Time-frequency represetation of radar signals using Doppler-Lag block searching Wigner-Ville distribution

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER Time-F equency Rep ese a ion o Rada Signals Using Dopple -Lag Block Sea ching Wigne -Ville Dis ibu ion Muhammad Noo Muhammad HAMDI, Ahmad Zu i SHA’AMERI Depa men o Elec onic and Compu e Enginee ing, School o Elec ical Enginee ing, Uni e si i Teknologi Malaysia, Joho Bah u, Joho , Malaysia muhdno[email p o ec ed], zu i@ ke.u m.my DOI: 10.15598/aeee. 16i3.2633 Abs ac . Rada signals a e ime- a ying signals whe e he signal pa ame e s change o e ime. Fo hese signals, Quad a ic Time-F equency Dis ibu ion (QTFD) o e s ad an ages o e classical spec um es- ima ion in e ms o equency and ime esolu ion bu i su e s hea ily om c oss- e ms. In gene a - ing accu a e Time-F equency Rep esen a ion (TFR), a ke nel unc ion mus be able o supp ess c oss- e ms while main aining au o- e ms ene gy especially in a non-coope a i e en i onmen whe e he pa ame e s o he ac ual signal a e unknown. Thus, a new signal- dependen QTFD is p oposed ha adap i ely es ima es he ke nel pa ame e s o a wide class o ada signals. The adap i e p ocedu e, Dopple -Lag Block Sea ching (DLBS) ke nel es ima ion was de eloped o se e his pu pose. Accu a e TFRs p oduced o all simula ed ada signals wi h Ins an aneous F equency (IF) es- ima ion pe o mance a e e i ied using Mon e Ca lo simula ion mee ing he equi emen s o he C ame - Rao Lowe Bound (CRLB) a SNR > 6 dB. Keywo ds Adap i e p ocedu e, au o- e ms, C ame -Rao lowe bound, c oss- e ms, ke nel unc ion, quad a ic ime- equency dis ibu ion. 1. In oduc ion Rada is widely used bo h in mili a y and non-mili a y applica ions such as acking missiles, ships, land ehi- cles and ai c a , ligh con ol sys em, ocean su eil- lance sys em and geological obse a ions. By de ini- ion, Low P obabili y o In e cep (LPI) ada s u i- lize special emi ed wa e o m ha has been speci ically designed o a oid de ec ion o in e cep ion by non- coope a i e in e cep ecei e [1]. This is achie ed by in eg a ing addi ional p ope ies such as ul a-low side- lobe, Ad anced Mul i unc ion Radio F equency Con- cep (AMRFC), wideband equency and minimum ansmi ed ene gy. The idea o LPI ada is o see and no be seen, meaning i mus ha e he capabili y o de ec a ge s like any ada while s aying in isible o elec onic econnaissance equipmen . In e cep ing LPI signals is no easy bu is no o ally impossible. Some o he impo an p ope ies equi ed in he mode n in e cep ecei e s in in e cep ing LPI signals a e channelized ecei e , u iliza ion o supe - he e odyne ecei e and sidelobe de ec ion capabili y [2] and [3]. Signal p ocessing algo i hms a e he im- po an componen s o mode n in e cep ecei e ha imp o e he de ec ion and analysis o LPI ada sig- nals. Example o me hods used o de ec ing and an- alyzing LPI ada signals a e adap i e ma ch il e - ing, pa allel il e a ays wi h highe o de s a is ics, Wigne -Ville Dis ibu ion (WVD), quad a u e mi o il e bank, and cyclos a iona y p ocessing [3]. A ime- a ying signal such as LPI ada signals, he spec al desc ip ion o which depends on ime is bes analyzed wi h Time-F equency Dis ibu ion (TFD). Among TFD classes, Quad a ic TFD (QTFD) is ap- p op ia ely used because i p o ides high- esolu ion ep esen a ion bo h in ime and equency [4]. C oss- e ms a e in oduced in QTFD due o he quad a ic na u e o he algo i hm, which makes i di icul o in e p e he ue signal cha ac e is ics and also ex- agge a es he e ec o noise [5]. Some o he ech- niques p oposed o p oduce accu a e TFR a e educed in e e ence dis ibu ion, ac ional Fou ie ans o m, Radon-Wigne dis ibu ion, adap i e c oss WVD and modi ied B-dis ibu ion [6] and [7]. