2012 IEEE 7th Sensor Array and Multichannel Signal Processing Workshop (SAM)

Structured De-Chirp for Compressive Sampling of LFM Waveforms

Bruce Pollock Nathan A. Goodman Department of Electrical and Computer Engineering School of Electrical and Computer Engineering The University of Arizona The University of Oklahoma Tucson, Arizona 85721 Norman, Oklahoma 73019 Email: [email protected] Email: [email protected]

Abstract—Stretch processing is often used by radar systems to signal’s reference range changes with time). This approach capture LFM waveforms. Although stretch processing reduces produces a unique time profile of beat frequencies for every sampling rate, it does so at the expense of increased data range in the desired swath, thus enabling compressive sam- collection period. Therefore, the time-bandwidth product of the de-chirped waveform is approximately constant, and stretch pling with the possibility of incorporating prior knowledge, processing cannot be considered a compressive technique. without sacrificing range swath, degrading range resolution, The sample rate of a stretch processor must match the or introducing ambiguities. After describing the hopping de- bandwidth of the beat frequencies created in the de-chirp process. chirp receiver, we consider a multichannel version where each Uniform sampling at lower rates will cause range ambiguities channel can potentially have a different hopping sequence. without an anti-aliasing filter, but an anti-aliasing filter removes part of the range swath. We propose a hopping de-chirp signal In Section II we introduce the compressively sampled (sensing matrix) that shifts regions of the range swath in and out stretch processing receiver and show through the use of a of the anti-aliasing filter’s bandwidth. This hopping produces hopping de-chirp reference signal that we can compressively a unique time profile of beat frequencies for each range in sample a LFM pulse. We examine the effects of random the swath and enables compressive sampling without sacrificing hopping and discuss an approach for maximizing the expected range swath or resolution and without introducing ambiguities. After describing the hopping de-chirp technique, we analyze energy from the target return through prior knowledge of the multichannel implementations where it is possible to use a target’s range distribution. We extend the receiver architecture different hopping sequence on each channel. to a multichannel system and show range/angle ambiguity functions for the random compressive sampling architecture. I.INTRODUCTION In Section III we show expected SNR results from the In traditional radar, linear frequency modulated (LFM) maximization process described in Section II. We make our waveforms are often captured with a de-chirp procedure called conclusions in Section IV. stretch processing [1]. Stretch processing allows the rate of II.COMPRESSIVELY SAMPLED STRETCH PROCESSING the analog-to-digital converter (ADC) to be reduced, at the RECEIVER expense of increased data collection time. Because the time- bandwidth product of the received signal is the product of the The modulated pulse is a common radar signal that provides data collection interval with the Nyquist sampling rate, the high bandwidth in a long pulse with plenty of energy for trade-off provided by stretch processing cannot be considered detection. One popular type of modulation is linear frequency a compressive sampling implementation. Indeed, in order to modulation (LFM)   realize a reduced sampling rate via stretch processing, the t − 0.5Tp 2 x(t) = rect ej(2πF0t+πγt ), radar pulse width must be longer than the propagation time Tp across the desired range swath, so the total number of data so called because the instantaneous frequency (F + γt) samples remains approximately the same. 0 changes linearly with time. The slope, γ called the chirp rate, In this paper, we consider the de-chirp process as part of B is given by γ = where B is the pulse bandwidth and Tp the sensing matrix formulation for compressive sensing (CS), Tp is the pulsewidth. but modify the de-chirp reference signal in a way that enables compressive sampling [2]. Typically, the sample rate must be A. Stretch Processing matched to the bandwidth of beat frequencies produced by The received pulse is shifted by a time τ proportional to the de-chirp process. If the de-chirped signal is sampled at target range, so that the signal becomes a lower rate, part of the range swath must either be filtered   t − 0.5Tp − τ 2 out by an anti-aliasing filter or range ambiguities will be r(t; τ) = αrect ejπγ(t−τ) e−j2πF0τ . (1) produced. To avoid this, we propose a hopping de-chirp signal Tp that sequentially shifts different regions of the range swath in The required ADC sample rate can be reduced by ”de- and out of the anti-aliasing filter’s bandwidth (i.e., the de-chirp chirping” the signal prior to sampling. This is done by

