Optimum Transmit Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise

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1 Copyright 2003 IEEE. Reprinted from IEEE TRANSACTIONS ON INFORMATION THEORY,VOL. 46, NO. 2, MARCH 2000 Optimum Transmit Receiver Design in the Presence of Signal-Dependent Interference and Channel Noise S. U. Pillai, Senior Member, IEEE, H. S. Oh, Student Member, IEEE, D. C. Youla, Fellow, IEEE, and J. R. Guerci, Senior Member, IEEE Abstract Optimal detection of a target return contaminated by signal-dependent interference, as well as additive channel noise, requires the design of a transmit pulse ( ) and a receiver impulse response ( ) jointly maximizing the output signal to interference plus noise ratio (SINR). Despite the highly nonlinear nature of this problem, it has been possible to show that ( ) may always be chosen minimum-phase. A full analysis concludes with the construction of an effective numerical procedure for the determination of optimal pairs ( ) that appears to converge satisfactorily for most values of input SINR. Extensive simulation reveals that the shape of ( ) can be a critical factor. In particular, the performance of a chirp-like pulse is often unacceptable, especially when the clutter and channel noise are low-pass dominant and comparable. Index Terms Clutter, optimum minimum phase pulse, radar, signal-dependent interference. I. INTRODUCTION Afundamental and ubiquitous design problem encountered in radar, sonar, and communication systems in general, is that of jointly optimizing the transmitter and receiver, given, perhaps, some knowledge of the channel. In the active sensor detection problem (e.g., radar, sonar, lidar), which is the primary focus of this paper, one is concerned with judiciously selecting the operating band, transmit waveform modulations, and receiver processing strategy, in order to maximize the probability of detecting the presence of a target, while maintaining a prescribed rate of false alarm [1]. In real-world applications, issues of cost, complexity, and reliability can often weigh heavily on the design process. Nevertheless, they are not static constraints. As technology progresses, new design-space domains can be explored. For example, recent improvements in low-noise linear amplifiers and high-speed digitally programmable arbitrary waveform generators, have allowed designers to consider the use of sophisticated pulse-shaping techniques heretofore considered impractical [2]. In the case of radar, a first generation of so-called matched illumination reception radars have been designed based on relaxing the Manuscript received December 20, 1997; revised September 28, This work was supported in part by the Office of Naval Research under Contract N J-1512P-5. S. U. Pillai and H. S. Oh are with the Department of Electrical Engineering, Polytechnic University, Brooklyn, N Y USA ( Pillai@fire.poly.edu).. D. C. Youla is with the Department of Electrical Engineering, Polytechnic University, Farmingdale, NY USA. He is now with DARPA, Arlington, VA J. R. Guerci was with SAIC, Arlington, VA 2220 USA. Communicated by E. Soljanin, Associate Editor for Coding Techniques. Publisher Item Identifier S (00) point target assumption common to all current surveillance radars [3] [5]. A point target, by definition, has a flat response (and linear phase) across the instantaneous operating band of the radar. Thus under this assumption, no attempt is made to shape (pre-emphasize) the transmit pulse. However, given a priori knowledge (deterministic or statistical) concerning the range extended (nonpoint) target scattering characteristics, an optimal pulse shape can be found which maximizes the energy reflected from the target. Specifically, it is obtained as an eigenfunction of a Fredholm integral operator of the first kind whose kernel is formed from the impulse response of the target [3], [4]. In the case of a channel corrupted solely by additive noise, it is well known that this solution maximizes the output signal to interference plus noise ratio (SINR) of an appropriately matched receiver [4], [5]. Unfortunately, in radar and many other active sensor scenarios, the above model is too simplistic. It should, for example, be expanded to include signal-dependent interference such as unwanted ground returns and environmental clutter. In this paper all these extra spurious components are labeled clutter and included as the output of a single filter with stochastic impulse response driven at the input by the chosen transmit signal Moreover, since clutter is most difficult to eliminate without the aid of Doppler, we only consider nonmoving targets. Our goal is to determine both and its companion receiver impulse response, so that the output SINR is maximized. Explicit analytical solutions are only available when additive channel noise effectively dominates interference or vice versa. However, owing to the nonlinearity of the equations, the case in which the two are comparable must be handled numerically. II. PROBLEM FORMULATION Fig. 1 displays all the essential features of our simplified model. A real finite-duration transmit pulse simultaneously illuminates a stationary target and surrounding clutter. The impulse response of the former, which we shall suppose known and both integrable and square-integrable, is real causal and deterministic, while that of the latter,, although real, is stochastic of known spectral density The target output and clutter interference combine additively with wide-sense stationary channel noise of known spectral density to provide the total input (1) /00$ IEEE