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 318 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER The applica ion o he F ac ional Fou ie T ans- o m (F FT) is p o en e ec i e in ep esen ing and p ese ing signal componen s in he Time-F equency (TF) plane such as Pulse Linea F equency Modula ion (PLFM) [8]. Besides ha , he echo o he mo ing a - ge o ai bo ne Syn he ic Ape u e Rada (SAR) can also be conside ed as a LFM signal. Thus, he F FT can be used o ep esen and analyze such signals. In he li e a u e, i was men ioned ha he F FT is equi - alen o a o a ion o he signal ei he in TF o ambi- gui y domain. The deg ee o he TF o a ion depends on he ac ional powe o F FT [9]. The applica ion o F FT can be u he expanded om his de ini ion o signal pa ame e s es ima ion and c oss- e m elimina- ion [10] and [11]. A echnique whe e a combina ion o a o a ed TF plane WVD wi h a sui able TF il e ing is compa ed wi h he DLBS-WVD. The es o he pape is o ganized as ollows. Sec- ion 2. p esen s he signal model and p oblem de - ini ion. The signal cha ac e is ics in he ime-lag and Dopple -lag domain a e discussed in Sec. 3. The ela ionship be ween F FT and TFR is also explained in his sec ion. The simula ion esul and discussion a e p esen ed in Sec. 4. while he ield ials esul s a e desc ibed in Sec. 5. 2. Signal Model and P oblem De ini ion The ou commonly used ada signal ypes u ilized o e i y he accu acy o ime- equency ep esen a ion p oduced by he Dopple -Lag Block Sea ching Wigne - Ville Dis ibu ion (DLBS-WVD) a e: Simple Pulse sig- nal (SP), 4 Cos as Coded pulse (CC4) signals, PLFM, and Con inuous Wa e Linea F equency Modula ion (CW-LFM). The signal pa ame e s a e desc ibed in Tab. 1. Excep o he SP signal, all he o he signals can be ca ego ized as LPI ada signal wa e o ms [1]. The signals a e assumed ha hey ha e been downcon- e ed om adio equency o in e media e equency whe e hey a e sampled a he Nyquis a e (sampling equency, s= 40 MHz). The use o sho pulse epe i ion pe iod, Tbe ween 5 o 20 µs is o simpli y he de elopmen o he ke nel es ima ion p ocedu e. Howe e , he ac ual ada sig- nals pa ame e s may a y acco ding o he applica ions and he de ec ion ange [12], [13] and [14]. Typically, Elec onic Suppo (ES) deals wi h a non- coope a i e en i onmen whe e p io knowledge o he ue signal cha ac e is ics – pulse epe i ion pe iod, e- quency agili ies, modula ion echniques, pulse wid h, pulse ampli ude – a e unknown. Adap i e ke nel im- p o es he TFR by es ima ing he ke nel pa ame e s acco ding o he pa e n o c oss- e ms which a y ac- Tab. 1: Signal Pa ame e s. Pulse epe i ion pe iod (T), pulse wid h (Tp), lowes equency ( min), highes equency ( max), bandwid h (BW ). Signal F equency Pa ame e s Time Pa ame e s Simple Pulse (SP) = 10 MHz T= 5 µs Tp= 1 µs 4 Cos as Coded pulse (CC4) 4 sub-pulse equencies b1= 4 MHz b2= 8 MHz b3=16 MHz b4= 12 MHz T= 16 µs Tp= 4 µs Pulse Linea FM (PLFM) min = 2 MHz max = 17 MHz BW = 15 MHz T= 9 µs Tp= 4 µs Con inuous Wa e Linea FM (CW-LFM) min = 2 MHz max = 10 MHz BW = 8 MHz T= 20 µs Tp= 10 µs co ding o he signal. The QTFD wi h he adap i e ke nel ensu es an accu a e TFR o a b oad class o signals. 