The Periodic Ambiguity Function Its Validity and Value

Transcription

1 The Peiodic Ambiguity Function Its Validity and Value Nadav Levanon Dept. of Electical Engineeing Systems Tel Aviv Univesity Tel Aviv, Isael Abstact The peiodic ambiguity function (PAF) elates to Woodwad's ambiguity function (AF) like peiodic autocoelation elates to autocoelation. AF is defined fo a finite signal, pocessed by its matched filte. PAF can handle peiodic signals with infinite (o lage) numbe of peiods, and a pocesso that is matched to fewe peiods. PAF suits pactical scenaios found in ada employing coheent pulse tain o CW wavefoms. This pape evisits the PAF and its popeties, demonstating its value and viability. I. IRODUCTION Woodwad's AF [1] extends the one dimensional autocoelation function of delay to a two-dimensional function of delay and Dopple. The PAF, suggested in [2] and studied in [3 6], pefoms a simila extension fo the peiodic autocoelation. It tuns out that fo some pactical ada scenes the PAF is moe elevant than the AF. Pominent examples ae continuous wave (CW) ada and pulse ada employing coheent pulse tain wavefoms. In both examples the tansmitte emits (and the taget etuns) moe peiods, of duation T, of the wavefom than the eceive pocesses coheently (Fig.1). The peiodicity of the identical tansmitted and efeence wavefoms implies that thei complex envelope obeys u() t = u( t + nt ), n =, ± 1, ± 2,... (1) As long as the delay is shote than the diffeence between the lengths of the tansmitted (o eceived) wavefom and the efeence wavefom, ( P N ) T, the output esponse of a coelato eceive, in the delay-dopple plane, is given by Tansmitted wavefom 1 2 P T 1 N Refeence wavefom Fig. 1 Relative lengths of signal and efeence 1 χ (, ν ) = u( t ) u ( t) exp ( j2πνt )dt (2) whee is assumed to be a constant and the delay ate of change is epesented by the Dopple shift ν. An impotant popety of the PAF, poved in [3], says that a PAF with N efeence peiods is elated to a PAF with a single efeence peiod by a univesal elationship, which is a function of Dopple only, sin ( ) ( ) ( NπνT ) χ, ν = χ T, ν (3) N sin ( πνt ) whee the PAF with a single efeence peiod is 1 T χt (, ν ) = u( t ) u ( t) exp( j2πνt) dt T 1 = T + u ( t + T ) u ( t) exp( j2πνt) T u ( t ) u ( t) exp( j2πνt)dt Anothe intuitively expected popety is that fo P N T the PAF is peiodic with peiod T. ( ) II. PAF AND AF OF A PHASE CODED SIGNAL An example of the diffeence between the AF, singlepeiod PAF and N-peiod PAF is shown in Figs. 2 to 5, which utilize a 19 element P4 wavefom. Fig. 2 displays the phase evolution of the wavefom. Fig. 3 displays it AF. Note the elatively high delay sidelobes at zeo-dopple, and the idgelike AF, indicating lack of Dopple esolution. The zeo- Dopple cut of the PAF of a single peiod (Fig. 4) exhibits no delay sidelobes due to the ideal peiodic autocoelation of P4 signals. The idge emains and epeats itself evey T. Dopple esolution is seen in Fig. 5, which is the PAF with a efeence wavefom containing 16 peiods. Note the Dopple ecuent lobes at ν =1 T (and its multiples). The fist Dopple null is seen at ν =1 ( 16T ). Still found in Fig. 4 ae Dopple sidelobes. Those can be mitigated by adding weight on eceive. dt (4) /1/$ IEEE 24

2 The PAF descibes the delay-dopple esponse when the efeence wavefom is identical to the tansmitted wavefom, except fo the numbe of peiods. Futhe deviations fom the AF and the PAF ae justified if the efeence signal diffes in othe ways. The fist modification is adding amplitude weighting to the efeence. Fig. 5 PAF of 16 peiods of a 19 element P4 wavefom Fig. 2 Phase evolution of a 19 element P4 wavefom Fig. 6 Amplitude and phase of 16 peiods of a 19 element P4 wavefom (Hamming amplitude weight.) Fig. 3 AF of a 19 element P4 wavefom Fig. 4 PAF of a 19 element P4 wavefom /1/$ IEEE Fig. 7 Delay-Dopple esponse with a efeence as in Fig. 6 25

