<0 [−ρk−λ, −ρk+λ] i hk= 0 −ρk+hkηk−λi hk>0 .(5.24) 5.2. OFDM Modula ion and he CS Algo i hm 117 We mus ensu e ha he closed in e al [−ρk−λ, −ρk+λ] con ains he ze o such ha hkis a global minimum, i.e., 0∈[−ρk−λ, −ρk+λ]⇒ −λ≤ρk≤λ . (5.25) Thus, inally ψs (hk, λ) =          (ρk+λ)/ηki ρk<−λ 0 i −λ≤ρk≤λ (ρk−λ)/ηki ρk> λ ,(5.26) which is he so h esholding ope a o ψs (hk, λ) wi h no maliza ion cons an 1/ηk. 5.2.1.3 Nes e o ’s Accele a ion Fac o The Nes e o ’s accele a ion ac o go e ns he dependency be ween wo suc- cessi e es ima ions, while ema kably educing he con e gence ime o he algo i hm [97]. The choice o he op imiza ion coe icien αiis no manda o y. By ollowing he pionee ing wo k [98], which inspi ed many o he wo ks, e.g., [96], αican be ecu si ely de ined as ξi+i=1 + q4ξ2 i+ 1 2,(5.27) αi=ξi−1 ξi+i ,(5.28) wi h ξ0= 1. Ano he choice simply based on he i e a ion index iis αi= (i−1)/(i+ 2) [97, 98]. Bo h solu ions desc ibe a cu e g owing om an ini ial alue (“ a ” om 1) up o 1. The associa ed plo s, wi h hei simila beha io s, can be seen in Fig. 5.1. The mos conse a i e choice is αi= 1, o which he dependency om he p e ious es ima ed alues is maximized, while he less conse a i e choice is αi= 0, comple ely o ge ing he p e ious es ima ed alues. In ui i ely, a small alue o αiis p e e able o he i s noisy i e a ions, 118 Chap e 5. Channel Es ima ion 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 i αi αi= (i−1) /(i+ 2) αi= (ξi−1) /ξi+i Figu e 5.1: The e olu ion o αio e i e a ions i o di e en app oaches. while a pa ame e αinea o one should be chosen when he eliabili y o he es ima ion inc eases. 5.2.2 Pilo Scheme The op imiza ion o a de e minis ic sensing ma ix o CS con igu a ions, such as LASSO, is up o now one o he mos s udied open p oblems in CS he- o y. In ac , he ypical pe o mance gua an ees o CS equi e p ope ies such as he es ic ed isome y p ope y [101, 38], o which explici cons uc ions a e no a ailable and e en checking he p ope y o a gi en andomly gen- e a ed ma ix is exponen ially complex [102]. On he o he hand, ensembles o andomly gene a ed ma ices ha e he p ope y o sa is ying hese p op- e ies wi h high p obabili y [38]. Hence, he e we eso o a pseudo- andom pilo placemen on he 2-dimensional ime- equency g id o ansmi ed sym- bols. Simula ion esul s ha e shown ha such andom placemen achie es wi h high p obabili y he bes pe o mance w. . . egula “la ice” placemen s (e.g., equally spaced combina ions o subca ie s o ime slo s), as usually speci ied in wi eless s anda ds [36]. An example o a andom pilo scheme is depic ed in Fig. 5.2. Mo eo e , gene ally, dis inc con igu a ions o a ixed numbe o 5.2. OFDM Modula ion and he CS Algo i hm 119 Figu e 5.2: Example o a andom pilo scheme o OFDM modula ion. pilo s, andomly placed wi hin he 2-dimensional g id, p o ides simila pe - o mance in e ms o channel es ima ion. I pilo s a e no placed andomly bu ollow some pe iodic pa e n, he algo i hm o sol ing he LASSO p o- duces a in e io esul s. This beha io is caused by he pe iodic sampling o a andom Fou ie ma ix (i.e., Ho Dhsp). This is he eason why commonly used pilo schemes (see, e.g., [36] and e e ences he ein), gene ally s uc u ed o pe iodic, a e no sui able o he CS-based es ima ion o OFDM sys ems (assuming ha he OFDM channel is ep esen ed by a Fou ie ma ix). O e - all, he aim i o maximize he o e all achie able a e unde andom pilo placemen . Hence, we can op imize he numbe o pilo s pe block o seek he op imal adeo be ween CSI es ima ion quali y and pilo o e head (see (5.41) in he ollowing and nume ical esul s in Sec. 5.4). 