3. Quad a ic Time-F equency Dis ibu ion The QTFD p oduces an ene gy ep esen a ion join ly o e he ime- equency plane. I is also conside ed as a ela ed class o il e ed WVDs wi h a speci ic ime–lag ke nel unc ion [15]. When exp essed wi h espec o he ime- equency ke nel and he WVD, he QTFD is w i en as [5] ρz( , ) = γ( , )∗ ∗ Wz( , ),(1) whe e γ( , )is a ime- equency ke nel and Wz( , ) is he WVD. The WVD can be de ined as Wz( , ) = ∞ Z −∞ Kz( , τ)e−j2π τ dτ, (2) whe e Kz( , τ)is he bilinea p oduc o Ins an aneous Au oco ela ion Func ion (IAF). The bilinea p oduc can be w i en as Kz( , τ) = z +τ 2z∗ −τ 2,(3) whe e z( )is he analy ical o m o he signal. The o mula ion o he QTFD wi h he ime-lag ke - nel is gi en as ρz( , ) = ∞ Z −∞ G( , τ)∗ Kz( , τ)e−j2π τ dτ, (4) whe e G( , τ)is he ime-lag ke nel. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 319 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER A sepa able ke nel o e s independen con ol o ime-smoo hing and equency-smoo hing o TFD which can be de ined in ime-lag unc ion as G( , τ) = g1( )g2(τ),(5) whe e g1( )is he smoo hing unc ion in ime and g2(τ) is he smoo hing unc ion in lag. By using a sepa able ke nel, he QTFD in Eq. (1) can be desc ibed as ρz( , ) = ∞ Z −∞ g1( )∗ ( )Kz( , τ)g2(τ)e−j2π τ dτ. (6) 3.1. Gene al Signal Cha ac e is ics in Time-Lag Domain The IAF de ined in Eq. (4) can be iewed as he co - ela ion o signal i sel o e successi e ime in e als. C oss- e ms a e in oduced due o he quad a ic na- u e o he IAF which can cause di icul y in he in- e p e a ion o he ue signal cha ac e is ics [16] and [17]. Ke nel unc ion as shown in Eq. (4) sol ed his p oblem. Va ious ke nel unc ions ga e ise o a ange o di - e en TFDs [6] and [15]. The gene al IAF de ini ion is p oduced by mul iplying he signal wi h i s conju- ga e as shown in Eq. (7). Each ype o inpu signal, z( ) esul s in di e en IAF de ini ion. Fo example, wo pulses o SP signal p oduce wo au o- e ms and wo c oss- e ms while wo pulses o CW-LFM p oduce ou au o- e ms and wel e c oss- e ms. The IAF ob- ained by subs i u ing he signal de ini ion in Tab. 1 in o Eq. (4) can be exp essed in he ollowing o m: Kz( , τ) = hz1 +τ 2+z2 +τ 2−Ti· ·hz∗ 1 −τ 2+z∗ 2 −τ 2−Ti =Kz,11 ( , τ) + Kz,22 ( −T, τ) | {z } au o− e ms + Kz,12  −T 2, τ +T+Kz,21  −T 2, τ −T | {z } c oss− e ms , (7) whe e z1( )is he i s pulse and z2( )is he second pulse o he signal. The accu acy o TFR depends on he capabili y o he algo i hm o p ese e he au o- e ms ene gy. C oss- e ms can be di ided in o wo ca ego ies: in a-pulse and in e -pulses. In a-pulse c oss- e ms occu as a esul o he co ela ion o signal be ween he sub-pulse o he signal while in e -pulse c oss- e ms a e he p oduc o he co ela ion be ween di e en pulses o he signal. Figu e 1 shows he ime-lag domain o CW-LFM sig- nal and p oduced by eplacing z1( )and z2( )in Eq. (7) 1,1 2,2 3,3 4,4 2,1 3,2 1,4 2,41,3 3,42,31,2 4,3 3,1 4,2 4,1 0 Tb 2Tb -Tb -2Tb -3Tb -4Tb 3Tb 4Tb τ 4Tb 3Tb 2Tb Tb Fig. 1: The ime-lag domain wo pulses o CW-LFM signal. wi h wo pulses o iangula CW-LFM signal whe e Tb is he sub-pulse du a ion. Fo iangula CW-LFM, he Tbis exac ly hal o he signal pulse. The do ed dia- mond shapes ep esen he in a-pulse c oss- e ms and he shaded diamond shapes ep esen he in e -pulse c oss- e ms. The emaining diamond shapes ep esen he au o- e ms o he signal. Gene ally, a ke nel ha is able o supp ess he c oss- e ms a |τ|> Tbis accep able excep o CW-LFM signal. Figu e 1 shows ha he e a e some c oss- e ms loca ed be ween 0≤τ≤Tbwhich could lead o in- accu a e TFR o he signal. Fo comple e c oss- e ms supp ession, he applied window o ke nel mus ma ch pe ec ly wi h he au o- e ms. To comple ely sepa a e he au o- e ms om he c oss- e ms in he ime-lag domain, al e na i e c oss- e ms supp ession echniques a e p oposed in he Dopple -lag domain [18]. 