2 578 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 Fig. 1. Simplified model of the essential features. to a receiver with real causal impulse response and output Let,, and Then if represents the signal component of the output and the component contributed by interference and noise, it follows readily that (2) Accordingly (10) a form which suggests an obvious use of Schwarz s inequality. All the same, the application of this inequality must be postponed because the inverse Fourier transform of 3 and 1 (3) is not necessarily causal. To overcome this difficulty, let where Our objective is to determine an admissible transmit receive pair which maximizes the output SINR (4) (11) Clearly, since and are analytic in and bounded in the closure and are causal and convolution gives SINR (5) at some prescribed detection instant, subject to an energy constraint on More precisely, we seek to achieve the iterated maximum and the first step is to maximize over with held fast. This is accomplished straightforwardly by working with the frequency-domain expression Owing to the relatively flat and wideband nature of,no harm is done to the physics of the problem by assuming and to be positive for 2 In addition, if the Paley-Wiener constraint [5] is imposed, there exists a function, real for real, unique up to sign and analytic together with its inverse in, such that 1 The processes w (t) and n(t) are assumed to be independent and h; i denotes ensemble average. (6) (7) (8) (9) But the Heaviside unit-step. Consequently, if Parseval s theorem applied to (12) yields Hence (Schwarz) with equality iff (12) (13) (14) (15) (16) (17) (18) 2 Certainly true when n(t) is modeled as white noise with constant spectral density G (!)= > 0 and G (!) is bounded. 3 a is the complex conjugate of a:

3 PILLAI et al.: OPTIMUM TRANSMIT RECEIVER DESIGN IN THE PRESENCE OF SIGNAL-DEPENDENT INTERFERENCE AND CHANNEL NOISE 579 any real nonzero normalization constant. The impulse response of this matched filter with transfer function given in (18) is real and causal. In addition, is a key result which allows us to employ the formula (19) (20) to optimize over Let Then and (21) Naturally, in view of (20), no generality is lost by limiting the causal transmit pulse to a duration of seconds. Maximization over is achieved by choosing an in (21) whose associated possesses the largest possible energy over the interval Unfortunately, when clutter is significant, the Wiener Hopf equation (22) imposes a nonlinear dependence of on that severely complicates the problem. It is, therefore, perhaps surprising to discover that in this setting nothing is sacrificed if an optimal is assumed to be minimum-phase! Proof: By definition, if where is optimal but nonminimum-phase, the Laplace transform (23) either possesses a real zero or a pair of complex-conjugate zeros and in To be definite, suppose is complex, introduce the regular all-pass We conclude that qualifies as an admissible transmit pulse. In fact, it is also optimal. For as is immediately seen from (22), and lead to the same canonical factor Thus and it suffices to establish the inequality (29) (30) Since, one may interpret as the output of a causal (energy-preserving) filter with transfer function and input [5]. Due to the causal nature of the filter, its response to the truncated input must agree with over the interval In particular (Parseval) (31) and (30) follows. This process of stripping away zeros in may be repeated until the resulting transmit pulse Fourier transform is minimum-phase. Q.E.D. III. THE OPTIMUM TRANSMIT PULSE A) In the absence of clutter, and (4) yields where is the known Wiener Hopf factor of From (21) (32) Let (24) (33) let as a zero of, and observe that the multiplicity of and rewrite (32) as (34) (25) Now introduce the kernel has been reduced by one. Furthermore,,, and (35) (26) and the associated linear operator defined by the mapping where 4 (36) (27) Then (easy details omitted) 5 Apparently, for implies while for implies is computed as the inner-product of and Since (37) (28) 4 Recall that 0u(0t)e $ 1 Re s > 0 j! 0 s and then invert (25) by convolution to obtain (27) and (26). 5 For any two functions (t) and (t) square-integrable over 0 t t (; ) = (t) (t) dt: Also, kk =(; ) :