3 Recall that coelation with the efeence is usually pefomed in a digital stage of the eceive, whee adding amplitude vaiation is elatively simple. Fig. 6 displays the amplitude and phase of 16 peiods of a 19 element P4 wavefom, with Hamming amplitude weight. The esulted delay-dopple esponse (not PAF anymoe) is shown in Fig. 7. Seen ae a significant eduction in Dopple sidelobes and a widening of the Dopple main and ecuent lobes. In [6] it was poved that fo the popula weight windows (Hamming, Hann, etc.), if the efeence and the weight window extend ove an intege numbe of peiods, the zeo- Dopple cut of the esponse emains sidelobe-fee. Adding amplitude weighting to the eceive's efeence becomes even moe impotant when the eceived signal's peiodicity is not pefect. This can happen, fo example, when the taget's etun is modulated by a otating antenna patten. Fo wavefoms with ideal peiodic autocoelation (like P4) this will e-intoduce sidelobes in the coelato esponse (-51 db in Fig. 8). Adding weight window to the efeence will geatly educe those sidelobes (-67 db in Fig. 9). Fig. 9 Weake cosscoelation sidelobes, weighted efeence, eflected signal is amplitude modulated Using the PAF we will demonstate attainable delay Dopple esponses when the coheent pocessing inteval (CPI) contains only one long peiod (Fig. 1), o when the same CPI contains eight shote peiods (Fig. 11). The fequency deviation will be the same in both cases. Fig. 8 Stonge cosscoelation sidelobes, unweighted efeence, eflected signal is amplitude modulated Fig. 1 Weighted single peiod LFM wavefom Fo most CW peiodic wavefoms, analytical expessions of the PAF ae difficult to deive. The esulted expessions ae usually tedious and do not povide qualitative insight. A simple altenative ae numeical calculations. MATLAB codes fo calculating and plotting PAFs ae listed in [7]. The softwae given thee can also plot AF and delay-dopple esponse (when the efeence diffes fom the signal). The PAFs dawn in this pape wee poduced using those pogams; so wee the many PAF plots found in [8,9]. III. USING PAF TO COMPARE TWO LFM-CW WAVEFORMS Linea-FM is the most common type of CW wavefom. It is used in low cost automotive ada as well as in advanced coastal ada. Fig. 11 Weighted 8 peiods LFM wavefom /1/$ IEEE 26

4 Note that the weight window, which applies to both the tansmitted signal and the efeence signal, is a squae oot of the Blackmann-Hais window. This implies using a matched filte; hence the delay-dopple esponse is indeed a PAF. In pactical ada a full (athe than squae oot) weight window will be implemented only in the eceive. The mismatch will entail small SNR loss, but the delay-dopple esponse will be simila to the PAF. Compaing Figs. 1 and 11, note that in both dawings the total duation of the CPI is 64 bits. The single fequency sweep in Fig. 1 is of that duation, while each sweep duation in Fig. 11 is 8 bits (fequency steps). The total fequency deviation, fo both signals, is theefoe 4 Δ f = (5) 64 implying a delay fist null, without weighting, at t b 1 null = = 1.6t b (6) Δf With amplitude weighting the mainlobe width would appoximately double, yielding 3.2t. null b The PAFs in Figs. 12 and 13 indeed exhibit that same delay esolution. The main diffeences between the two PAFs ae in Dopple esolution and unambiguous delay. The PAF of the single fequency sweep (Fig. 12) shows no Dopple esolution and (not seen) an unambiguous delay equal to the sweep duation (64 bit). The PAF of the 8 fequency sweeps (Fig. 13) shows Dopple esolution equal to the invese of the CPI ( = 1 64t b ), Dopple ambiguity equal to the invese of the epetition peiod ( = 1 8t b ) and delay ambiguity equal to the shote sweep duation (8 bit). The PAFs in the LFM example demonstate what pefomances can be obtained fom a peiodic LFM-CW wavefom, if pocessed by a matched filte. Othe, less optimal pocessos, like stetch pocessing, often used with a single sweep LFM, may be simple to implement but will achieve pooe esponse. When the eceive is matched to a Dopple shifted vesion of its nominal signal, its delay-dopple esponse is not a PAF anymoe. Such a esponse, fo the 8 peiods LFM wavefom, is given in Fig. 14. While not obvious fom the dawing, the esponse is not symmetical with espect to the oigin, which is an impotant popety of a PAF. Fig. 12 PAF of a weighted single peiod LFM wavefom Fig. 13 PAF of weighted 8 peiods LFM wavefom Fig. 14 Delay-Dopple esponse of a weighted filte matched to a Dopple shifted vesion of the 8 peiods LFM wavefom IV. CODED PULSE TRAIN The last example of analyzing peiodic signals involves a peiodic pulse tain, in which both peiodicity and a mismatch efeence ae necessay in ode to extact the special featues of the wavefom. The example follows an Ipatov-coded pulse tain descibed in [1]. The envelope of the binay- coded wavefom, the efeence and the esulted peiodic cosscoelation ae shown in Fig. 15. The combination of inte-pulse Ipatov coding and a mismatched efeence, esults in mitigating the coss-coelation peaks at the pulse epetition intevals. A section of the peiodic delay-dopple esponse is shown in Fig. 16. It shows how well the mitigation holds with Dopple. Dopple sidelobes would have been educed if the efeence was amplitude weighted /1/$ IEEE 27