3.2. Rada Signal Cha ac e is ics in Ambigui y Domain The AF is ela ed o he IAF by he Fou ie ans o m wi h espec o ime as shown in he ollowing equa ion: Az( , τ) = FT → [Kz( , τ)] = ∞ Z −∞ Kz( , τ)e−j2π d . (8) The comple e signal equa ions o SP, CC4, PLFM and CW-LFM signals in he ambigui y domain can be ound in [19]. The AF o he CW-LFM signal is illus- a ed in Fig. 2. The AF ep esen s wo pulses CW-LFM signal whe e he au o- e m a e ep esen ed by he g ey shaded shapes, in e -pulse c oss- e ms by he blue shaded shapes and in a-pulse c oss- e ms by he o ange shaded shapes. Simila colo codes a e used in Fig. 3, c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 320 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 0 Tp T -Tp -T -T-Tp T+Tp τ ` Δ NTΔ NT T-Tp G2( ) G2(- ) g1(-τ ) g1(τ ) (1,1),(2,2), (3,3),(4,4) (1,3),(2,3), (1,4),(2,4) (3,1),(3,2), (4,1),(4,2) (4,3),(2,1) (3,4),(1,2) Fig. 2: The AF o wo pulses o CW-LFM. Fig. 4 and Fig. 5. F om Fig. 1, he au o- e ms a e loca ed a τ≤ |Tp|while mos o he in a-pulse c oss- e ms a e posi ioned e y close o he au o- e ms. On he o he hand, he in e -pulse c oss- e ms a e loca ed be ween T−Tp≤τ≤T+Tp. I is wo h men ioning he e ha he ene gy o he in a-pulse c oss- e ms o CW-LFM is signi ican ly lowe compa ed o i s au o- e ms. Thus, a small po ion o he in a-pulse c oss- e ms in he AF does no cause a majo deg ada ion in he TFR. The p ocedu e o supp essing in e -pulse c oss- e ms is much simple in he Dopple -lag domain com- pa ed o he ime-lag domain because he loca ion be- ween he in e -pulse c oss- e ms and au o- e ms a e well sepa a ed. Thus, se ing a lag window, g1(τ)a Tpis su icien o sepa a ing he au o- e ms om he in e -pulse c oss- e ms. Figu e 3 shows he AF o wo pulses o PLFM sig- nal. The posi ion o he au o- e ms is a τ≤ |Tp|and he in e -pulse c oss- e ms a e loca ed a T−Tp≤τ≤ T+Tp. The cha ac e is ics o PLFM a he AF a e simila wi h he CW-LFM excep he e a e no in a- pulse c oss- e ms o PLFM signal. Due o his, he same c oss- e ms supp ession p ocedu e in CW-LFM signal is applicable o PLFM signal. Figu e 4 shows he AF o wo pulses o SP signal. No e ha he au o- e m and in e -pulse c oss- e ms loca ion a e simila wi h he PLFM and CW-LFM sig- nals which a e a τ≤ |Tp|and T−Tp≤τ≤T+Tp espec i ely. The only di e ence is he shape o he au o- e ms and he c oss- e ms. Thus, a simila p oce- 0 Tp T -Tp -T -T-Tp T+Tp τ ∆ NT ` -∆ NT T-Tp G2( ) G2(- ) g1(-τ ) g1(τ ) (1,1),(2,2) (2,1) (1,2) Fig. 3: The AF o wo pulses o PLFM signal. 0 Tp T -Tp -T -T-Tp T+Tp τ 1/Tp -1/Tp T-Tpg1(τ ) g1(-τ ) G2( ) G2(- ) Fig. 4: The AF o wo pulses o SP signal. du e o a enua e he c oss- e ms o he PLFM signal can be applied o he SP signal. The AF o one pulse o he CC4 signal is shown in Fig. 5 whe e he au o- e ms o he signal a e loca ed a τ≤ |Tb|while he in a-pulse c oss- e ms a e sca e ed on he ambigui y plane a −4Tb≤τ≤4Tb. Se ing a lag window a τ≤ |Tb|is enough o ex ac he au o- e ms om he c oss- e ms in he ambigui y domain. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 321 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER 0 τ Tb -Tb 2Tb 3Tb 4Tb -4Tb -3Tb -2Tb Δ N*T Δ 2N*T Δ 3N*T -Δ N*T -Δ 2N*T-Δ 3N*T G2( ) G2(- ) g1(-τ ) g1(τ ) (1,1) (3,4) (2,1) (4,2) (3,1) (4,2) (4,1) (4,3)(1,2) (1,3) (2,4) (1,4) (2,4) Fig. 5: The AF o one pulse o he CC4 signal. 