4 580 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 is symmetric, completely continuous, and nonnegative-definite. As such [6], the integral equation (38) admits a denumerable set of eigenfunction solutions of, orthonormal 6 and kernel mean-square complete over The eigenvalues are positive and when arranged as a monotonically nonincreasing sequence (39) have zero as sole possible limit point. In view of (37) (40) Concomitantly, for specified pulse energy (41) is attained by choosing (42) The receiver transfer function matched to this optimum transmit signal is given by in which (43) (44) establish the correct asymptotic performance limits in the presence of overwhelming signal-dependent clutter. C) In the most general practical situation, neither channel noise nor clutter is negligible and the equation (50) for the Wiener Hopf factor permits no simplification. Nonetheless, we have been able to construct a successful iterative procedure for the determination of an optimal pair that extends the integral equation approach developed in Part A and seems to converge robustly over a wide range of input SINR. Given the target impulse response, or its transform, and the spectral densities and either analytically or in numerical form, choose and the energy of the desired optimum pulse and proceed to Step 1). 1) For start with any real causal function of duration and energy 2) Let and find the Wiener Hopf solution 7 of the equation by using footnote 7 as a guide. 3) Let and compute (51) (52) and (45) Equations (41) (45) are exact for zero clutter and include the results obtained in [3] and [4] as a special case. B) At the other extreme, is negligible compared to and (22) reduces to But may be assumed minimum-phase, whence (46) where is the Wiener Hopf factor of In particular, (21) now makes it clear that to the same order of approximation (47) In other words, when clutter is truly predominant, nonlinear thresholding occurs, so that as determined by (47), is independent of (48) And as expected (49) varies inversely with the choice of Although the above conclusions are in part qualitative due to our (physically unjustified) total disregard of channel noise in arriving at (46), they do 6 ( ; )= : (53) 4) Find the largest eigenvalue of the integral equation (54) and the corresponding normalized eigenfunction Compute the coefficient 5) Define the error at stage by 8 and invoke the update rule 6) Let (55) (56) (57) 7 Write L (j!) = jl (j!)je and observe that (!) is the Hilbert transform of ln jl (j!)j = 1 ln [G (!) +G (!)jf (!)j ] 2 etc. 8 All square roots are positive.

5 PILLAI et al.: OPTIMUM TRANSMIT RECEIVER DESIGN IN THE PRESENCE OF SIGNAL-DEPENDENT INTERFERENCE AND CHANNEL NOISE 581 Fig. 2. The optimum transmit waveform and its companion receiver impulse response for a short rectangular target impulse response in clutter and noise. and go back to Step 2) with replaced by, and repeat until is acceptably small. Then (58) 1) Explanation: At stage, the iterate is an admissible transmit pulse of energy which produces the output signal to interference plus noise ratio The optimum receiver impulse response accordingly, and is computed Evidently (60) (59) with equality iff (61) where as.

