Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 Artile Loally Optimal Radar Waveform Design for Deteting Doubly Spread Targets in Colored Noise Zhenghan Zhu 1, *, Steven Kay 1 and R. S. Raghavan 1 Department of Eletrial,Computer and Biomedial Engineering, University of Rhode Island, Kingston, RI, 881 USA; {zzhu,kay}@ele.uri.edu Air Fore Researh Laboratory, Wright-Patterson Air Fore Base, Dayton, OH 45433 USA; ramahandran.raghavan@us.af.mil * Correspondene: zzhu@ele.uri.edu; Tel.: +1-41-874-56 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 Abstrat: Radar transmit signal design is a ritial fator for the radar performane. In this paper, we investigate the problem of radar signal waveform design under the small signal power onditions for deteting a doubly spread target, whose impulse response an be modeled as a random proess, in a olored noise environment. The doubly spread target spans multiple range bins (range-spread) and its impulse response is time-varying due to flutuation (hene also Doppler-spread), suh that the target impulse response is both time-seletive and frequeny-seletive. Instead of adopting the onventional assumption that the target is wide-sense stationary unorrelated sattering, we assume that the target impulse response is both wide-sense stationary in range and in time to aount for the possible orrelation between the impulse responses orresponding to lose range intervals. The loally most powerful detetor, whih is asymptotially optimal for small signal ases, is then derived for deteting suh targets. The signal waveform is optimized to maximizing the detetion performane of the detetor or equivalently maximizing the Kullbak-Leibler divergene. Numerial simulations validate the effetiveness of the proposed waveform design for the small signal power onditions and performane of optimum waveform design are shown in omparison to the frequeny modulated waveform. Keywords: radar; transmit signal waveform design; doubly spread; extended target; flutuation; Kullbak-Leibler divergene; loally most powerful detetor; olored noise 18 19 1 3 4 5 6 7 8 9 3 31 3 33 34 35 1. Introdution Radar transmit signal waveform design is an important problem and ative researh area as the transmit signal ritially affets a radar system s performane [1-18]. It is also ategorized as a type of waveform diversity problem [9]. Many methods have been proposed for radar waveform optimization suh as maximizing mutual information (MI), minimizing mean square error, relative entropy, and maximizing output-signal-to-noise ratio (SNR) []. For example, Bell advaned the waveform design by proposing maximizing MI between the target ensemble and reeived data [13]. Yang et al. applied the minimum mean square error metri and the MI metri to multiple-input multiple-output (MIMO) target reognition and lassifiation in [14]. Romero et al. studied optimizing SNR and MI for deteting targets of different types in [15]. Tang et al. studied using KLD and MI for MIMO radar waveform design in [17]. Aubry et al. developed knowledge-aided transmit signal in signal-dependent lutter [16]. Demaio et al. onsidered designing waveform under similarity onstraint to ahieving good ambiguity properties in [18]. Among the existing literature, the targets are typially assumed as a point target or an extended target. In this paper, we study a more ompliated ase when the extended target is flutuating, also alled doubly spread target [3]. The doubly spread target an be moving or stati. We onsider designing optimal radar waveform for deteting suh targets in olored noise. Limited work has been devoted to this type of target detetion waveform design. There are several areas that this type of 17 by the author(s). Distributed under a Creative Commons CC BY liense.
