Intrinsic Processing Gains in Noise Radar

Transcription

1 Intrinsic Processing Gains in Noise Radar Brian D. Rigling Department of Electrical Engineering Wright State University 4 Russ Engineering Center 364 Colonel Glenn ighway Dayton, O brigling@cs.wright.edu Abstract In continuous wave noise radar, pseudo-pulse compression is typically accomplished through a simple crosscorrelation of the received and reference signals. We consider the use of adaptive filtering algorithms in pseudo-pulse compression for air-to-ground noise radar. Through analysis and simulations, we compare the SNR performance of cross-correlation to channel estimation via the LMS algorithm. The LMS algorithm was found to provide improved performance, relative to cross-correlation, as the number of observed targets increases. I. INTRODUCTION Whether engaging moving or stationary targets, the widely used solution to all-weather, air-to-ground surveillance has long been pulsed monostatic RF radar. The historical appeal of monostatic operation is its simplicity - the system must only geo-locate and coordinate a single platform. For reasons of stealth and other operational advantages, more attention has recently been focused on bistatic systems, wherein a passive receiver of relatively low cost may be used to observe scenes from closer range. This passive receiver may be teamed with a higher cost transmitting platform at a safe standoff distance, or it may exploit illuminators of opportunity such as local television and radio transmitters or even over-passing satellites. Recent advances in uninhabited aerial vehicle (UAV) technology are making deployment of multiple low-cost platforms to interrogate a single scene more appealing. This multistatic network of sensors may operate passively by relying on illumination from non-cooperative transmitters and/or remotelylocated cooperative transmitters, or nodes within the multistatic network may alternately operate as transmitters and receivers. Multi-function operation to allow multiple modes to operate concurrently (e.g., synthetic aperture radar (SAR), ground moving target indication (GMTI), communications) is also a desirable feature. Design and operation of a multistatic/multi-function system, consisting of multiple low-cost receivers as well as multiple cooperative and non-cooperative transmitters, poses significant challenges beyond the design of a traditional monostatic system architecture. Commonly used pulsed waveforms such as the LFM-chirp are poorly suited to multistatic operation due to mutual interference problems and a lack of adaptability in terms of range-doppler ambiguity and multi-function operability. As an alternative, noise waveforms offer significantly greater versatility. Current approaches to noise radar have some apparent disadvantages. The most common implementation involves digitiing and recording the transmitted waveform and the received signal. The recorded continuous wave signals may then be segmented into pseudo-pulses and compressed by correlation in software [1], []. This approach is demanding on both memory and computational resources. In addition, as the number of targets in the observed scene grows, the intrinsic signal-to-noise ratio (SNR) of the correlator output steadily degrades. Specifically, the thumb-tack ambiguity function of a noise waveform has side lobes that behave statistically like white noise and add in RMS to any thermal noise that is present. Superposition of multiple target returns cause their respective side lobes to correspondingly add in RMS, thus degrading the apparent SNR of the system without any change in the thermal noise level. This effect can be especially detrimental when observing returns from heterogeneous scatterers, such that side lobes from strong reflectors can incrementally swamp the response of weaker returns. An alternative architecture, based on wide-band wireless communications technology, is adaptive noise radar (ANR). As shown in [3], the least-mean squared (LMS) [4] algorithms developed for adaptive channel equaliation/identification may also be used for range profile estimation in noise radar pseudopulse compression. This is accomplished by treating the complex-valued range profile as a wireless channel, modelled by a finite impulse response (FIR) filter. As emerging wireless technology will allow channel estimates to be updated continuously, this ANR system architecture may output compressed pseudo-pulses at a rate up to the received signal analog-todigital conversion (ADC) rate. The LMS algorithm is known to converge to the least squares solution in the mean [5], thereby implying a potential improvement in intrinsic SNR. Specifically, the error in the estimated FIR filter weights is dependent only on the autocorrelation properties of the transmitted signal and the level of additive noise in the receiver; the estimated weight errors are independent of the optimal filter solution and thus independent of the number and composition of target returns. Simulations reported on in [6] showed that the intrinsic SNR performance of ANR did not vary with the number of scatterers present. In contrast, the simulated intrinsic SNR of cross-correlation appeared to decrease as the number of target returns grew.