5.2.3 Recei ed Samples Exp ession — Real and App oxima ed Channel Condi ions Wi hou en e ing in o de ails o he comple e inpu -ou pu de i a ion o a CP OFDM sys em which can be ound in Sec ion 1.2, we only p o ide he ecei ed 120 Chap e 5. Channel Es ima ion sample exp ession, which is y[n, m] = 1 M P−1 X p=0 hpej2πνpnT M−1 X m0=0 xn, m0e−j2πm0∆ τp M−1 X i=0 ej2πi M νp ∆ ej2πi(m0−m) M (5.29) ≈1 M P−1 X p=0 hpej2πνpnT M−1 X m0=0 xn, m0e−j2πm0∆ τp M−1 X i=0 ej2πi(m0−m) M = P−1 X p=0 hpej2πνpnT e−j2πτpm∆ x[n, m].(5.30) By conside ing eal and app oxima ed channel condi ions, he ecei ed samples a ime ins an nand subca ie ma e espec i ely gi en by (5.29) and (5.30), in which he ICI- ee app oxima ion ollows he assump ion νmax/∆ 1, and he las equali y ollows by using he o hogonali y p ope y. No e ha he exp ession (5.30) is equi alen o (5.6), meaning ha he ICI- ee assump ion has been aken in o accoun o he algo i hmic design. When compa ing OTFS and OFDM, he channel model in (5.30) is conside ed, such ha he absence o ICI assump ion holds. Howe e , by ocusing he a en ion on OFDM only, he pe o mance compa ison is ex ended o he case o eal channel condi ions, i.e., wi h inpu -ou pu ela ion (5.29), showing he pe o mance deg ada ion when he ICI- ee assump ion is a o be sa is ied. Clea ly, a simila analysis o OTFS is meaningless, being he wa e o m no sensi i e o he magni ude o delay and Dopple shi s. 5.3 OTFS Modula ion and he P oposed Es ima ion Algo i hm By neglec ing, bu keeping in mind, he comple e de i a ion o he OTFS inpu -ou pu ela ion, which can be ound in Chap e 1, we can p oceed u he desc ibing he channel es ima ion p ocedu e, which is inspi ed by [86]. The inpu -ou pu ela ion exp essed in ma ix, shown he i s ime in (1.44), is 5.3. OTFS Modula ion and he P oposed Es ima ion Algo i hm121 epo ed he e o con enience y=  P−1 X p=0 hpΨp x+z,(5.31) whe e zdeno es he AWGN wi h ze o mean and co a iance ma ix σ2INM . No e ha Ψpimplici ly akes in o accoun a Dopple -delay pai (τp, νp), i.e., Ψp,Ψp(τp, νp). Wi hou en e ing in o ma hema ical de ails which can be ound in Sec. 1.3, he e ec o he channel o symbols a anged in blocks is sho ly desc ibed in he ollowing. Conside a block composed by all ze o magni ude symbols bu one non- ze o, ha ing enough ene gy o be well dis inguishable, w. . . o he noise loo , and posi ioned anywhe e wi hin he block. No e ha he posi ion o he sym- bol does no in luence he esul , since he channel shi e ec is ci cula wi hin he block, as p o ed by he esul s o Sec. 1.3. This block is hus ansmi - ed o e he ime- equency selec i e channel in (5.3). A he Rx, mos o he ene gy concen a es in a poin o he block (o , mo e p ecisely, a poin pe mul- ipa h componen ), while dissipa ing o he su ounding posi ions, acco ding o Fig. 1.4, whe e and example o blocks o ansmi ed symbols and ecei ed samples blocks a e depic ed. The in ui i e es ima ion o he pai s (τp, νp), o each mul ipa h componen , ollows by sea ching he peaks wi hin he ecei ed samples g id, successi ely associa ed o an es ima e (ˆτp,ˆνp) (as sugges ed in [86]). Howe e , his in ui i e es ima ion p ocedu e is only able o p o ide he in ege pa s o he Dopple and delay shi s, associa ed o he Dopple -delay g id poin , whe e a peak is de ec ed, collec ing enough ene gy. The ac ional pa s o delay and Dopple shi s a e, ins ead, associa ed o he dissipa ion o he ene gy a ound he peak poin s (see Fig. 1.4), and ha e o be ea ed and analyzed sepa a ely. The app oxima ion o he channel beha io o in ege Dopple and delay shi s, as done in [86], allows his in ui i e es ima ion p ocedu e o wo k co - ec ly, bu only unde such non- ealis ic channel condi ions. Thus, based on he ML es ima o p oposed in Sec. 1.3, he idea o [86] is ex ended, and a 122 Chap e 5. Channel Es ima ion 25 50 32 64 0 10 20 NM  Figu e 5.3: Example o a pilo scheme o OTFS modula ion. Wi hin he block o dimension N×M he cen e ed pilo wi h high ene gy εis well dis in- guishable, and su ounded i s by ze