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 of 15 36 37 38 39 4 41 4 43 44 45 46 47 48 49 5 51 5 53 54 55 56 57 58 59 6 61 6 63 64 65 66 67 68 target is enountered in suh as when the details of the target are of interest, e.g., mapping radar, and when the target is rotating [3]. In [], both the target and reverberation are modeled as doubly spread and expression of the signal-to-interferene ratio is derived. It is worth pointing out that the mathematial model of a doubly spread target is similar to that of a doubly dispersive ommuniations hannel [3][4] given the similarities between radar and ommuniations. A doubly spread target/hannel an be understood as a linear and time-varying (LTV) system in that the refleted signal is a superposition of all refleted signal from different ranges and the target response at eah range hanges versus time. The time-varying harateristis of a radar target may be aused by its flutuation [3]. The return from eah range is assumed as a sample funtion of a stationary zero- mean omplex Gaussian random proess [3]. On the other hand, the returns from different intervals have been often assumed to be statistially independent [3]. Together, the target/hannel is assumed to be of wide-sense stationary unorrelated sattering (WSSUS) [3][4], whih was introdued by Bello [5] and has been widely used ever sine. The WSSUS assumption greatly simplifies the statistial haraterization of LTV ommuniations hannel and radar targets. However, the unorrelated assumption of returns from different intervals may be invalid in pratie beause target omponents that are lose to eah other often result from the same physial satterer and will hene be orrelated [4]. In addition, filters, antennas, and windowing operations at the transmit and/or reeive side ause some extra time and frequeny dispersion that results in orrelations of the spreading funtion [4]. Therefore, the target response is assumed WSS in both time diretion and range in this paper and the transmit signal is designed aording to the power spetral density of the target. It is worth strenghtening that the results of this paper may not apply to the ase when the target impulse response is modeled as a deterministi funtion instead of a random proess. The paper is organized as follows. In Setion, the refleted signal from a moving doubly spread target is derived for a pulsed transmit signal. Setion 3 derives the power spetral density of the reeived data. And the detetion performane is posed in the frequeny domain. The loally most powerful detetor and optimal waveform solution are derived in Setion 4. To evaluate the effetiveness of the derived waveform, several numerial simulations are given in Setion 5. Lastly, Setion 6 draws the onlusions.. Modeling of the Pulsed Transmit Signal and Reeived Data Throughout the paper, the transmit signal is denoted as s(t); the refleted signal orresponding to s(t) is denoted as r(t); additive noise is denoted as w(t); θ denotes propagation attenuation; and the reeived data is denoted as x(t). The detetion problem an be written as a hypothesis testing problem as H : x(t) = w(t); target absent H 1 : x(t) = θr(t) + w(t); target present where the refleted signal is r(t) = s(t τ)h(t, τ)dτ; and h(t, τ) is the target impulse response (TIR), whih desribes the target s response as a funtion of time τ due to an impulse at time t τ, and τ is the single-trip delay assoiated with the propagation of the transmit signal to a target (also temporal range). Speifially, we onsider to design a pulsed transmit signal instead of a ontinuous-waveform signal sine the goal is to deteting a moving target [9]. The pulsed waveform transmit signal s(t) at the baseband is of the form: s(t) = a k p(t kt r ) (1) k=
