Comression Waveforms for Non-Coherent Radar Uri Peer and Nadav Levanon el Aviv University P. O. Bo 39, el Aviv, 69978 Israel nadav@eng.tau.ac.il Abstract - Non-coherent ulse comression (NCPC) was suggested recently []. It was described using on-off eying (OOK) signals based on Manchester-coded binary ulse comression sequences (e.g., Barer, Iatov). he resent aer eands the discussion on waveform choice for both eriodic and a-eriodic cases, and on detection erformances of this method. OOK transmitter and a receiver based on enveloe-detection, suggested for the NCPC system, are simler to imlement than a binary hase-coded transmitter and a coherent receiver with I&Q synchronous detector, required for coherent ulse comression. NCPC can be used in simle radar systems where Doler information is not required, in directdetection laser radar systems and in ultra wide band (UWB) radar. Non-coherent rocessing has drawbacs in cases of reflections from multi-scatterer targets. he drawbacs and means of mitigating them are considered in section II. I. SCHEMAICS AND WAVEFORMS We shall eamine the receiving scheme shown in Fig.. his schematic erforms correlation rocessing by utilizing a finite imulse resonse (FIR) filter, shown in the figure as a vector of constants (b i, i =.n) which multily the history of the received signal, after enveloe detection (square-law, =, or linear-law, =). In order to achieve high range resolution with low sidelobes, as well as good detection erformances, there should be some comlementary relationshi between the transmitted signal and the coefficients of the FIR stored in the receiver. Noting the transmitted vector as a and the FIR filter (of the same length) as b, without loss of generality we can first imose a normalization requirement, n = a b = () where a and b are the elements of a and b resectively. In order to maintain low noise average at the outut of the receiver, the FIR should be a band-ass filter (BPF) namely BPF Rectified outut Outut b n b Fig.. Receiver bloc diagram b b = = () n b n = wo simle eamles of NCPC signals are shown in Fig. (a-eriodic signal) and Fig. 6 (eriodic signal). In the first eamle (Fig. ), the transmitted signal (blac) is based on the transmitted sequence a, of length n=56, which is a Manchester-coded (, ) MPSL 8 sequence [, able 6.3]: a = { } in the transmitted sequence a is reresented by a transmitted ulse in the corresonding time slot, while is reresented by a missing ulse. In the receiver the reflected ulses are enveloe detected and cross-correlated, using the FIR filter, with a reference waveform (red) which is based on a reference comlementary sequence b, where: ~ b = m b = 8b = a = { ---- --- --- - () --- - ---- --- --- -- -} m is the number of ''s in the transmitted sequence a and b ~ is the unnormalized b. In other words the reference signal differs from the transmitted signal by inserting negative ulses at the locations corresonding to in the transmitted sequence. he lower sublot of Fig. shows the outcome of the crosscorrelation between the two signals a and b. It maintains the general low ea sidelobe ratio ( 8) found in the autocorrelation of the original MPSL 8 signal, ecet for the two negative near sidelobes, whose sum is almost equal to the height of the mainlobe. Note that the mainlobe width deends on the width of the ulses (transmitted and reference) rather than on the duration of a sequence element. Narrowing the ulses will narrow the mainlobe width, but will require wider bandwidth, hence more noise at the inut to the enveloe detector. In Fig the cross-correlation vector c was multilied by m (=8), for resentation uroses only. For Manchester-coded binary sequences the reference vector b contains a small set of values (in our eamle: /8,-/8), which further simlifies the receiver. In a-eriodic cases filter b can be longer than signal a. In that case the normalization in () will be relaced by the requirement that the crosscorrelation vector c will get a value of at zero delay. (3)