o pilo s ( he hollow zone) and a e by in o ma ion symbols (he e, o con enience, wi h uni ene gy). eliable es ima ion algo i hm wo king unde ealis ic channels is p o ided. 5.3.1 Pilo Scheme A block o N×M ansmi ed symbols con ains bo h in o ma ion bea ing symbols and pilo s. The a angemen o pilo s consis s o a ec angula egion placed wi hin he block (no necessa ily in he middle, since he channel e ec is ci cula ) con aining wo ypes o symbols (see Fig. 5.3):  Ze o Pilo s: Placed be ween in o ma ion symbols and non-ze o pilo s o gua an ee as less in e e ence as possible be ween hem. The Di ich- le ke nel unc ions appea ing in he OTFS inpu -ou pu ela ion (see Sec. 1.3 and in pa icula (1.44)) a e apidly dec easing non-nega i e unc ions (see Fig. 1.3 and 1.5), hence, pe ec o hogonali y be ween in o ma ion symbols and pilo s canno be achie ed, bu , a leas , he Dopple -delay ISI can be educed. 5.3. OTFS Modula ion and he P oposed Es ima ion Algo i hm123 25 50 32 64 0 2 4 N M  Figu e 5.4: Example o a pilo scheme o OTFS modula ion. Wi hin he block o dimension N×M he pla eau o pilo s wi h high ene gy εis well dis in- guishable, and su ounded i s by ze o pilo s ( he hollow zone) and a e by in o ma ion symbols (he e, o con enience, wi h uni ene gy).  Peak Pilo : A single pilo symbol wi h high ene gy, collec ing he ene gy o all ze o pilo s, is placed a he g id cen e . I s shi s in he Dopple - delay g id a e used o p o ide he ini ial coa se es ima ion o he Dopple - delay pai s, which esul s o be as and simple. Gi en his pilo a angemen , he numbe o pilo symbols has o be op imized o ma ch he op imal pe o mance-o e head adeo , while keeping cons an he o al block ene gy. Gi en some pa icula sys em se ups, such as communica ion including non-linea ampli ie no cons an channel beha iou s o e block symbols in ime, basing he es ima ion algo i hm on jus one symbol ( he peak pilo ) could no be a sa e app oach. In such cases, he pilo scheme can be ex ended, o ins ance, by conside ing a ec angula egion composed o ze o pilo s and a pla eau on non-ze o pilo s (see Fig. 5.4), ins ead o jus he peak pilo . We would like o s ess he ac ha he successi e algo i hm design is no only sui able o he a o emen ioned pilo con igu a ion, bu is able o p o ide eli- 130 Chap e 5. Channel Es ima ion o p agma ic capaci y. Fo ins ance, by conside ing a CP o leng h T/4, being T he symbol ime, he loss is T T+T/4=T 5T/4=4 5= 0.8 = 20% .(5.42) This means ha , wi h a modula ion o ca dinali y C, while he maximum achie able a e is log2Cbi s/symbol, OFDM sa u a es a 0.8·log2Cbi s/symbol. The o e all loss akes in o accoun bo h he pilo o e head and he p esence o a CP and/o GI (also o leng h T/4, o consis ency). Ano he impo an aspec is he de ini ion o he numbe o pilo s |P| w. . . he ambien dimension G. Many ea u es a e in luenced by his choice. Conside i s OFDM modula ion. I is well known in he CS li e a u e ha he minimum numbe o pilo s (o measu emen s, om CS li e a u e) o eco e a spa se signal is gi en by he loga i hmic scaling ac o [105] |P| ≥ Plog G P,(5.43) whe e Phe e ep esen he numbe o non-ze o componen s o he ec o o be es ima ed. Thus, gi en a mul ipa h channel wi h Ppa hs and a sensing ma ix o dimension G(de ined in (5.7)), he (asymp o ically) minimum numbe o pilo symbols necessa y o sol e he minimiza ion p oblem in (5.14) is gi en by (5.43). Due o he loga i hmic scaling, he unc ion is slowly inc easing, e en i he g anula i y o he sensing ma ix and i s dimension g ows (see Sec ion 5.2.1.1). Howe e , while (5.43) p o ides a lowe bound on he dimension, he op imal pe o mance migh be achie ed o a numbe o pilo s di e en om he minimum. Fo his eason, i he numbe o pilo s used o a gi en se up (i.e., o ixed Pand G) is well abo e he lowe limi , we can s a e ha while