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 3 of 15 Figure 1. An illustration of the pulses and orresponding refleted signal where T r is the pulse repetition interval (PRI) and p(t), the omplex signature pulse of the waveform with a duration T p, a k s are omplex-valued oeffiients, and K is the number of pulses in a oherent integration interval. Let p k (t) = a k p(t kt r ) () then transmit signal an be writen as s(t) = p k (t) (3) k= The baksattered signal orresponding to p k (t) is denoted as r k (t), and therefore r(t) an be written as r(t) = r k (t). k= 69 7 71 7 73 74 75 76 77 78 79 8 81 8 83 84 85 86 87 An illustration of the first two pulses p (t), p 1 (t) and a general p k (t) and their orresponding refleted signal r k (t) is given in Figure 1. The relationship between the refleted signal r(t) and the transmit signal s(t) are based on the following set of assumptions. A1: The pulse duration T p is far smaller than PRI T r ; that is, T p T r. The extended target moves in a linear diretion and as suh the temporal length T g of the target is a onstant over the oherent integration interval. The time duration of responses from all ranges to a transmitted pulse is smaller than the PRI, so that the returns from adjaent pulses do not overlap; that is,t p + T g T r. A: The phase of the refleted signal for eah transmitted pulse is assumed to be onstant and hanges on a pulse-to-pulse basis due to target movement. This is the stop-and-go approximation [3]. The initial phase of r k (t) whih is the signal return for the k th pulse is πkf d T r, where F d is the Doppler frequeny. A range dependent omponent of the phase is added to eah pulse return as shown next. A3: The flutuation auses the statistial harateristi of TIR h(t, τ) hanges from pulse to pulse, so when the k th pulse illuminates the target, equivalently kt r t (k + 1)T r, h(t, τ) = h(t k, τ) = h k (τ) where t k = kt r + R with R denoting the distane of the nearest point of the target and the radar, denoting the speed of light. So h k (τ) represents TIR when the k th pulse illuminating the target at time kt r + R. To aount for Doppler effet, the TIR h(t k, τ) an be expressed as h k (τ) = e jπf dt k h LP (t k, τ), so that h LP (t k, τ) is low pass in t k. With the above assumptions, it is shown in Appendix A that the refelted signal r k (t) for k =, 1,..., K 1 is Tg r k (t) = a k e jπf dt k p(t kt r τ)h LP (t k, τ)dτ (4)
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 4 of 15 when kt r + R t kt r + R + T g + T p and zero otherwise. Furthermore, we sample at t = i t, where t is determined by the bandwidth of eah transmitted pulse p(t). Let the disrete time representation of p(t) be denoted by p[m] and eah pulse of duration T p is represented by N p samples. Similarly, eah PRI has N r samples. The extended target of length T g is sampled into N g samples. The lowpass TIR h LP (t k, τ) is represented as h LP [k, l] in disrete time for k =, 1,, K 1 and l =, 1,, N g 1. The refleted signal for a given pulse has a length N = N p + N g 1. The reeived signal for the k th pulse (See Appendix A for derivation) r k [n] = a k e jπ f dk N g 1 p[n l]h LP [k, l] (5) l= 88 89 9 91 with f d = F d T r. Equivalently, we an write the refleted data as a matrix R with its (k, n) element being r[k, n] = r k [n]. In literature, k =, 1,, K 1 is also alled slow time index and n =, 1,, N 1 is alled fast time index. Similarly, the reeived data an be denoted as x[k, n] and the additive noise w[k, n]. Then we an write the detetion problem in a disrete time form as follows: H : x[k, n] = w[k, n] H 1 : x[k, n] = θr[k, n] + w[k, n] (6) 9 93 94 95 96 97 98 3. Detetion Problem Formulation We fous on designing the signature pulse p[m] by letting a = a 1 = = a = 1. Then r[k, n] redues to N g 1 r[k, n] = e jπ f dk p[n l]h LP [k, l]. l= Different from the WSSUS assumption, we assume that the disrete-time lowpass TIR h LP [k, l] is a -D wide sense stationary (WSS) random proess both in slow time and fast time and has a -D autoorrelation matrix R hh [ k, l] with a orresponding -D power spetral density (PSD) denoted as P h (η, φ). Let the autoorrelation matrix of r[k, n] denoted as R rr [k, n, k, n], and it is proved in Appendix B that when a single signal pulse duration is far shorter than the extended target length (N p N g ), we have R rr [k, n, k, n] = e jπ f d k N p 1 N p 1 m= m = p[m]p [m ]R hh [ k, n (m m )] (7) 99 1 11 1 As shown, the autoorrelation R rr [k, n, k, n] only depends upon k and n and therefore r[k, n] is also a -D WSS proess both in slow time k and fast time n. Intuitively speaking, if a -D WSS proess is filtered by a 1-D (one of the two dimensions) linear time invariant filter, the output is still a -D WSS proess. The autoorrelation an hene be simply denoted as R rr [ k, n] instead. Furthermore, the -D PSD P r (η, φ) of the refleted signal r[k, n] is shown in Appendix B to be P r (η, φ) = P h (η f d, φ) S(φ) (8) 13 14 15 16 17 where S(φ) is the transmit signal energy spetral density (ESD) of a single pulse p[m]. It says that along the slow time k diretion, equivalently η diretion in frequeny domain, the refleted signal PSD is a shift of target PSD by Doppler and along the fast time n diretion, equivalently φ diretion in frequeny domain, the refleted signal PSD is a simple multipliation of the transmit signal ESD and target PSD in that diretion.