3.3. Es ima ion o Ke nel Pa ame e s Many s udies we e conduc ed o es ima e he sui able ke nel pa ame e s o educing he in e e ence in he TFR. Adap i e op imal ke nel TFR (AOK-TFR) [20] is one o he ea lies echniques ha inco po a es window in he AF and is able o p oduce accu a e TFR e en a SNR o 0 dB. Howe e , he capabili y o he me hod is limi ed o mul i-componen LFM signals. Adap i e Op imal Ke nel Smoo h-Windowed Wigne -Ville Dis ibu ion (AOK-SWWVD) [17] and Adap i e Smoo hed Windowed c oss WVD (ASW- WVD) [15] a e also able o p oduce accu a e TFR a a low SNR bu hese solu ions a e limi ed o digi ally modula ed signals such as FSK and PSK signals. All he me hods men ioned abo e despi e p o en eliable o signal ep esen a ion, hey lacked he capabili y o co e a b oade class o signals. Thus, he Dopple -Lag Block Sea ching (DLBS) p ocedu e is in oduced in his sec ion o p oduce an accu a e TFR a low SNR while co e ing a wide class o signals. Be o e desc ibing he adap i e p ocedu es, i is c u- cial o i s discuss he ou ambigui y unc ion quad- an s as shown in Fig. 6. In he DLBS app oach, i is su icien o es ima e he ke nel pa ame e s only in he Q1quad an due o he symme ical p ope ies o he ambigui y domain. One o he AF p ope ies exploi ed is he maximum ene gy ha occu s a he o igin o he ambigui y domain. As de ined in [21], he AF is highes a he o igin in compa ison o he o he pa s o he ambigui y domain acco ding o he ollowing inequali y: |Az( , τ)|≤|Az(0,0)|.(9) Thus, he DLBS ini ia es he sea ch a he o igin and checks o a signi ican d op in ene gy in Dopple and lag. This is pe o med by ma ching he e e ence block, 0 τ Q1 Q2 Q3 Q4 Fig. 6: The quad an di ision in he ambigui y domain. Az(0,0) and he analyzed blocks, Az(λ1, λ2)in he AF domain o ob ain he ke nel pa ame e s. The DLBS algo i hm can be exp essed as ollows: ∆Az=|Az(0,0) −Az(λ1, λ2)|,(10) whe e ∆Azis he ene gy di e ence be ween Az(0,0) and Az(λ1, λ2)which can be de ined as Az(0,0) = ∞ Z −∞ ∞ Z −∞ wa( , τ)Az( , τ)d dτ. Az(λ1, λ2) = ∞ Z −∞ ∞ Z −∞ wa( −λ1, τ −λ2)Az( , τ)d dτ, (11) 0≤λ1<∞,0≤λ2<∞, whe e wa( , τ)is he analysis window, λ1is he ins an Dopple and λ2is he ins an lag. The analysis window can be desc ibed as [16] wa( , τ) = a τa ,(12) 0≤ ≤ a<1 Tp ,0≤τ≤τa< Tp, whe e τaand aa e he analysis window size in e ms o lag and Dopple espec i ely, Tpis he signal pulse wid h. In o de o DLBS o accu a ely es ima e he signal pulse wid h, he analysis window mus be se smalle han he expec ed signal pulse wid h. All he analyzed blocks ha ha e ene gy di e ence, ∆zabo e he h eshold alue, Az, hd a e conside ed as a block wi h a minimum numbe o au o- e ms and can be excluded om he gene a ion o he TFR. Figu e 7 shows he Q1quad an o PLFM signal and used as an example o DLBS p ocedu e. The DLBS sea ch p ocedu es a e as ollows: c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 322 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER τ 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 Tp 1/4Tp 1/2Tp 3/4Tp 5/4Tp 5 10 15 20 21 22 23 24 25 1/Tp 1/0.5Tp Figu e 7 Fig. 7: The Q1 o PLFM in DLBS me hod. •Fo m he Dopple -lag unc ion, Az( , τ) o he Q1 quad an using Eq. (8). •Compu e he o al ene gy o block 1 as a e e ence block, Az(0,0) and se a h eshold alue, Az, hd. •Compu e he ene gy di e ence o block 2, Az(1/0.5Tp,1/4Tp). Since he ene gy di e ence is below he h eshold alue con inue o he nex block on he igh . •The ene gy di e ence o block 3, Az(1/0.75Tp,1/4Tp)is ound o be abo e he h eshold alue which means a his pa icula block he au o- e ms ene gy is low. Hence, he e alua ion o ow (1/4)Tpis s opped and he e alua ion mo es o he nex ow, (1/2)Tp. •The e alua ion o new ow always s a s a one column be o e he las e alua ed column because i will ensu e he e alua ion s a whe e he au o- e ms ene