inc easing he dimension G, he op imal numbe o pilo s sligh ly changes. Hence, he pilo loss in (5.41) ends o ze o, while inc easing he ambien dimension G, which is di ec ly linked o he block size NM (assuming, o simplici y, on-g id pa hs shi s and neglec ing he e inemen solu ion depic ed in Alg. 3), and OFDM bene i s om la ge blocks. No e ha his is no a p ecise quan i a i e analysis bu i jus gi es a quali a i e idea o in ui ion 5.4. Compa ison in Te ms o P ag ama ic Capaci y 131 on how la ge he numbe o pilo s pe block should be, knowing ha he a o emen ioned CS-based condi ions a e always gi en up o cons an ac o s ha depend on he speci ic p oblem, SNR, shape o he sensing ma ix, and o he a iables. Fo OTFS modula ion, he pilo scheme p esen ed in Sec. 5.3.1 and i s es ima ion algo i hm a e independen o he block dimension, since he shi o he peak pilo , and so he ough es ima ion, is only associa ed o he maxi- mum Dopple and delay o he channel de ined in (5.2). Hence, i ollows ha he pilo o e head ends o ze o while inc easing he block dimension, hence, as o OFDM, also OTFS bene i s om la ge blocks. Mo eo e , i he dimen- sion o he block inc eases, one can se mo e pilo s o ze o o aise he powe o he peak pilo , allowing he de ec ion o low powe sca e ing componen s in he h eshold-based app oach (as explained a he beginning o Sec. 5.4). Howe e , limi s on he block dimension comes, i s , om impo an es ic- ions on OTFS de ec ion compu a ional complexi y (as seen in Chap e 4 and [46]), and hen om he block ading assump ion, which b eaks down i he block becomes oo la ge (see Chap e 1). Thus, ealis ic block sizes ha e o be conside ed in bo h di ec ions. Mo eo e , as an icipa ed in Sec. 5.2, in o de o es ic o he classical low-complexi y symbol-by-symbol MMSE es ima ion o OFDM we ha e ne- glec ed he ICI. As al eady seen in (5.29), he ICI depends on he a io be ween he subca ie spacing ∆ and he maximum Dopple shi in oduced by he channel. In o de o ha e negligible ICI he necessa y condi ion is ∆ νmax, o , equi alen ly, νmax/∆ 1. Since ∆ =B/M, wi h B o al bandwid h, he condi ion may no be sa is ied when he numbe o subca ie s Mbe- comes oo la ge, e en o mode a e Dopple . He e, we insis on neglec ing ICI and conside he ange o sys em pa ame e s o which his assump ion is in- deed i ually exac . Fu he mo e, we no ice ha while OFDM incu s in his addi ional limi a ion, OTFS emains no sensi i e o he Dopple shi . 132 Chap e 5. Channel Es ima ion Table 5.1: Sys em pa ame e s c= 5.89 [GHz] M= 64 B= 10 [MHz] N= 50 ∆ =B/M = 156.25 [kHz] T= 1/∆ = 6.4 [µs] 5.4.1 Simula ion Resul s In he ollowing igu es, we plo he p agma ic capaci y s. SNR o OTFS and OFDM modula ions wi h quad a u e phase-shi keying (QPSK) modula ed symbols, o a ime- equency mul ipa h channel wi h Pcomponen s a ec ed by AWGN, and unde non-pe ec CSI. The channel es ima ion has been pe - o med wi h he pilo schemes and he algo i hms o Sec. 5.2 o OFDM and Sec. 5.3 o OTFS. As a e e ence benchma k, we plo he AWGN (symme ic) capaci y Csym AWGN o QPSK modula ion, which esul s o be no achie able due o he pilo o e head. The sys em pa ame e s a e lis ed in Table 5.1. Fig. 5.5 shows he pe o mance o OFDM o a di e en numbe o pilo symbols. Fo he case P= 1, i easy o no e ha he pe o mance sligh ly changes o di e en pilo o e heads, whose pe cen age is indica ed in he legend. On he o he hand, as sugges ed by (5.43) and associa ed discussion, i he numbe o non-ze o componen s o be es ima ed inc eases, i.e., wi h P > 1, he channel es ima ion algo i hm needs mo e pilo s o wo k e icien ly. Gi en hese esul s, om now on, we will conside a pilo o e head o 3.125%, achie ing, in ou se up, he bes adeo be ween es ima ion accu acy and achie able p agma ic capaci y ( o any numbe o sca e ing componen s). Fig. 