2 y mth Scatterer r m r t (τ) Transmitter Path x it is reflected by scatterers within the area of illumination. Some of these reflected signals are observed by the antenna on the receiving platform, and are recorded in the form of phase history data. Given a transmitted signal û(t), we assume that u(t) = û(t τ (t)) is a copy of the transmitted signal delayed by the round-trip time to scene center. We may then represent the received signal as r r (τ) Receiver Path Fig. 1. Top view of a bistatic data collection geometry. The x-y plane is the ground plane. This paper extends the results of [6] by analytically predicting the SNR performance of cross-correlation and adaptive noise radar. We validate this analysis with Monte Carlo simulations of one-dimensional range profiles. For large scenes, particularly with many strong target returns, the ANR algorithm is found to give an appreciable improvement over crosscorrelation. The remainder of this paper is organied as follows. Section II describes our model for bistatic air-to-ground noise radar. Section III details, and analyes the performance of, two receiver processing architectures for noise radar: crosscorrelation and adaptive noise radar. Section IV presents simulation examples that validate our analysis. Section V summaries our results and conclusions. The notation used in this discussion is as follows. Column vectors are denoted by underlined lower-case letters (e.g., x). The conjugate transpose of a vector or matrix is denoted by ( ) (e.g., x, X ), and complex conjugation is denoted by ( ) (e.g., x ). II. BISTATIC NOISE RADAR Consider the bistatic SAR data collection geometry shown in Fig. 1. The center of the scene to be imaged is located at the origin of coordinates, and the ground plane is the x- y plane. A scatterer within that scene is located at r m = (x m, y m, m ). At a given time τ, the location of the transmitter is r t (τ) = (x t (τ), y t (τ), t (τ)), and the location of the receiver is r r (τ) = (x r (τ), y r (τ), r (τ)). The transmitter moves along its flight path and continuously radiates energy in the direction of the scene center. We assume that the transmitted signal is a noise waveform with uniform power over the frequency range f [f c W/, f c + W/], where f c and W represent the center frequency and bandwidth of the transmitted waveform, respectively. The radiated energy travels from the transmitter to the scene of interest, where d(t) = m A m u(t τ m (t)) + (t) (1) where (t) is band-limited, Gaussian noise and A m is the complex reflectivity of each scatterer in the scene, and τ m (t) = r t(t) r m (t) + r r (t) r m (t) c r t(t) + r r (t) c is the time delay of each scatterer relative to that of a return from scene center. The variable c represents the speed of light. At the receiver, d(t) is quadrature demodulated and sampled at a rate F s > W, such that the digitied output may be represented as d(n) = ( )) A m u (n τ m + (n) m nfs = K/ k= K/ h (k)u(n k) + (n) () = h (n) u(n) + (n). (3) Satisfying Nyquist allows h (n) to model the response of the scene as a FIR filter, commonly referred to as a range profile, which is convolved with the transmitted signal. The complex value of each tap of this filter is equal to the sum of the reflectivities of all of the scattering centers at the range corresponding to that filter tap. The number of taps in this filter, or the number of range gates in the profile, is computed as K = F s r/c (4) where r is the range swath of the illuminated scene and represents rounding up to the nearest integer. The time it takes for round-trip traversal of the range swath (T = r/c) is analogous to the delay spread of a wireless channel. III. PROCESSING ARCITECTURES In A/G imaging applications, one seeks to estimate the reflectivity of each scatterer in the scene as a function of range and cross-range (Doppler). Typically, this is accomplished by first compressing the received signal to achieve range resolution (i.e., range profiles) at a number of aimuth sample points. Coherent processing across many pseudo-pulses then produces cross-range resolution.

3 A. Cross-correlation For the received signal model in (3), assuming that the scatterers lie at a uniform set of ranges allows one to estimate a range profile via cross-correlation. If the received and reference signals, d(n) and u(n), are segmented into pseudopulses of Q samples, the estimated reflectivity at the pth range gate is computed as ĥ q (p) = = q+q 1 n=q K/ k= K/ u (n p)d(n) h (k)ρ u (p k) + ρ u (p), p K (5) where ρ u ( ) is the autocorrelation function of u(n), and ρ u ( ) is the cross-correlation function of the noise signal (n) and the transmitted signal u(n). This compressed pseudo-pulse corresponds to a sample in aimuth at slow time t = q/f s. As the transmit and receive platforms move relative to the targets in the scene, the values of h (k) will change. These changes are observed by repeating the cross-correlation processing (5) at regular intervals in the data collection to synthesie compressed pseudo-pulses. The spacing between pseudo-pulses (i.e., samples in aimuth or slow time) is Q/F s. Each of these sample points in slow time will use a different segment of the received data. The maximum allowable interval between pseudo-pulses