978-1-4673-1071-0/12/$31.00 ©2012 IEEE 37 multiplying the received signal by the conjugate of a reference In other words, the expected signal energy for a given hopping signal. The reference signal x0(t) is offset by τ0, the center sequence is calculated by computing the energy retained by delay of the range swath under interrogation the hopping sequence for a given range, and then average over 2 the pdf of the target range. Define the time-bandwidth product jπγ(t−τ0) x0(t) = e . (TBWP) of the received signal over Ts as K = bTsBc. Divide Multiplying the received signal by the conjugate reference the range swath into a bank of K range bins defined by τi = ∗ j2πγ(τ0−τ)t Ts (x0(t)r(t; τ) ∝ e ) produces a beat frequency i K + τmin for i = 0,...,K − 1. The probability that a target R τi+1 falls in a particular range bin is pi = pτ (τ)dτ. Likewise, (γ(τ0 − τ) ) that is proportional to range, relative to the ref- τi erence [1]. The resulting bandwidth (called the IF bandwidth) define the TBWP of the radar pulse as P = bTpBc and divide generated by all ranges in the swath after de-chirp is the received signal into m = 0,...,P −1 hop intervals. Here, the de-chirp waveform stays at a particular hop for an interval BIF = γTs = B(Ts/Tp) (2) Tp of P . A shorter hop interval risks unintentionally modulating where Ts = τmax − τmin is time width of the range swath. the signal such that it is no longer identifiable as hopping LFM. Equation (4) becomes B. Compressive Sampling (m+1)T K−1 P −1 τ + p Equation (2) shows that as long as T is smaller than the Z i P s X X j2πγ(βm+i+τ0−τi)t 2 E¯s = Es pi |h(t)∗e | dt. pulsewidth, the Nyquist sample bandwidth (BIF ) will be less mTp τi+ than the bandwidth of the transmit pulse. If we wish to sample i=0 m=0 P at a lower rate or interrogate a larger swath but keep the same Using Parseval’s Theorem [3], the expression can be written sample rate, we must find a way to modulate the highest beat in the frequency domain, but because the time integration frequencies into the sample bandwidth. This can be done by has been approximated with discrete intervals, we can only ”hopping” the reference signal. The beat frequencies of the express it as an inequality. By swapping the integration with de-chirped signal will hop with the reference so that different a summation and substituting n for m + i and summing the ranges will have unique hopping patterns, thus eliminating intervals over Tp +Ts, we can integrate over frequency in each range aliasing while ensuring at least part of the return pulse interval independently. Define Q = min(K,P ) as the number will shift into the sample bandwidth for any given range. The range bins present in any one time interval, then price paid for this reduced-rate implementation is that not all P +K−1 Z ∞ Q−1 ranges can be shifted within the sample bandwidth at all times; γTp X X E¯ ≤ E p H(f)X (f)df, (5) therefore, some energy will be lost when the IF signal passes s s P n−i n,i n=0 −∞ i=0 through the anti-aliasing filter prior to sampling. Let a hopping de-chirp reference signal be given by where Xn,i(f) = δ (f − γ(βn + τ0 − τn−i)) and δ(f) is the 2 Dirac delta function over frequency. x (t) = ejπγ(t+β(t)−τ0) 0 Figures 1 and 2 show the process of de-chirping a signal where β(t) represents a piecewise hop function that is constant using a uniform randomly hopping de-chirp reference. The for the time interval tn−1 ≤ t < tn. At the end of each time left panel shows instantaneous frequency versus time for the interval, β(t) hops to a new constant value. The tn’s form a set hopping de-chirp reference and the reflected signal while the of disjoint time intervals that cover the full sampling period right panels shows the de-chirped beat frequency versus time. of duration (Ts + Tp). The set of βn = β(tn) can either be As seen in Figure 1, for uniformly random hopping γβn, randomly generated or deterministically designed in order to and Ts > Tp the reference range for any hop should be capture some attribute of the received signal. within the range swath. After de-chirp, the beat frequencies In (1) we model the received signal as a finite-energy, for any range will be uniformly distributed over B. The de- bandlimited continuous-time pulse, parameterized by time-of- chirp operation acts as an addition in frequency of two random arrival τ and amplitude α so that r(t; τ) = αx(t−τ). In order variables (the hopping sequence and the random target range), to quantify the average SNR loss incurred by our compressive each uniformly distributed across B. Since target range and implementation (Note that all compressive RF receivers will hopping sequence are statistically independent, the resulting incur SNR loss), let τ be a random variable with pdf pτ (τ). distribution on the beat frequencies is triangular, where the The received signal αx(t − τ) is now a random process and base of the triangle is across twice the original bandwidth. The the output of the de-chirping process and anti-aliasing filter is SNR loss during each hop interval is found by integrating the ∗ beat frequency distribution over the sample bandwidth. This y(t; τ) = h(t) ∗ (x0(t) · αx(t − τ)) . (3) results in each range suffering the same average loss in SNR. where h(t) represents an ideal low pass filter with cutoff For Ts < Tp, the story is somewhat different. Observe 2 frequency matched to half the ADC rate. Defining Es = |α| , in Figure 2 that each of the possible returns has a constant the expected energy in the received signal is frequency offset from the center of BIF . Because of this, the " # γτn−i will have the same distribution as τ except scaled by Z τ+Tp ¯ 2 γ and shifted to the center of BIF . Therefore, an efficient Es = E(τ) |y(t; τ)| dt . (4) τ hopping pattern hops uniformly around BIF . The expected