gy is highe . A he ow (1/2)Tp he e alua ion s a s a block 7. •The e alua ion o ow (1/2)Tps a s om block 7 o he igh di ec ion un il i eaches block 9. Since he ene gy di e ence o block 9 is abo e he h eshold alue, he sea ch block mo es o he nex ow, (3/4)Tpand s a a block 13. •S eps om 3 o 5 a e epea ed o he o he ows un il eached ow (5/4)Tp. The ene gy di e ence o all he blocks in his ow a e ound o be abo e he h eshold alue and indica e ha he e alua- ion o PLFM signal is comple ed. •The ke nel pa ame e is es ima ed by choosing he loca ion o las e alua ed block ha has ene gy di e ence below he h eshold alue. Fo his ex- ample, he las e alua ed block is block 19 and i s loca ion ep esen he Dopple and lag pa ame e s which is 1/Tpand Tp espec i ely. The same p ocedu e is applicable o he o he sig- nals as hei p ope ies in he AF a e simila wi h he PLFM signal. The h eshold alue o 0.3is chosen because i p o ides he bes Dopple and lag window wid h o mos o he signal [19]. 3.4. F ac ional Fou ie T ans o m in C oss-Te ms Reduc ion The F FT is ac ually a gene alized o m o Fou ie T ans o m (FT) wi h he α- h o de o ac ional powe [22] and is bes exp essed wi h he help o ans o ma- ion ke nel. I x( )is he signal, hen he F FT o x( ) is gi en as [9] Xα(u) = ∞ Z −∞ x( )Kα( , u)d , (13) whe e Kα( , u)is he ans o ma ion ke nel and can be exp essed as Kα( , u)= 1−jco α 2πexpj 2+u2 2co α− u csc α .(14) Equa ion (14) is alid i αis no mul iple o π. How- e e , i αis a mul iple o 2π, he ke nel becomes δ( −u). Fo (α+π)a mul iple o 2π, he ke nel becomes δ( +u). The ex ension o he F FT o he TFR o he signal is he o a ed e sion o he WVD o he o iginal signal by he angle θwhich is gi en as ρxα( , ) = R−θ{ρx( , )},(15) whe e Rθ{} is he ope a o which o a es he TF plane clockwise. I he ime- a ying signal is linea ly sepa- able in he ime- equency plane, he c oss- e ms can be sepa a ed wi h he app op ia e o a ion angle and sui able TF il e ing echnique [5]. 3.5. IF Es ima e Ins an aneous F equency (IF) es ima ion can be used o desc ibe he equency cha ac e is ics o he signal. No mally, a good es ima o has o be consis en while s a is ically and compu a ionally e icien [23]. Rao and Taylo p o ed ha WVD peak based IF es ima ion is op imal o linea FM signals o mode a e and high SNR al hough he es ima o pe o mance deg ades sig- ni ican ly a low SNR [24]. Peak based IF es ima o can be exp essed as a g max ρ( , ),(16) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 323 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER whe e i( )is he IF and ρ( , )is he TFR. Peak based IF es ima o is able o p oduce a decen es ima ion as i s capabili y o localize ene gy along he IF law [23]. Fo he pu pose o measu ing he pe o mance o any unbiased pa ame e es ima o , he C ame –Rao Lowe Bound (CRLB) is equen ly used because i can p o- ide he heo e ical limi o he a iance o he es ima- o . The mos e icien es ima o is he one ha can achie e he lowe bound on he a iance [25]. The gene al o mula ion o CRLB o IF es ima e is [15] a ˆ l≥24 (2π)2γN (N2−1),(17) whe e Nis he a e age window wid h, and γis he SNR. Wi h he assump ion ha he ac ual IF o he signal is known, and he signal is in disc e e o m, he a iance o he IF es ima ed om measu emen can be exp essed as [15] a ˆ i=1 N N−1 X n=0 ˆ i(n)− i(n)2,(18) whe e Nis he o al numbe o samples, ˆ i(n)is he ac ual IF and i(n)is he ac ual IF. 4. Resul s and Discussion This sec ion desc ibes he TFRs and IF es ima es o SP, CC4, LFM and CW-LFM signals using he DLBS- WVD and F FT ollowed by he pe o mance o he IF es ima o benchma ked wi h he CRLB. The pe - o mance o DLBS-WVD is p esen ed a SNR o 5dB. This alue is chosen because SNR abo e 10 dB is con- side ed as high SNR om p e ious wo k on IF es ima- ion [15]. 