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 5 of 15 18 19 11 The assumption that the additive noise for eah refleted pulse are unorrelated implies that w[k 1, n] is unorrelated from w[k, m] if k 1 = k and that w[k, n] has a PSD along the fast time diretion, we an pose the detetion problem in frequeny domain as: H : P x (η, φ) = for all η (9) H 1 : P x (η, φ) = θp r (η, φ) + = θp h (η f d, φ) S(φ) + (1) 111 11 113 114 115 116 117 118 119 1 11 1 13 where P x (η, φ) denotes the -D PSD of the reeived data. Note that P h (η, φ) and are assumed known and θ > is unknown. And in this paper we assume that f d is also unknown. The waveform design problem is to design the pulse ESD S(φ). 4. The Optimal Waveform Solution To design the waveform, we begin by deriving the Loally most powerful detetor, whih is an asymptotially optimal detetion for small signal ases [19]. Assume that the observed data X is of size K N. The asymptoti expression for the Log-likelihood funtion of hypothesis H or H 1 is given by the following with the appropriate expression for P x (η, φ) substituted from equations (9) or (1) respetively [1]. ln p(x) = KN ln π KN [ ln P x (η, φ) + I ] x(η, φ) (11) P x (η, φ) where I x (η, φ) is the -D Periodogram whih is the squared value of the -D Disrete Fourier Transform of X for frequeny (η, φ) and when devided by KN, it an be viewed as the estimate of the reeived data s -D PSD. Then with (1), we have under H 1 ln p(x; θ) θ = KN [ Ph (η f d, φ) S(φ) P x (η, φ) and the Fisher information matrix of θ an be found as [19] I x(η, φ)p h (η f d, φ) S(φ) ] Px (1) (η, φ) ( Ph (η f d, φ) S(φ) ) (13) I(θ) = KN P x (η, φ) 14 The Loally Most Powerful (LMP) test statisti is [19] T LMP = = = ln p(x;θ) θ I(θ) θ= [ P h (η f d,φ) S(φ) KN P w I x(η,φ)p h (η f d,φ) S(φ) (φ) Pw(φ) KN ( P h (η f d,φ) S(φ) ] ) (14) KN Ph (η f d,φ) S(φ) I x (η,φ) ( ) (15) Ph (η f d,φ) S(φ)
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 6 of 15 15 16 17 As shown, (15) represents the LMP test statisti at a given Doppler shift and to implement the LMP detetor, the information of Doppler f d is needed. To maximize the detetion performane with respet to the transmitted signal, we need to only maximize the defletion oeffiient, whih is derived [19] as d LMP = θ I(θ ) θ = = KNθ ( Ph (η f d, φ) S(φ) ), (16) 13 18 where θ is the true value of θ under H, whih is zero. Note that the defletion oeffiient does not 19 depend on Doppler f d. In [], it has been shown that maximizing the Kullbak-Libeler divergene 13 (KLD) between the probability of density funtion of reeived data when target present and that of 131 target absent is the orret metri to use for deteting random targets. A omparison of equation (16) and equation (A9) in Appendix C shows that the KLD D(p 1 p ) 1 d LMP for doubly spread targets. Note that different from the waveform design for range-spread target [], for the doubly spread target, 133 134 135 136 137 we first need to integrate the target PSD (squared) along the slow time diretion (representing the flutuations) to produe a single value for a ertain φ l, whih produes k= P h (η k f d, φ l ). The waveform design problem is to maximize the KLD with the energy onstraint N 1 l= S(φ l) = E, whih an be expressed as follows. max S(φl ) N 1 l= k= P h (η k f d,φ l ) Pw(φ l ) S(φ l ) 4 s.t. N 1 l= S(φ l) = E (17) 138 139 14 141 14 143 144 145 The objetive funtion is a onvex funtion on a onvex set. The optimal solution is to put all energy into the frequeny bin φ l whih makes the term, denoted as (φ l ) = k= P h (η [ ] k f d,φ l ) = Pw(φ Ph (η k,φ l ) l ) k= P w (φ l ) the maximum among all l s. Note that (φ l ) does not depend on f d and hene the Doppler does not affet the optimal design solution. The reason is that Doppler auses the target PSD to be shifted along the η diretion (representing flutuation), whih is the diretion we integrate the target PSD over. In a speial ase when the target PSD is separable; that is the -D target PSD