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.