4.1. TFR and IF Es ima e Pe o mance By using he DLBS-WVD, he TFR plo and IF es i- ma e o he signals using he pa ame e s p esen ed in Tab. 1 is p oduced. The ene gy o au o- e ms compo- nen s is highes a he o igin and dec ease posi ioned away om he o igin. Due o his, DLBS-WVD has a di icul y o p ese e he en i e ene gy o he signal componen s especially when he au o- e ms a e loca ed a om he o igin. This p oblem becomes ob ious o PLFM and CW-LFM signals compa ed o SP and CC4 signals as discussed in [19]. A small po ion o he au o- e ms componen s a e supp essed o PLFM and CW-LFM signals which p oduced mino e o s in he TFR and IF es ima e. (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-F equency Rep esen a ion F equency (Hz) 00.002 0.004 0.006 0.008 0.01 0.012 0 5 10 15 x 106 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 x 10 6 Ins an aneous F eqeuncy Time (ms) F equency (Hz) (a) TFR using DLBS. (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-F equency Rep esen a ion F equency (Hz) 00.002 0.004 0.006 0.008 0.01 0.012 0 5 10 15 x 106 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 x 10 6Ins an aneous F equency Time (ms) F equency (Hz) (b) IF using DLBS. Fig. 8: TFR plo and IF es ima e o wo pulses SP signal. Figu e 8 shows he TFR plo and IF es ima e o a wo pulses SP signal. The IF o he signal is accu- a ely es ima ed a 10 MHz. This is con ibu ed om a clean TFR o he signal due o he success ul sup- p ession o he c oss- e ms in he AF domain. The TFR and IF es ima e o a wo pulses CC4 signal is shown in Fig. 9. The equency componen s o he signal a e co ec ly es ima ed gi en ha he Cos as sequence o he signal is [1 2 4 3]. In his pape , he CC4 signal is used o illus a e he unc ionali y o he DLBS-WVD on he Cos as coded class o signals. Figu e 10 shows he analysis esul s o a wo pulses PLFM signal. The equency componen o he signal is es ima ed om 2.27–16.75 MHz while Tand Tpa e 13 ms and 4ms espec i ely. The ime componen s o he signal a e accu a ely es ima ed bu he equency componen o he signal is es ima ed wi h 1.6% e - c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 324 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER (b) IF using DLBS (a) TFR using DLBS Time-F equency Rep esen a ion F equenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 5 10 15 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 2 4 6 8 10 12 14 16 x 10 6 Ins an aneous F eqeuncy Time (ms) F equency (Hz) (a) TFR using DLBS. (b) IF using DLBS Time-F equency Rep esen a ion F equenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 5 10 15 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 2 4 6 8 10 12 14 16 x 10 6 (a) TFR using DLBS Ins an aneous F equency Time (ms) F equency (Hz) (b) IF using DLBS. Fig. 9: TFR plo and IF es ima e o wo pulses CC4 signal. o . This e o is due o ine i able supp ession o he small pa o signal au o- e ms in he ambigui y do- main. Howe e , he es ima ion e o is oo small and can be conside ed insigni ican . The TFR and IF es ima e is shown in Fig. 11 o a wo pulses CW-LFM signal. The minimum and max- imum equency o he signal is es ima ed a 2.19 MHz and 5.83 MHz espec i ely. The e is abou 4.2% e o in he equency componen es ima ion which is sligh ly highe han he LFM signal. This is due o he in a- pulse c oss- e ms in he CW-LFM signal which educes he ene gy concen a ion o he signal componen