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 jh(t)∗e j 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 1 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).