38 These samples are generated at a rate sufficiently high (in the Nyquist sense) for the transmit pulse, the hopping de-chirp reference, and the de-chirped signal. Note that the de-chirp output can have bandwidth greater than the original pulse due to the hopping reference (consistent with the tails of the beat frequency pdf described earlier). The received signal becomes T Fig. 1. An example of a uniformly random hopping de-chirp. Reference r(τ) = [r(tn; τ)]n=0,...,D−1. (7) signal ”hops” are shown vs. time, as well as the beat frequency time profile that results from the de-chirping process. Here Ts > Tp, so that the beat The simulated de-chirp reference (sampled over the same rate frequency will vary over B the bandwidth of the received signal and period) is placed along the diagonal of the square matrix X0, thus acting as a mapping between spaces of equal size. In conventional stretch processing, the sample space of y would be of dimension N = b(Tp + Ts) min(BIF ,B)c. Here, K acts to map the higher dimensional space into a space of size M = b(Tp + Ts)Fsc with y a vector of size M. N The compression ratio is then C = M . After compression, a matched filter is formed from the hypothesized delay τi, H −1 H Fig. 2. An example of a uniformly random de-chirp, its frequency ”hops” ζi = ((KK ) si) y (8) plotted vs. time and the results of de-chirping on a typical receive pulse, its s = Kr(τ ) i K beat frequencies plotted against time. Here Ts < Tp, so that de-chirp will where i i and represents one of filters from (5). vary over BIF the implied bandwidth of range swath at any given moment C. Multichannel Compressive Sampling For a multichannel system with 1×L array manifold vector jψ j(L−1)ψ loss is again found by finding the distribution of beat frequen- a(ψ) = [1 e ··· e ] where ψ is the per-element cies that results from uniformly distributed hops and the range angle-of-arrival dependent phase shift due to a target signal pdf, and then integrating over the sample bandwidth. In this incident on a uniform linear array [4]. Modify (1) to account case, however, each range can suffer a different loss in SNR, for a’s phase progression, then r(t; τ, ψ) = a(ψ)r(t; τ). The especially at the endpoints of the swath. discrete-time version is R(τ, ψ) = r(τ)a(ψ), and (6) becomes Define the compression ratio C as the ratio of the IF H Y = KR(τ, ψ) = HX0 R(τ, ψ). (9) Bandwidth to the compressive receiver’s sample rate C = B /F . For random de-chirp, with any range distribution, The channel responses are the columns of Y, a M ×L matrix. IF s In the case where K is identical across channels, the single the loss in SNR is at least C. For uniformly distributed target channel data vector from (6) is sufficient to form the multi- range, the expected loss in SNR will never be better than C channel data matrix Y = ya(ψ). However, when each channel even for custom-designed hopping sequences. For non-uniform has a different kernel K , we must separate the compression distributions it’s possible to decrease the expected loss. l effects. The pre-dechirping signal on each channel is given by Though expressed as an inequality, (5) can be used to j(l−1)ψ maximize the expected energy for a given range distribution rl(τ, ψ) = r(τ)al(ψ) = r(τ)e . and thus the expected SNR. At the n’th interval β is chosen n The temporal measurements on the lth channel are so as to maximize the integration in (5) over frequency. This H will maximize the expected energy in the sample bandwidth yl = Klrl(τ, ψ) = HX0,lrl(τ, ψ) across the range swath, the equivalent of enhancing the most where XH is the lth channel de-chirp reference signal matrix, likely ranges. 0,l and Y = [y1 ... yL] is the M × L data matrix. To express the de-chirping procedure in a matrix form con- The effects of uniformly random independent sampling sistent with the CS literature, we form the column vector y = kernels (i.e., independent hopping sequences on each channel) [y ... y ]T from samples of (3) where y = y(t; τ)| , 0 N n t=n/Fs at C = 4 are seen in Figure 3. The ambiguity function and samples are taken at rate Fs. Thus, the sensing-matrix H A(τ, Ψ) = vec(Y(τ = τ0, ψ = 0)) vec(Y(τ, ψ)) is shown based representation of (3) becomes for a six channel system, along with cuts through τ = τ0 y = Kr(τ) = HXH r(τ), (6) and ψ = 0. The mainlobe response is centered at (τ0, 0) as 0 expected; however, distortion is visible in the sidelobes due where H represents filtering followed by sampling at rate to the random peaks and nulls of the independent kernels. Fs. The de-chirp signal x0(t), either designed or random, Although the mainlobe of this range-angle ambiguity function is embedded along the diagonal of the matrix-operator X0, is well formed, random high sidelobes could leak across the with the off-diagonal elements left as zero. The de-chirp, filter bank causing uncompensated range-dependent distortion. filtering, and sampling processes can be combined into the In other words, Figure 3 shows that using different compres- measurement matrix-operator K. We model the analog signal sion kernels on each channel causes the range-angle ambiguity r(t; τ) via samples over the range swath plus the pulsewidth. function to be no longer separable in range and angle.