s and p oduce e o s in he IF es ima ion. Time (ms) (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-F equency Rep esen a ion F equenc y (Hz) 00.005 0.01 0.015 0.02 0.025 0 5 10 15 x 10 6 0 5 10 15 x 10 -3 5 10 15 x 10 6 Ins an aneous F eqeuncy F equency (Hz) (a) TFR using DLBS. Time (ms) (b) IF using DLBS Time (ms) (a) TFR using DLBS Time-F equency Rep esen a ion F equenc y (Hz) 00.005 0.01 0.015 0.02 0.025 0 5 10 15 x 10 6 0 5 10 15 x 10 -3 5 10 15 x 10 6Ins an aneous F equency F equency (Hz) (b) IF using DLBS. Fig. 10: TFR plo and IF es ima e o wo pulses PLFM signal. 4.2. TFR o WVD Using F FT Using he same signal as in he p e ious sec ion, he pe o mance o F FT in p oducing an accu a e TFR is e alua ed. The IF es ima e o he signal using F FT is simila o he esul om he p e ious sec ion gi en ha he c oss- e ms a e success ully emo ed. I is im- po an o pe o m a TF o a ion in such a way ha he ac ional equency spec um is mos compac be- o e applying any TF il e ing p ocedu e. The ech- nique used in de e mining he co ec o de o F FT is sea ching scheme [26]. This me hod de e mines he op- imal ac ional powe by e alua ing he compac ness o he ac ional equency spec um. The ac ional powe ha p o ides he mos compac ac ional spec- um will be used o TF il e ing. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 325 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER Time (ms) (b) IF using DLBS (a) TFR using DLBS Time-F equency Rep esen a ion F equenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 2 4 6 8 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6x 10 6 Ins an aneous F eqeuncy F equency (Hz) (a) TFR using DLBS. Time (ms) (b) IF using DLBS Time-F equency Rep esen a ion F equenc y (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0.035 2 4 6 8 x 10 6 00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 1 2 3 4 5 6x 10 6 (a) TFR using DLBS Ins an aneous F equency F equency (Hz) (b) IF using DLBS. Fig. 11: TFR plo and IF es ima e o wo pulses CW-LFM signal. The SP signal does no equi e any TF o a ion since he equency spec um is mos compac a ac ional powe ze o. The c oss- e ms ha a e loca ed be ween he signal componen s can be easily emo ed by apply- ing a TF il e ing as shown in Fig. 12(a) wi h a ed do ed line esul ing in as shown in Fig. 12(b) a c oss- e ms ee TFR o he SP signal. Figu e 13 shows a TFR plo o wo pulses o CC4 signal which is hea ily co up ed by c oss- e ms. Fo e e y pulse o CC4 signal, he e a e ou signal com- ponen s ep esen ing ou di e en equencies as indi- ca ed by ed do ed boxes. The localiza ion o c oss- e ms ha a e e y close o he au o- e ms esul in s ong deg ada ion o signal powe . The e is no sui - able o a ion angle ha can be used o comple ely e- mo e all he c oss- e ms. Fu he mo e, he smea ing e ec om he c oss- e ms u he educe he quali y o he signal componen s especially a a equency o TFR o O iginal Signal F equency (Hz) Time (ms) 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) (b) F equency (Hz) Time (ms) TFR A e Fil e a ion 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) TFR o O iginal Signal F equency (Hz) Time (ms) 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (a) (b) F equency (Hz) Time (ms) TFR A e Fil e a ion 00.002 0.004 0.006 0.008 0.01 0.012 0 2 4 6 8 10 12 14 16 18 x 10 6 (b) Fig. 12: TFR plo o wo pulses SP signal. TFR o O iginal Signal F equency (Hz) Time (ms) 00.005 0.01 0.015 0.02 0.025 0.03 0 2 4 6 8 10 12 14 16 18 x 10 6 Fig. 13: TFR plo o wo pulses CC4 signal. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 326