is separable in slow time (representing flutuation harateristis) and in fast time (target impulse response at ertain slow time) suh that P h (η, φ) = P h1 (η)p h (φ). Then, we have k= P h (η k, φ l ) P w(φ l ) = k= P h 1 (η k )P h (φ l ) P w(φ l ) = P h (φ l ) Pw(φ l ) k= P h 1 (η k ) (18) and the optimal solution is to put all energy into the frequeny bin φ l where P h(φ l ) 146 is the maximum among all l s, whih is the same result for nonfluutation ase (singly-spread) in P w (φ []. l ) 147 148 149 15 151 15 153 154 155 156 5. Simulations In this setion, we set up several numerial simulations to evaluate the performane of the proposed waveform and ompare it with the linear modulated frequeny (LFM) waveform, whih is widely used in pratie due to its easiness in implementation. The detetor employed is the derived LMP detetor for both waveforms. The -D TIR is a 3 3 two dimensional random proess whih means that there are K = 3 pulses sent and the extended target has a length N g = 3. While the transmit signal signature pulse p[m] has a length N p = 8. Then N = N g + N p 1 = 39. The rest of the simulation setup details an be found in Appendix D. In the first simulation, we onsider a ase where the key term that deides the waveform design, (φ l ) = k= P h (η k,φ l ) Pw(φ l ), is suh as shown in the
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 7 of 15 ( ).4. The term ( ) -.5 -.4 -.3 -. -.1.1..3.4.5 S( ) 4 LMP-based waveform ESD -.5 -.4 -.3 -. -.1.1..3.4.5 S( ).4. LFM waveform ESD -.5 -.4 -.3 -. -.1.1..3.4.5 Figure. Simulation 1 setup and the waveform designs 157 158 159 16 161 16 163 164 165 166 167 168 169 17 171 17 173 174 175 176 177 178 179 18 181 18 183 184 top subfigure of Figure. And the signal energy is θe = 4.16. The LMP-based waveform is given in the middle subfigure of Figure and the LFM waveform is given in the bottom subfigure of Figure. The detetion performanes of the two waveforms, represented in the reeiver operating harateristis (ROC), are given in Figure 3. It shows that the LMP-based waveform substantially outperforms the LFM waveform. In simulation, we onsider an extreme ase when the term (φ l ) is flat as shown in the top subfigure of Figure 4. Reall that the LMP-based waveform puts all the signal energy into the frequeny bin where (φ l ) is the maximum. For the LMP-based waveform it is equivalent to put the signal energy into any frequeny bin sine (φ l ) is the same value for all frequeny bin φ l. For illustration, the frequeny bin φ l =.365 is hosen. The LMP-based waveform is shown in the middle subfigure of Figure 4. The LFM waveform (shown in the bottom subfigure of Figure 4) and the signal energy θe = 4.16 are kept the same as the previous simulations. Figure 5 shows the detetion performane omparison between the two waveforms. The LMP-based waveform still outperforms the LFM waveform in this ase; although the differene between the two waveforms performane is smaller ompared to that of Simulation 1. 6. Conlusions In this paper, we onsidered the optimal radar waveform design for deteting a moving doubly spread target, both range-spread and Doppler-spread (due to flutuating), in a olored noise environment for the small signal power ondition. The impulse response of the target is assumed to be a two-dimensional (slow time and fast time) wide-sense stationary random proess. The optimal waveform is derived by maximizing the defletion oeffiient of the loally most powerful detetor or equivalently maximizing the Kullbak-Leibler divergene. The optimal signal waveform is shown to be putting all the signal energy in the frequeny bin where the ratio of the summed squared target power along slow time diretion over the squared noise power is maximum. Its performane is ompared to the onventional LFM waveform. The performanes of both waveforms depend on the target PSD and noise PSD. Numerial simulations show that the LMP-based waveform generally outperform the LFM waveform in terms of detetion performane. When the target PSD, relative to noise PSD, is highly seletive in frequeny, the LMP-based waveform generally an ahieve a substantial performane
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 8 of 15 1 ROC.9.8.7.6 P D.5.4.3 LMP-based waveform LFM waveform..1..4.6.8 1 P FA Figure 3. The performanes of the two waveforms in Simulation 1 4 1-3 ( ) The term ( ) -.5 -.4 -.3 -. -.1.1..3.4.5 4 S( ) LMP-based waveform ESD -.5 -.4 -.3 -. -.1.1..3.4.5.4 S( ). LFM waveform ESD -.5 -.4 -.3 -. -.1.1..3.4.5 Figure 4. Simulation setup and the waveform design