6 582 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 Fig. 3. The optimum transmit waveform and its companion receiver impulse response for a given target impulse response, clutter, and noise spectra. Consequently, the square-root of (62) is a meaningful measure of the error at stage and plays a central role in the recursion adopted in (57). 9 Note that the fixed energy requirement is maintained by the specific choice of denominator. 9 Predicated on the assumption that convergence to the optimum f (t) will be realized by forcing the difference E 0 (f ;T f ) to go to zero. A fully rigorous proof is still lacking. IV. NUMERICAL RESULTS With clutter absent and channel noise flat, and Thus and it follows from (11) that (63) Of course, for an ideal stationary point target,, a real nonzero constant,,, and (64) where is the energy of the transmit pulse In this limiting case, the shape of is theoretically irrelevant and may be

7 PILLAI et al.: OPTIMUM TRANSMIT RECEIVER DESIGN IN THE PRESENCE OF SIGNAL-DEPENDENT INTERFERENCE AND CHANNEL NOISE 583 Fig. 4. The optimum transmit waveforms for various values of energy. chosen chirp-like to combine the need for increased range and enhanced resolution made possible by the compressive properties of the matched receiver impulse response (65) On the other hand, when signal-dependent clutter is present and comparable to channel noise, extensive simulation confirms that the chirp is almost invariably suboptimal and its use often entails a drastic reduction in output SINR. To understand the data presented in Figs. 2 and 3 it is only necessary to recognize that the output SINR (66) computed at stage by the algorithm described in Part C of Section III is portrayed in the form depicted in Figs. 2 (f) and 3 (f). In Fig. 2, the target is selected to have a short rectangular impulse response and the spectral densities and are both low-pass dominant (as is common in many radar scenarios).

8 584 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 The heavy and superimposed dotted curves in Figs. 2 (e), (f) and 3 (e), (f) are obtained, respectively, by using the chirp in (c) and the equi-energy signal in (d) as initialization for the algorithm. The chirp achieves a largest possible output SINR of 3 db, the lowest point on the solid curve in (f). Observe that this is 14.6 db below the 11.6-dB SINR attainable with the optimum transmit pulse shown in (g)! Even more dramatic is the decibel gain over the conventional matched filter for chirp. Indeed, if, substitution of into (7) yields SINR (67) which computes to db and represents a jarring db deterioration in performance below the optimum. Furthermore, the output SINR produced by the arbitrarily chosen equi-energy initial waveform in (d) already exceeds that of the chirp by 6.57 db. Nevertheless, as seen from (g) and (h), the algorithm converges to the same optimal pair irrespective of starting point. According to Fig. 3, a change in the shapes of,, and does not qualitatively alter any of the main conclusions reached in our examination of Fig. 2. In Fig. 4(c) (g), is computed as a function of transmit pulse energy and respectively. The corresponding maximum output SINR s are found as the ordinates of the solid curve in Fig. 4 (h). At, the latter becomes asymptotic to the theoretical upper limit of 21.3 db calculated with the help of (47) and (48). However, the lower dashed curve, which is a plot of the largest possible output SINR achievable with the scaled chirp, appears to reach a point slightly below this saturation value at This considerably slower rise as a function of is undoubtedly owed to the shape of and the nonminimum-phase character of 10 V. CONCLUSIONS Unlike the classical radar case, the choice of transmit pulse shape can be critically important for the detection of extended targets in the presence of additive channel noise and signal-dependent clutter. It appears from our analysis and numerical experience that an optimal transmit receive pair which realizes (68) exists and is uniquely determined up to a real constant nonzero multiple of by a specification of target impulse response, nontrivial spectral densities,, and transmit signal energy In this regard, the chirp, which has been designed for other purposes, can perform very poorly, especially when clutter and noise are comparable. The issue is now open for further study and the algorithm described in Section III of this paper should prove to be a valuable research tool. REFERENCES [1] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, [2] Di Franco and Rubin, Radar Detection. Norwood, MA: Artech House, [3] J. H. H. Chalk, The optimum pulse shape for pulse communication, in Proc. Inst. Elec. Eng. London, vol. 87, 1950, pp [4] J. R. Guerci and P. Grieve, Optimum matched illumination-reception radars, U.S. Patent , June 1992, and U.S. Patent , Dec [5] A. Papoulis, Signal Analysis. New York: McGraw-Hill, [6] F. G. Tricomi, Integral Equations. New York: Interscience, We have verified numerically that ln jf (!)j and (!) are not Hilbert transforms.