Fig.. o: ransmitted (blac) and reference (red) signals, based on Manchester-coded MPSL 8. Bottom: Cross-correlation between the transmitted and reference signals. II. SENSIIVIY OF NCPC O MULIPLE-SCAERERS Before roceeding to the eriodic signal eamle, we ause to discuss a major difficulty that may hamer non-linear detection. It affects both eriodic and a-eriodic waveforms, but will be demonstrated using the a-eriodic signal introduced in the revious section. Coherent receiver, matched or mismatched, rocessing a reflected coherent comressed ulse, is only slightly affected by the resence of two or more scatterers. Consider coherent transmission of the OOK signal whose comle enveloe was reresented by the sequence a and a receiver that erforms coherent synchronous detection of that comle enveloe, and cross-correlate it with b. Assume also that the received reflected signal results from two scatterers yielding the received, noise-free, comle enveloe: u u u () t = u() t + u () t () t = a() t () t = α e( jβ ) a( t τ ) where α is a real ositive number that reresents the relative intensity, β is the relative hase in radians and τ is the delay difference between the two reflections. he linear rocessing erformed in the coherent receiver yields the outut v L ( t) = u() t b() t = a() t b() t + α e( jβ ) a( t τ ) b() t (8) where reresents cross-correlation. If the cross-correlation between a and b ehibits low easidelobes ratio (e.g., in the Manchester-coded MPSL-8 signal, (5) (6) (7) PSLR=/), and if the delay difference τ is larger than one bit duration, then the magnitude of the outut v L () t will include the original two mainlobes, searated by τ, whose normalized ea values P and P would be bounded by α PSLR P + α PSLR (9) α PSLR P α + PSLR () In contrast, the non-coherent rocessor erforms enveloe detection rior to the cross-correlation oeration. Assuming = in Fig., the outut of the non-linear rocessor would be v NL () t b() t = a() t + α e( jβ ) a( t ) b() t ( t) = u τ () With this ind of rocessor the effect on the two mainlobes of the cross-correlation could be more drastic, and could not be bounded as in (9) and (). A comarison between linear and non-linear rocessing is shown in Figs. 3 and. In that eamle τ = 8.5t b and α =. 9. he hase difference β is 3.5 radians (= ). he ulse-width is half the bit duration. he to sublot of Fig. 3 shows the first reflected signal. he second sublot shows the magnitude of the second signal, searated by 8.5 bits and slightly attenuated. he two signals add coherently at the antenna and the magnitude of their sum is shown in the third sublot of Fig. 3. We will come bac to the bottom sublot shortly. he to sublot of Fig. shows the magnitude of the outut of a synchronous coherent detector that erforms what was described in equation (8).
Fig. 3. Signals reflected from two scatterers Fig.. Detection oututs of the signals in Fig. 3.
Note that each one of the two eas is hardly affected by the resence of the other reflection, and maintains its relative strength (8. instead of 8 and 5.7 instead of 5.) as redicted by (9) and (). he second sublot of Fig. shows the outut of the non-linear detector that erforms equation (). Comaring it to the single scatterer case (bottom sublot of Fig. ) we note considerable degradation of erformances. A ossible remedy that can mitigate the degradation caused by multile scatterers is to add random hase coding to the transmitted ulses (bits). As a matter of fact, such random interulse hase modulation is inherent in some ractical transmitters, e.g., lasers, magnetrons. Detection erformances in a single scatterer scenario will not be affected by such hase modulation, because enveloe detection is transarent to hase coding. However, reflections from two scatterers, saced in delay by several bits, add coherently at the receiving antenna, and are liely to average out when a different and random hase modulates each bit. An eamle of the magnitude of the resulted signal is shown in the bottom sublot of Fig. 3 and the outut of the non-linear detector is shown in the bottom sublot of Fig.. For that secific scenario, the imrovement caused by the added random hase coding is rather rominent. In order to quantify the contribution of random hase coding on transmit, we erformed a Monte-Carlo simulation, whose results are summarized in Fig. 5. In that simulated situation the