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 9 of 15 1 ROC.9.8.7 P D.6.5 LMP-based waveform LFM waveform.4.3..1.1..3.4.5.6.7.8.9 1 P FA Figure 5. The performanes of the two waveforms in simulation 185 186 187 188 189 improvement. However, these results may apply to the detetion problem only and no aount has been taken on other important onsiderations for a single radar suh as range resolution. Aknowledgments: This work was supported by the Sensors Diretorate of the Air Fore Researh Laboratory (AFRL/RYMD) under ontrat FA865-1-D-1376 4 to Defense Engineering Corporation. Appendix A Derivation of refleted signal r(t) for pulsed transmit signal In this setion, we derive the refleted signal r(t). First, we have r k (t) = p k (t + kt r + R τ)h k (τ)dτ (A1) Shifting r k (t) by kt r + R suh as r k (t + kt r + R ) = p k (t τ)h k (τ)dτ (A) where R denotes the distane of the nearest point of the target and the radar, and is the speed of light and h k (τ) is the target impulse response (TIR) for the k th pulse illuminating the target at time kt r + R, We denote the illuminating time point t k = kt r + R for k =, 1,, K 1. Then, we have h k (τ) = h(kt r + R, τ) = h(t k, τ) (A3) 19 191 19 where the funtion h(t k, τ) represents the target impulse response at time t k and temporal range τ. This is the assumption A3 in Setion II. Also the TIR length when the target is illuminated by the k th pulse is assumed the same for all k s and is denoted as T g (assumption A1 in Setion II). Then we have r k (t + kt r + R ) = Tg p k (t τ)h(t k, τ)dτ (A4) where t T g + T p
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 1 of 15 If we let then we have r k (t ) = Tg t = t + kt r + R p k (t kt r R τ)h(t k, τ)dτ when kt r + R t kt r + R + T g + T p and r k (t ) = otherwise. Furthermore, we have r(t) = Tg a k p(t kt r R τ)h(t k= k, τ)dτ (A5) (A6) (A7) 193 where the range bin R is assumed fixed for all K pulses during oherent proessing interval (CPI). To aount for Doppler (assumption A3 in Setion II) let h(t k, τ) = e jπf dt k h LP (t k, τ) (A8) so that h LP (t, τ) is lowpass in t. Then we have r(t) = Tg a k e jπf dt k p(t kt r R τ)h LP (t k= k, τ)dτ (A9) if we referene time to the beginning of return signal at t = R r(t) = we have { k= a ke jπf Tg dt k p(t kt r τ)h LP (t k, τ)dτ, t (K 1)T r + T g + T p, otherwise 194 195 Next we onsider the problem in disrete time by sampling at a time resolution t, whih is determined by the bandwidth of p(t). r[n] = r(n t) = = Tg a k e jπf dt k p(n t kt r τ)h LP (t k, τ)dτ k= a k e jπf d(kt r + R ) k= Tg p(n t kt r τ)h LP (t k, τ)dτ But the phase fator e jπf R d an be ombined with h LP, that is we an let h LP (t k, τ) = e jπf R d h LP (t k, τ) (A1) Also we have assumed that h LP (t, τ) is WSS in τ, so we an omit phase term that ontains R in h LP (t k, τ) to yield Tg r[n] = a k e jπf dkt r p(n t kt r τ)h LP (t k, τ)dτ k= Assume that a pulse interval T r = N r t and τ = l t for l N g 1, where T g = N g t we have r[n] k= f d {}}{ N g 1 a k e jπk F d T r p(n t kt r l t)h LP (kt r, l t) t l= Now let p(n t kt r l t) = p[n l kn r ] (A11)
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 11 of 15 and h LP [k, l] = h LP (kt r, l t) t (A1) r[n] = a k e jπ f dk k= N g 1 p[n l kn r ]h LP [k, l] l= for n =, 1,, (K 1)N r + N p + N g 1. If we let n = n kn r ; that is, we referene the sequene to the begining of eah transmit pulse time, we have r k [n ] = r[n + kn r ] = a k e jπ f dk N g 1 p[n l]h LP [k, l]. l= (A13) we have for the reeived pulse for the k th transmission r k [n] = a k e jπ f dk N g 1 p[n l]h LP [k, l] l= (A14) 196 where k =, 1,, K 1 slow time n =, 1,, N p + N g 1 fast time f d = F d T r Doppler effet 197 Appendix B The Autoorrelation Matrix and Power Spetral Density of Refleted Signal r(k, n) 198 199 Appendix B.1 Derivation of the autoorrelation matrix R rr ( k, n) From the definition of autoorrelation matrix, we have R rr [k, n, k, n] = E(r[k + k, n + n]r [k, n]) ( = E = e jπ f d k E = e jπ f d k = e jπ f d k e jπ f d(k+ k) ( Ng 1 N g 1 l= N g 1 N g 1 l= l = N g 1 N g 1 l= l = N g 1 N g 1 p[n + n l]h LP [k + k, l]e jπ f dk l= l = l = p[n + n l]p [n l ]h LP [k + k, l]h LP [k, l ] p[n + n l]p [n l ]E ( h LP [k + k, l]h LP [k, l ] ) p[n + n l]p [n l ]R hh [ k, l l ] p [n l ]h LP [k, l ] ) ) (A15) 1 Reall that for one single pulse of the transmit signal, the transmit signal p[m] is only nonzero when m N p 1 where N p is the length of a single pulse. Hene, a term in R rr [k, n, k, n] is only nonzero if n + n l N p 1 n l N p 1 (A16)