is determined by the Doppler bandwidth of the targets and clutter in the scene. As shown in Fig., the complex-valued range profile estimates are passed downstream for Doppler processing. Doppler filtering across pulses, within each range bin, produces cross-range or Doppler resolution, which is the key product of a SAR [7], [8] or GMTI. The intrinsic SNR characteristics of images resulting from Doppler processing of range profiles estimated by (5) are determined by the signal autocorrelation properties and the thermal noise power. We assume that the noise radar signal and the thermal noise signal are mutually independent, -mean Gaussian random processes that are band-limited with uniform power over the band of frequencies f [f c W/, f c +W/]. Given a coherent integration time of T coh seconds, the coherent processing gain of a target is (W T coh ), while the side lobe and thermal noises levels only experience a gain of W T coh for a net SNR improvement of W T coh. For a scene containing M scattering centers, each with a received signal power of σd, the output image SNR of each scatterer in decibels is SNR 1 = 1 log 1 ( σ d W T coh Mσ d + σ ). (6) The contribution of thermal noise to the noise floor is σ = E{ (n) }, the receiver thermal noise power, and Mσd represents the RMS addition of the side lobes for the M target returns where σd = E{ A m }. Note that the intrinsic noise floor of the compressed signal (i.e., with σ = ) increases linearly with the number of scattering centers. This is due to the fact that the compressed Transmitted u(n) q Round trip Time Delay Scene Response (n) Delayed Reference d(n) Received Noise Radar Receiver Correlator Doppler Processing Range Profile Estimates Fig.. Noise radar receiver-correlator architecture. Cross-correlating a copy of the transmitted signal with the received signal provides range profile estimates. Transmitted u(n) q Round trip Time Delay Scene Response (n) Delayed Reference d(n) Received Adaptive Channel ID e(n) Doppler Processing Range Profile Estimates Fig. 3. The adaptive noise radar receiver architecture computes range profile estimates through adaptive channel ID processing. signal is equal to the superposition of M shifted replicas of the signal autocorrelation function. As M grows large (e.g., for large SAR images containing millions of pixels), the intrinsic noise of the signal autocorrelation function can dominate the image SNR. B. Adaptive Noise Radar Algorithms used for adaptive wireless channel identification (i.e., the LMS algorithm) allow a complex range profile to be estimated with computations of the same order as crosscorrelation. As shown in [3], the adaptive noise radar algorithm uses the LMS algorithm for adaptive channel identification to accomplish noise radar pseudo-pulse compression. This is done by treating the complex-valued range profile, which has the vector form h n = [h()... h(k 1)] T, as a FIR filter to be estimated. Fig. 3 shows the block diagram for the ANR system architecture, which makes use of LMS filtering for noise radar signal compression. Thus, the update equation for ANR is ĥ n+1 = ĥn + µu(n)e (n) (7) where µ is a step-sie, and e(n) = d(n) ĥ n u(n) is an error signal forced to ero by the LMS algorithm. The transmitted signal is represented in vector notation by u(n) = [u(n),..., u(n K + 1)] T. The ANR process adapts the estimated range profiles ĥn continuously, rather than processing the digitied data in blocks of Q samples. The output of the LMS algorithm is equal to the least squares result in the mean [4], thereby conferring some degree of improvement in intrinsic SNR. The errors in the adapted filter weights h n = ĥn h n appear as noise in the ANR-estimated range profile. Stochastic analysis techniques for the LMS algorithm can approximate the expected covariance of these errors. Use of independence assumptions and small step sie approximations allow the

4 correlation matrix of the adapted filter weight error vector R = E{ h n h n } to be computed through the recursive approximation [4, Equation (9.6)] 5 Cross Correlation Wiener Solution LMS R h (n + 1) = [I µr u ] R [I µr u ] + µ R u σ (8) where R u = σ ui is the autocorrelation matrix of the transmitted signal. Note that (8) is independent of the channel to be estimated, and therefore unlike cross-correlation, the noise present in the estimated range profile is independent of the true range profile and thus is independent of the number of target returns. At steady state, we assume R h ( ) = lim n R = lim n R h (n + 1). Equation (8) represents the uncertainty in adapted filter estimates on a sample-by-sample basis. In the ANR algorithm, a compressed pseudo-pulse is estimated by averaging over many updates of the LMS algorithm. This averaging provides an improvement in SNR and reduces the update rate of the LMS algorithm to a pseudo-pulse repetition frequency appropriate to a radar application. For small step sies µ, the LMS update (7) indicates that consecutive filter estimates will be highly correlated. Given R u diagonal, examining (8) indicates that the weight error correlation matrix R = σ h I will also be diagonal. Based on the mean behavior of the LMS algorithm [5], E{ h n+1 } = [I µr u ]E{ h n }, the correlation coefficient of