relative intensity of the second reflection was.9. he hase coding was random from bit to bit and changed from run to run. runs were erformed for each choice of sacing between the two reflections. hey differed by the reflections hase difference β, drawn from a uniform robability density function (PDF). Random hase coding was added only in the NCPC + rnd hase case (solid, blac). he thresholds were set so that each one of the two detectors (coherent and noncoherent) will yield the same robability of false alarm, P FA =.. he signal-to-noise ratios (SNR) were set to yield P =. D 95 in a single reflector case. Indeed that is the robability of detection observed in Fig. 5, for all three cases, when the sacing is larger than the signal length of 8 bits. he required SNR for non-coherent detection was.9db larger than for the coherent detection. his SNR loss is due to the difference between coherent and non-coherent detectors. here is an additional loss caused by the mismatch. In a coherent system the original hase-coded MPSL signal could have been transmitted, for which a matched receiver yields good resonse. Fig. 5 shows that for coherent detection (dotted, red), there is ractically no degradation when the searation is longer than one bit. With non-coherent (enveloe) detection, when no hase coding is added (dash, blue), the degradation in P D increases as the searation decreases, reaching P D. 67 for a searation of one bit. When random hase coding was added (solid, blac) the robability of detection is u again, fluctuating between.85 and the desired P value of.95. D Fig. 5. Detection erformances of coherent and non-coherent detection of Manchester-coded MPSL 8 signal, with and without random hase coding.
he conclusion is that adding random hase modulation to the transmitted ulses is advantageous for multi-scatterer or etended targets, but has no effect on detecting single-scatterer targets. As ointed out already, there are situations when the hase of the individual ulses is inherently changing randomly from ulse to ulse. III. PERIODIC WAVEFORM For the second eamle (eriodic signal) a elements Iatov code [3], [ (Sec. 6.5)] was Manchester-coded to get a desired transmitted sequence a. hen a reference sequence b was found in order to yield a secific cross-correlation. a={} () b ={q q r r r r s s s s s s q q s s s s s s s s s s q q s s s s r r s s r r q q s s s s -s s r r s s} (3) where q =5/8, r = /8, s = 7/8. he eriodic signal, reference, and cross-correlation are shown in Fig.6. In the middle sublot the reference signal was multilied by 8 for resentation uroses only. he lower sublot shows that the erfect cross-correlation found in the original Iatov code is maintained, ecet for the two negative sidelobes. Of the many ossible search algorithms we have adoted the Hill Climbing method to our alication, and used it successfully to find the desired sequences. his method was used in [] to find otimal sequences for mismatch filters. It is a secial case of the simulated annealing search method. For the eriodic case, with a desired cross-correlation c, transmitted sequence a and reference sequence b, the erformance function is as follows ( c) = ca( A A) ( A A) [ ] A c bb f a = () where and eriodic, a a an = an a an A (5) a an a ( A ) b = ca A (6) In the a-eriodic case, b is the minimum ISL mismatched filter of length l for the sequence a, and the erformance function is f a, l ISL level of the cross - correlation (7) ( ) a eriodic = where the cross-correlation vector c will be normalized to yield at zero delay, in order to satisfy (). We found it advantageous to eclude the first near sidelobe (on both sides) from the ISL minimization. hese two correlation sidelobes are inherently negative and will be removed by the one-way rectifier. Fig. 7 demonstrates the sidelobe reduction achieved by using a long (8 element) mismatched filter to the 56 element Manchester-coded MPSL 8. he ea sidelobe droed from -.9 db to -.3 db. he added SNR loss (not shown) was. db. Fig.6. One eriod of Manchester-coded Iatov : Signal, reference and cross-correlation. IV. FINDING COMPLEMENARY SEQUENCES Manchester coding a nown binary code is only one family of ossible transmission sequences. Its advantage is maintaining the original code roerties (i.e., low-sidelobe binary code will become low-sidelobe OOK sequence). In order to find the transmitted sequence a and the reference sequence b for the general case, we have defined erformance functions both for the eriodic and the a-eriodic case. Once a erformance function is defined, one searches for a sequence that will bring the erformance to a maimum. Fig.7. Normalized rectified outut with 56 element biolar (red) and 8 element minimum-isl (blac) filters