39 Fig. 4. A plot of SNR vs. the standard deviation of a Gaussian distributed Fig. 3. A plot of the ambiguity surface of Y centered on τ0 is shown on target range measured in range bins for both a 2:1 and 4:1 compression (one the left with cuts through range (ψ = 0) and angle (τ = τ0) shown to the range bin = 15m). Shown are the maximum expected SNR from the designed right. de-chirp plus the average SNR and SNR given a uniformly random de-chirp for a signal that would be 20dB SNR with no compression.

III.SIMULATION RESULTS We use (10) and the distribution of τ to compute the We use a Bayesian signal model, where the time-of-arrival expected SNR (the blue line) for each point on the x-axis. The τ is a random variable governed by a known distribution i expected SNR increases with decreasing uncertainty about the y = K (r(τi) + n) . target location, until it reaches it’s maximum value of 20dB. Thus showing that a hopping de-chirp measurement kernel can The noise vector n is sampled as in (7) from a complex be designed to exploit prior information (the distribution of Gaussian random process distributed as n(t) ∼ CN(0, σ2 ). n τ) about target locations. For example, the hopping sequence SNR can be measured by examining the peak value of might be optimized to emphasize roads and de-emphasize the noise free measurement to the noise increase through the shadowed ranges. system. For the single channel system, the variance is given The expected SNR from a uniformly random de-chirp (the by σ2 = [n∗(t)n(t)], the expected SNR is n E black line) is also shown. Here we predicted a loss of 4 dB in ∗ E(τi)[ζi ζi] the left panel and 6.3 dB in the right (based on the expected E[SNR] = ∗ (10) E[ζnζn] distribution after de-chirp). The slightly higher losses than expected in the right hand graph are again from modulation ζ = ((KKH )−1s )H Kn where n i . For uniformly distributed (high hopping rates). 1 Es delays this results in a best case of [SNR] = 2 . As the E C σn delay becomes less uncertain, we expect the SNR to increase IV. CONCLUSION Es to its uncompressed level 2 . We have evaluated the effects of random as well as struc- σn Figure 4 shows the effects on SNR for the Ts > Tp case. tured hopping for a possible implementation of compressing The simulation was run with a 10Mhz LFM waveform with sampling stretched processing. We have explored the possibil- 2 20dB amplitude and σn = 1. The compression is set to ity of extending this concept to a multi-channel architecture, C = 2 in the left panel and C = 4 in the right. The range and shown that while independent kernels can be compen- swath is twice the pulsewidth, 20µs and 10µs respectively. sated in mainlobe, in the sidelobe distortion as a function of Two different types of de-chirp reference signals were used, range will remain. We have shown that the process described one from the process described in II-B for expected energy in II-B is capable of increasing expected SNR based on prior maximization and the other, a uniformly randomly distributed information; however, the effects from switching at high rates de-chirp. With no compression (Nyquist rate sampling) we have yet to be incorporated. It is expected that degradation expect a 20dB SNR for both average and expected SNR (losses in SNR should occur if de-chirps are formed that switch at are with respect to this value). every possible time interval. Further research will incorporate The delay (τ) is distributed as Gaussian with mean τ0 and modulation effects as well as adding constraints that will standard deviation varying along the x-axis and scaled to range minimize the number of switches within the algorithm. bins. For each point along the x-axis we form a new reference ACKNOWLEDGMENT for both types of de-chirps, designed and random. A target is placed separately in each range bin in the swath, and the SNR We acknowledge support from the Defense Advanced Re- is computed for both de-chirps at that range-bin. search Projects Agency via grant #N66001-10-1-4079. The average SNR (the red line in both graphs) is found by REFERENCES taking the average SNR across the swath using the designed [1] M. A. Richards, Fundamentals of Radar Signal Processing. New York: de-chirp. We expect a loss of 3dB in the left panel and 6dB McGraw-Hill, 2005. in the right (based on the compression ratio). However, it [2] E. J. Candes and M. B. Wakin, “An introduction to compressive sensing,” is slightly higher (1-2dB) in both graphs due to modulation IEEE Signal Processing Magazine, pp. 20–30, March 2008. [3] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. effects creating by ”hopping” over very short time intervals New Jersey: Prentice-Hall, Inc., 1989. (many successive hops at short time intervals will cause signal [4] H. L. V. Trees, Optimum Array Processing, part iv of detection, estimation energy to leak out of the sample bandwidth). and modulation theory ed. New York: John Wiley and Sons, Inc., 2002.

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