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 1 of 15 3 4 That is, for given n and n, l and l an only take values in the following sopes otherwise R rr [k, n, k, n] will be zero n + n + 1 N p l n + n N g 1 n + 1 N p l n N g 1 (A17) 5 6 7 If assume that N p N g, that is, eah transmit pulse length is small ompared with the TIR length in temporal range number N g, then the valid sope of N p 1 n N g 1 is approximate to the whole sope n N g + N p 1 we have R rr [k, n, k, n] = e jπ f d k n+ n l=n+ n+1 N p n l =n+1 N p p[n + n l]p [n l ]R hh [ k, l l ] (A18) 8 Now let m = n + n l and m = n l then we have m N p 1 m N p 1 l l = n (m m ) (A19) 9 and R rr [k, n, k, n] = e jπ f d k N p 1 N p 1 m= m = p[m]p [m ]R hh [ k, n (m m )] (A) 1 11 Appendix B. Derivation of PSD P r (η, φ) Next the relationship between the refleted data PSD and target PSD is derived. First, we rewrite R rr [ k, n] = e jπ f d k m= m = p[m]p [m ]R hh [ k, n (m m )] (A1) 1 13 where p[m] is nonzero only if m [, N p 1]. autoorrelation with its PSD. First, we replae the target impulse response R rr [ k, n] = e jπ f d k = e jπ f d k = e jπ f d k = e jπ f d k = e jπ f d k = e jπ f d k = η φ m= m= m = m = η φ m= m = p[m]p [m ]R hh [ k, n (m m )] p[m]p [m ] P h (η, φ)e jπη k e jπφ( n (m m )) dφdη η φ η φ m= m = p[m]e jπφm η φ m= m = η φ p[m]p [m ]P h (η, φ)e jπη k e jπφ( n (m m )) dφdη p[m]e jπφm p [m ]e jπφm P h (η, φ)e jπη k e jπφ n dφdη S(φ) P h (η, φ)e jπη k e jπφ n dφdη S(φ) P h (η, φ)e jπ(η+ f d) k e jπφ n dφdη p [m ]e jπφm P h (η, φ)e jπη k e jπφ n dφdη (A)
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 13 of 15 14 15 16 where S(φ) = F{p[m]} with F{ } denoting the Fourier transform is the signal energy spetrum of a single pulse. As seen if let η = η + f d then the above beomes R rr [ k, n] = = η φ η φ Hene, we have that the -D PSD of the r[k, n] is S(φ) P h (η, φ)e jπ(η+ f d) k e jπφ n dφdη S(φ) P h (η f d, φ)e jπη k e jπφ n dφdη (A3) P r (η, φ) = S(φ) P h (η f d, φ) (A4) 17 18 The same result an be obtained by taking -D Fourier transform of the above autoorrelation to produe the -D PSD of r[k, n] as follows. P r (η, φ) = R rr [ k, n]e jπη k jπφ n k n = e jπ f d k k n = k n f 1 f 1 f P( f ) P h ( f 1, f )e jπ f 1 k e jπ f n d f d f 1 exp( jπη k jπφ n) f P( f ) P h ( f 1, f )e jπ f 1 k e jπ f n exp( jπ(η f d ) k jπφ n)d f d f 1 = P h (η f d, φ) S(φ) (A5) 19 1 Appendix C The relationship between Kullbak Leibler divergene and defletion oeffiient We next alulate the KLD as follows. ln p 1 (X) ln p (X) = KN [ [ ln P 1 (η, φ) ln P (η, φ) + I x(η, φ) P 1 (η, φ) I ] x(η, φ) (A6) P (η, φ) where P 1 (η, φ) is P x (η, φ) under H 1 and P (η, φ) is P x (η, φ) under H. Then the KLD D(p 1 (X) p (X)), also simplified as D(p 1 p ) at times, an be found as D(p 1 p ) = p 1 (X)[ln p 1 (X) ln p (X)]dX = KN [ ln P 1 (η, φ) ln P (η, φ) + P 1(η, φ) P 1 (η, φ) P ] 1(η, φ) P (η, φ) = KN [ ln P 1(η, φ) P (η, φ) + 1 P ] 1(η, φ) P (η, φ) = KN [ ln θp h(η f d, φ) S(φ) + + 1 θp h(η f d, φ) S(φ) ] + = KN [ θph (η f d, φ) S(φ) ln (1 + θp h(η f d, φ) S(φ) )] (A7) In disrete-time expression, let η k = K k for k =, 1,, K 1, η = K 1 and φ l = N l for l =, 1,, N 3 1, φ = 1 4 then the KLD an be written as N D(p 1 p ) = 1 k= N 1 [ θph (η k f d, φ l ) S(φ l ) ln P l= w (φ l ) (1 + θp h(η k f d, φ l ) S(φ l ) )] (A8) P w (φ l ) 5 6 We onsider the small signal ase (θ is small). By employing the Taylor expansion ln(1 + x) x 1 x, D(p 1 p ) for the small signal ase is approximately