5.6 shows he pe o mance o OTFS wi h di e en de ec ion algo- i hms. The MP so -ou pu de ec ion app oach o [46] is able o almos achie e he AWGN capaci y unde non-pe ec CSI o a low numbe o sca e ing componen s, oge he wi h a ema kable educ ion o he compu a ional com- plexi y [46]. Howe e , in line wi h he esul s o [46] and Chap e 4, he de ec- 5.4. Compa ison in Te ms o P ag ama ic Capaci y 133 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbi s symboli 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 2 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 3 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 4 Figu e 5.5: P agma ic Capaci y s. SNR o OFDM modula ion wi h mul ipa h componen s Pand di e en pilo o e head (whose pe cen age is indica ed in he legend). 134 Chap e 5. Channel Es ima ion −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbi s symboli LMMSE / 5.28% G/ 5.28% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 2 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN −50 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 3 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 4 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN Figu e 5.6: P agma ic Capaci y s. SNR o OTFS modula ion wi h mul ipa h componen s P, di e en de ec ion algo i hms, i.e., LMMSE and MP app oach o [46], o a ixed pilo o e head, in pe cen age 5.28%. 5.4. Compa ison in Te ms o P ag ama ic Capaci y 135 o pe o mance dec eases wi h inc easing mul ipa h. No e ha he small loss w. . . Csym AWGN unde non-pe ec CSI is an indica o o he pe o mance o he channel es ima ion algo i hm, which esul s o be e y accu a e (o he wise he cu e would ha e de ia ed om he benchma k). In ligh o hese esul s, we see ha he e is no eason o adop he LMMSE es ima o o OTFS, which esul s in highe complexi y and wo se pe o mance (see Chap e 4 and [46] o a mo e de ailed analysis). Hence, om now on, o he compa ison wi h OFDM modula ion, we conside he MP “Ma ix-G” app oach o Chap e 4. In Fig. 5.7, we plo he p agma ic capaci y s. SNR o OFDM and OTFS unde he con igu a ions men ioned abo e. Fi s o all, i is possible o no e ha pe o mance sligh ly dec ease while inc easing he numbe o mul ipa h componen s, p o ing he obus ness o he pilo schemes and he algo i hms p oposed. The pilo o e head has been chosen ad hoc o bo h modula ions. Fo OTFS, as said in Sec. 5.3.1, he peak cen e ed pilo collec s all he ene gy o su ounding ze o pilo s, and he pe cen age o o e head (indica ed in he igu es) is also an indica o o he peak pilo ene gy. Fo OFDM, as poin ed ou be o e, he op imal adeo be ween es ima ion pe o mance and pilo o e head has been ound by b u e- o ce sea ch o e a sui able se o possibili ies (some o hem a e isible in Fig. 5.5). While he pe o mance o he wo modula ions is simila , he p esence o a pe symbol CP o OFDM ema kably de e io a es he p agma ic capaci y, while a pe block GI o OTFS in oduces a negligible loss. In Fig. 5.8, we plo he p agma ic capaci y o OFDM o a ixed alue o SNR, i.e., 18 dB, while changing he a io be ween he maximum Dopple shi and he subca ie spacing, i.e., νmax/∆ , aking in o accoun a di e en numbe o subca ie s M(N= 50 o all cases). In his case, he ecei ed samples a e ob ained by conside ing a eal channel aking in o accoun he ICI, i.e., (5.29), while he channel es ima ion wo ks unde he hypo hesis o an ideal in e e ence- ee channel. In ui i ely, he pe o mance deg ades when he ICI becomes signi ican . No e ha he es ima ion pe o mance o he LASSO sol e is independen o he numbe o subca ie Mand, o his eason, 136 Chap e 5. Channel Es ima ion −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbi s symboli −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 2 SNR [dB] −50 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 3 SNR [dB] IPC hbi s symboli −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 4 SNR [dB] MOD OFDM OTFS (G)Csym AWGN Pilo O e head 3.125% 5.28% - OFDM CP OTFS GI (G) Figu e 5.7: The cu es ake in o accoun he loss ela ed o he p esence o pilo s in he block o N×Msymbols. The OFDM CP cu e includes he CP o e head which is 0.25 o he symbol ime, while he OTFS GI cu e includes a GI o he en i e block. A QPSK modula ion is used. The legend indica es he pe cen age o pilo symbols. 