5 Fig. 15 Two peiods out of a high PRF pulse tain with inte-pulse coding based on Ipatov 24 sequence: Signal's envelope (top), efeence (middle), peiodic coss-coelation (bottom). Fig. 16 Delay-Dopple peiodic esponse of M=16 peiods of a signal based on an Ipatov 24 sequence. Zoom on < 1.1T and ν 1.2 MT V. CONCLUSIONS The peiodic vesion of Woodwad's ambiguity function was evisited in ode to show how impotant and useful it is fo the analysis of CW wavefoms and othe peiodic signals that ae not necessaily CW. PAF analysis also applies to finite peiodic coheent signals wheneve the eceiving filte is matched to fewe peiods than the finite numbe of peiods coheently tansmitted. In cases when a mismatched efeence is used (e.g., to educe Dopple sidelobes) the delay-dopple esponse is an extension of the peiodic coss-coelation athe than the peiodic autocoelation. In those cases the tem PAF cannot be used, despite the similaity. REFERENCES [1] P. M. Woodwad, Pobability and Infomation Theoy with Applications to Rada, London: Pegamon Pess, [2] A V. Nenashev, Popeties of the peiodic ambiguity function fo polyphase quadatic sequences, Radioelectonics and Communications Systems, vol. 29 (4), pp , [3] N. Levanon and A. Feedman, Peiodic ambiguity function of CW signals with pefect peiodic autocoelation, IEEE Tans. on Aeospace and Electonic Systems, vol. AES-28 (2), pp , [4] N. Levanon, CW altenatives to the coheent pulse tain signals and pocessos, IEEE Tans. on Aeospace and Electonics Systems, vol. AES-29 (1), pp , [5] A. Feedman and N. Levanon, Popeties of the peiodic ambiguity function, IEEE Tans. on Aeospace and Electonic Systems, vol. AES-3 (3), pp , [6] B. Getz and N. Levanon, Weight effects on the peiodic ambiguity function, IEEE Tans. on Aeospace and Electonic Systems, vol. AES-31 (1), pp , [7] N. Levanon and E. Mozeson, Rada Signals, Hoboken, NJ: Wiley, 24. [8] P. E. Pace, Detecting and Classifying Low Pobability of Inteface Rada, 2 nd edition, Nowood, MA: Atech House, 29. [9] M. Jankiaman, Design of Multi-Fequency CW Rada, Rayleigh, NC: SciTech, 27. [1] N. Levanon, Mitigating ange ambiguity in high PRF ada using binay inte-pulse coding, IEEE Tans. on Aeospace and Electonic Systems, vol. AES-45 (2), pp , /1/$ IEEE 28