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 14 of 15 P h (, )..15 PSD.1.5.5.5 -.5 -.5 Figure A1. Simulation 1 target PSD P h (η, φ) D(p 1 p ) 1 4 = θ 4 = θ 4 k= k= N 1 l= N 1 l= N 1 l= ( θph (η k f d, φ l ) S(φ l ) ) P w (φ l ) ( ) Ph (η k f d, φ l ) S(φ P w (φ l ) l ) k= P h (η k f d, φ l ) Pw(φ S(φ l ) l ) 4 (A9) 7 Appendix D Simulation Setup Details It is nontrivial to setup the simulations arried out in Setion V. This appendix arguments several details of generating the simulation data. To generate a two dimensional random TIR h LP (k, l), we employed a -D autoregressive (AR) model. Note we let K = N g = 3; that is, the number of the transmitted pulses K are the same with the extended target length N g. The -D AR parameter is with order (, ) and the oeffiient matrix is A = 1..1.1.5.75.5.75.5 8 with the exitation noise σh =.1; Also with the parameters we have the target -D PSD P h(η, φ) for 9.5 η.5 and.5 φ.5 as shown in Figure A1. Note that we assume the real-valued 3 data, hene the -D dimensional PSD has the symmetry property P h (η, φ) = P h ( η, φ). Also the 31 olored noise is generated with 1-D AR proess with the order being and the oeffiients being B = [1.3.5] and the exitation noise σw 3 = 1. [ ] With the PSD P h (η, φ) and, we an aulate the term (φ l ) = Ph (η k,φ l ) 33 k= P w (φ l ) and it is 34 shown in the top subfigure of Figure. As seen, the LMP-based waveform is to put all energy into the 35 bin φ =.5 where the term (φ l ) ahieves the maximum. Note that the transmit pulse length N p is 36 hosen to be 8 whih is muh less than the target length N g and the signal energy θe = 4.16.,
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 1 Deember 17 doi:1.944/preprints171.74.v1 15 of 15 37 38 39 4 41 4 43 44 45 46 47 48 49 5 51 5 53 54 55 56 57 58 59 6 61 6 63 64 65 66 67 68 69 7 71 7 73 Referenes 1. L. Ziomek, A sattering funtion approah to underwater aousti detetion and signal design, The Pennsylvania State University, Dissertation, 1981.. Z. Zhu, S. Kay and R. S. Raghavan, Information-theoreti optimal radar waveform design, IEEE Signal Proessing Letters, vol. 4, no.3, pp. 74-78, Marh 17. 3. H. L. Van Trees, Detetion, Estimation, and Modulation Theory, Vol. III, New York: J. Wiley, 1971. 4. G. Matz and F. Hlawatsh, Fundamentals of Time-Varying Communiation Channels, F. Hlawatsh and G. Matz, Eds. New York: Aademi Press, 11. 5. P. A. Bello, Charaterization of randomly time-variant linear hannels, IEEE Trans. Comm. Syst., no. 11, pp. 36 393, 1963. 6. G. Matz, On non-wssus wireless fading hannels, IEEE Transations on Wireless Communiations, vol. 4, no.5, pp.465 478, Sept. 5. 7. S. M. Karbasi, A. Aubry, A. De Maio, and M. H. Bastani, Robust transmit ode and reeive filter design for extended targets in lutter, IEEE Transations on Signal Proessing, vol. 63, no. 8, pp. 1965-1976, Apr. 15. 8. M. M. Naghsh, M. Soltanalian, P. Stoia, and M.M. Hashemi, Radar Code Design for Detetion of Moving Targets, IEEE Transations on Aerospae and Eletroni Systems, vol. 5, no. 4, pp. 76 778, Ot. 14. 9. S. Blunt and E. Mokole, An overview of radar waveform diversity, IEEE Aerospae and Eletroni Systems Magazine, vol. 31, pp. 4, Nov. 16. 1. A. De Maio, and A. Farina, New trends in oded waveform design for radar appliations, A Leture Series on Waveform Diversity for Advaned Radar Systems, July 9. 11. S. M. Karbasi, A. Aubry, A. De Maio, and M. H. Bastani, Robust transmit ode and reeive filter design for extended targets in lutter, IEEE Transations on Signal Proessing, vol. 63, no. 8, pp. 1965-1976, Apr. 15. 1. P. Whittle, On stationary proesses in the plane, Biometrika, 41, pp. 434-449, 1954. 13. M. R. Bell, Information theory and radar waveform design, IEEE Trans. Inf. Theory, vol. 39, no. 5, pp. 1578 1597, Sept. 1993. 14. Y. Yang and R. S. Blum, MIMO radar waveform design based on mutual information and minimum mean-square error estimation, IEEE Trans. Aerosp. Eletron. Syst., vol. 43, pp. 33 343, Jan. 7. 15. R.A. Romero, J. Bae, and N. A. Goodman, Theory and appliation of SNR and mutual information mathed illumination waveforms, IEEE Trans. Aerosp. Eletron. Syst.,, vol. 47, no., pp. 91 97, April 11. 16. A. Aubry, A. DeMaio, A. Farina and M. Wiks, Knowledge-aided (potentially ognitive) transmit signal and reeive filter design in signal-dependent lutter, IEEE Transations on Aerospae and Eletroni Systems, vol. 49, no. 1, pp.93 117, Jan. 13 17. B. Tang, M. M. Naghsh, and J. Tang, Relative entropy-based waveform design for MIMO radar detetion in the presene of lutter and interferene, IEEE Trans. Signal Proess., vol. 63, no. 14, pp. 3783 3796, Jul. 15. 18. A. De Maio, S. De Niola, Y. Huang, Z.-Q. Luo, and S. Zhang, Design of phase odes for radar performane optimization with a similarity onstraint, IEEE Trans. Signal Proess., vol. 57, no., pp. 61 61, Feb. 9. 19. S. Kay, Fundamentals of Statistial Signal Proessing: Detetion, Prentie-Hall, Englewood Cliffs, NJ, 1998.