V. DEECION PERFORMANCES he a-eriodic eamle (Manchester-coded MPSL 8) will serve to quantify the detection roerties, using theory and simulations. Detection erformances, corresonding to the ea of the cross-correlation outut, were calculated assuming square-law enveloe detection, a non-fluctuating, single scatterer target, and additive white Gaussian noise (AWGN). Let the received signal be a row comle vector, ( σ ), u N(,σ ) r = Aa + v + ju, =,,..., n, v N, (8) where A is the non-fluctuating amlitude of the reflected ulses, v and u are the in-hase and quadrature-hase comonents of the noise, and all three defined at the outut of the BPF. Both v and u are taen from zero-mean Gaussian robability density function (PDF), with standard deviation σ (a function of the BPF bandwidth). he inut single-ulse signal-to-noise ratio (SNR) is define as A SNR = (9) σ he outut of the square-law enveloe detector is obtained from (8) s = ( A a + v ) + u, =,,..., n () and the ea of the cross-correlation will be = s *b () where () is the transose oeration. An aroimate PDF of can be calculated using the central limit theorem. We will get a Gaussian PDF ( ) N ( µ, σ ) () µ = A (3) where (3) holds because of roerties () and (), and var ( ) = σ = where W bwb () = diag ([ 8σ (.5 + a SNR),, 8σ (.5 + a SNR) ]) Without target (noise only), is still a normally distributed r.v. ( A ) N(, σ ) = A = A = µ (5) µ = (6) A = ( A = ) = σ bb var (7) In the secial case in which the reference is comosed out of two value set {/m, -/m} and the transmitted sequence is comosed out of {,}, equation () taes the following form: σ 8σ n var( ) = σ = ( nσ + ma ) = + msnr (8) m m Furthermore, if n = m, as in a Manchester-coded sequence, (8) will become n 8σ var( ) = σ = ( SNR) n m m + (9) = he PDFs in () to (9) were also numerically calculated for the first eamle (Manchester-coded MPSL 8 sequence) using Monte-Carlo simulations. Both numerical and theoretical results are lotted on to of each other in Fig. 8. Note that the PDFs derived from theory and from the histogram agree very well in the noise-only case and to a lesser degree when signal is resent. Fig. 8. PDFs corresonding to non-coherent ulse comression. o: signal + noise, Bottom: noise only he histograms of and were obtained with noise Monte-Carlo runs. he histograms were created usingσ =. 5. o get the desired robability of false alarm ( P FA =.) the threshold was set at.3. o get the desired robability of detection ( P =.95 D ) required single-ulse SNR of.9db. Similar analysis can be erformed in the eriodic case. here however it is ossible to integrate (non-coherently) many eriods, and achieve further SNR gain. ACKNOWLEDGMEN Section II was romted by very helful comments from J. Mie Baden and Marvin N. Cohen, based on their field results with the ill-fated Intraulse Polarization Agile Radar (IPAR) [5]. We are indebted to Mie Baden for many ensuing discussions and suggestions. REFERENCES [] N. Levanon, Noncoherent ulse comression,. IEEE ransactions on Aerosace and Electronic Systems, vol.,. 756-765, Aril 6. [] N. Levanon, and E. Mozeson, Radar Signals. New Yor, NY: Wiley,. [3] V. P. Iatov, and B. V. Fedorov, Regular binary sequences with small losses in suressing sidelobes, Radioelectron. a. Commun. Syst. (Radioeletronica), vol. 7, no. 3,. 9-33, 98. [] K. R. Grie, J. A. Ritcey, and J. J. Burlingame, "Poly-hase codes and otimal filters for multile user ranging," IEEE ransactions on Aerosace and Electronic Systems, vol. 3,. 75-767, Aril 995. [5] M. N. Cohen, B. Perry, and J. M. Baden, "Preliminary analysis of IPAR erformances," Proceedings of the 98 IEEE National Radar Conference, 98,. 37-.