5.5. Conclusions 137 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 νmax/∆ IPC hbi s symboli M= 64 M= 128 M= 256 M= 512 Csym AWGN Figu e 5.8: P agma ic Capaci y s. νmax/∆ a io o OFDM modula ion wi h P= 1, a SNR = 18 dB. wha e e he choice o νmax and M, he pe o mance o OFDM depends only on hei a io. Fig. 5.8 shows ha he p agma ic capaci y pe o mance s a s dec easing signi ican ly o νmax/∆ ≃0.15. Almos he same beha io is shown o di e en numbe o subca ie s M(no epo ed he e o he sake o space limi a ion), excep o he pe cen age o pilo loss due o di e en block dimensions, suppo ing wha s a ed abo e. Howe e , as poin ed ou in Table 5.2, while he pe o mance is almos cons an , he maximum ole able Dopple (o eloci y), in e sely p opo ional o M, is no . Fo hese easons, as expec ed, OFDM is no independen o he block dimension and he sys em has o be de ined p ope ly o ope a e in he ange whe e he ICI is negligible. 5.5 Conclusions We ca ied ou a ai compa ison be ween OFDM and OTFS modula ion o - ma s in e ms o maximum achie able a e o p ac ical sepa a ed de ec ion and decoding, quan i ied by he p agma ic capaci y measu ed a he so - 138 Chap e 5. Channel Es ima ion νmax/∆ 0.1 0.15 0.2 M 64 1431 [km/h] 2147 [km/h] 2863 [km/h] 128 715 [km/h] 1073 [km/h] 1431 [km/h] 256 357 [km/h] 536 [km/h] 715 [km/h] 512 178 [km/h] 268 [km/h] 357 [km/h] Table 5.2: Maximum suppo able eloci y w. . . he a io νmax/∆ o OFDM modula ion wi h subca ie spacing ∆ =B/M and ca ie equency c= 5.89 GHz. The eloci y is gi en by =c·νmax/(2 c). de ec o ou pu . We conside ed wo pilo schemes and channel es ima ion algo i hms each one speci ically sui ed o he gi en modula ion scheme. Bo h pilo and CSI es ima ion schemes a e able o achie e e y good pe o mance (nea genie- aided) unde ime- a ying communica ion channel in he spa si y egime o a small numbe o numbe o mul ipa h componen s. This conclusion is ully suppo ed by nume ical esul s, whe e simula ion cu es achie e he heo e - ical benchma k unde non-pe ec CSI, p o ing he quali y o he p oposed app oaches. OTFS achie es a be e communica ion a e mainly because o he p esence o a pe block gua d in e al a he han a pe symbol cyclic p e ix as in OFDM. This o cou se comes a he cos o a mo e complex channel es ima ion scheme, wo king on la ge block-wise ope a ions. In e ms o so -ou pu da a de ec ion, he use o he message passing so -ou pu de ec o o Chap e 4 yields cons an pe symbol complexi y o OTFS, which is he same scaling law o symbol-by-symbol MMSE de ec ion o OFDM. Al hough we do no claim ha he complexi y o he wo de ec o s 5.5. Conclusions 139 is iden ical, in ac he ac ual complexi y di e o some implemen a ion-based cons an . Finally, we can obse e ha OTFS is indeed e y insensi i e o he mag- ni ude o he Dopple shi s, while he pe o mance o OFDM deg ades sig- ni ican ly e en unde small- o-mode a e Dopple alues i he numbe o sub- ca ie s inc eases. The e o e, OTFS is e ec i ely a good candida e o high- mobili y sys ems in u al en i onmen s (e.g., high speed ains [106]) o ae ial en i onmen s (e.g., UAVs [107]), whe e Dopple shi s may be la ge, and he p opaga ion channel con ains ypically he line-o -sigh and a ew o he e- lec ion componen s (e.g., g ound e lec ion, hills, la ge buildings), and i is he e o e spa se in he Dopple -delay domain.