successive errors for the pth filter weight is approximately p = E{ h n(p) h n+1 (p)} E{ h n (p) } 1 µσ u K 1 K when µ (Kσu) 1. Averaging Q filter estimates at steady state thus results in an error variance for a single weight of [ Q(1 p) p(1 p k ave = σh Q ) Q (1 p) 1 ]. (1) Q Equations (8 1) allow one to approximate the SNR performance of the ANR algorithm by recursively computing the steady state filter error correlation R h ( ) and then including the effects of averaging to obtain k ave. The predicted SNR is then ( ) σ SNR = 1 log d W T coh 1, (11) k ave Q which includes additional gain due to Doppler processing. IV. SIMULATION RESULTS To validate the results of our analysis, Monte Carlo simulations were performed. Each simulated scene contained M homogeneous scatterers that were observed for N = W T coh independent, coherent samples, as described by (3). The amplitude of each scatterer return was A m = 1 (i.e., σ d = 1), and the noise power was σ = 1. In each simulation, crosscorrelation and ANR provided estimated range profiles, which were RMS averaged over 1 iterations. Given N = 1, figures 4 and 5 show the RMS-averaged range profile estimates for M = 1 and 8, respectively. (9) Range Profile (db) Range Gate Fig. 4. Estimated range profiles for a single target using cross-correlation and LMS channel identification. Range Profile (db) Cross Correlation LMS Range Gate Fig. 5. Estimated range profiles for eight targets using cross-correlation and LMS channel identification. The RMS-averaged range profiles allow an effective SNR to be estimated for each algorithm and each simulation scenario. Figure 6 plots the predicted SNR performance curves, equations (6) and (11), along with the results of the Monte Carlo simulations. The predicted and simulated SNR values for cross-correlation match perfectly. owever, there is a slight discrepancy between the simulated and predicted values for ANR. This mismatch is due to the small-step sie assumptions used to derive the filter error correlation matrix (8). It was found experimentally that as the step sie is decreased the predicted and simulated performance curves move into agreement. In both the simulated and predicted performance analysis, it was found that the ANR signal-to-noise ratio does not degrade with an increased number of scatterer returns, thus giving an improvement in intrinsic SNR over cross-correlation. V. CONCLUSION In the pursuit of multi-function operability and exploitation of noncooperative transmitters, bistatic/multistatic radar and noise waveforms are of growing interest in the defense community. Through analysis and simulation, the performance

5 SNR (db) Simulated LMS Simulated X Correlation M = Number of Samples Fig. 6. SNR performance of the cross-correlation and LMS algorithms as a function of the number of processed signal samples. CNR (db) GMTI performance M = 4 M = Number of Targets Fig. 7. Predicted SNR performance of a GMTI with 3 M bandwidth and a 3 millisecond coherent integration time, using cross-correlation and the LMS algorithm. of adaptive noise radar processing was compared to that of cross-correlation. As predicted, simulations showed that the SNR performance of the ANR algorithm was independent of the number of scatterer returns in a scene, while the SNR resulting from cross-correlation was found to worsen with an increased number of returns. This performance difference becomes significant when observing large scenes containing many targets. Figures 7 and 8 show the possible impact of ANR processing as opposed to simple cross-correlation for nominal GMTI and SAR systems, respectively. As the number of targets observed by the GMTI increases, the SNR resulting from cross-correlation decreases, while the SNR of ANR remains unchanged. For a large SAR image consisting of heterogeneous target returns, the predicted CNR of low return areas decreases as the number of strong clutter returns increases, thus resulting in a loss of image contrast. REFERENCES [1] P. B. J. Jordan and B. Kiani, Correlation-Based Measurement Systems. West Sussex, England: Ellis orwood Limited, [] J.K. Kayani and S.F. Russell, A computationally efficient correlator for pseudo-random correlation systems, Spread Spectrum Techniques and Applications, IEEE Sixth International Symposium on, pp , Sep. [3] B. Rigling, Adaptive Filtering for Air-To-Ground Surveillance, In Algorithms for Synthetic Aperture Radar Imagery XI, Edmund G. Zelnio, editor, Proceedings of SPIE, Apr 4. [4] S. aykin, Adaptive Filter Theory, 3rd ed. Upper Saddle River, NJ: Prentice-all, [5] A. Sayed, Fundamentals of Adaptive Filtering. oboken, NJ: John Wiley & Sons, Inc., 3. [6] B.D. Rigling, Performance Prediction in Adaptive Noise Radar, Thirty- Eighth Asilomar Conf. on s, Systems & Computers, Nov. 4. [7] W.G. Carrara, R.S. Goodman and R.M. Majewski, Spotlight Synthetic Aperture Radar: Processing Algorithms. Norwood, MA: Artech ouse, [8] C.V. Jakowat, D.E. Wahl and P.. Eichel, Spotlight-Mode Synthetic Aperture Radar: A Processing Approach. Boston, MA: Kluwer Academic Publishers, SAR performance CNR (db) Number of Targets Fig. 8. Predicted CNR performance of a SAR with 45 M bandwidth and a 6 second coherent integration time, using cross-correlation and the LMS algorithm.