Spatial diversity in radars-models and detection performance

SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 1 Spatial Diversity in Radars - Models and Detection Performance Eran Fishler† , Alex Haimovich† , Rick Blum‡ , Len Ciminio , Dmitry Chizhik∗ , Reinaldo Valenzuela∗ The work of Eran Fishler was supported by the New Jersey Center for Wireless Telecommunications. The work of Alexander Haimovich was supported in part by the Air Force Office of Scientific Research under Grant No. F49620-03-1-0161. The work of Rick Blum is based on research supported by the Air Force Research Laboratory under Grant No. F49620-03-1-0214. † New Jersey Institute of Technology, Newark, NJ 07102, e-mail: {eran.fishler, haimovic}@njit.edu ‡ Lehigh University, Bethlehem, PA 18015-3084, e-mail: [email protected] o University of Delaware, Newark, DE 19716, e-mail: [email protected] * Bell Labs - Lucent Technologies, e-mail: chizhik,[email protected] April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 2 Abstract Inspired by recent advances in multiple-input multiple-output (MIMO) communications, this proposal introduces the statistical MIMO radar concept. To our knowledge, this is the first time that the statistical MIMO is being proposed for radar. The fundamental difference between statistical MIMO and other radar array systems is that the latter seek to maximize the coherent processing gain, while statistical MIMO radar capitalizes on the diversity of target scattering to improve radar performance. Coherent processing is made possible by highly correlated signals at the receiver array, whereas in statistical MIMO radar, the signals received by the array elements are uncorrelated. Radar targets generally consist of many small elemental scatterers that are fused by the radar waveform and the processing at the receiver, to result in echoes with fluctuating amplitude and phase. It is well known that in conventional radar, slow fluctuations of the target radar cross section (RCS) result in target fades that degrade radar performance. By spacing the antenna elements at the transmitter and at the receiver such that the target angular spread is manifested, the MIMO radar can exploit the spatial diversity of target scatterers opening the way to a variety of new techniques that can improve radar performance. In this paper, we focus on the application of the target spatial diversity to improve detection performance. The optimal detector in the NeymanPearson sense is developed and analyzed for the statistical MIMO radar. It is shown that the optimal detector consists of non-coherent processing of the receiver sensors’ outputs and that for cases of practical interest, detection performance is superior to that obtained through coherent processing. An optimal detector invariant to the signal and noise levels is also developed and analyzed. In this case as well, statistical MIMO radar provides great improvements over other types of array radars. I. Introduction Radar theory has been a vibrant scientific field for the last fifty years or so [1], [2], [3]. Radar theory deals with many different and diverse problems. However, the two most important problems are the detection and range estimation problems. The importance of these two problems is not limited to radars, and other engineering disciplines like sonar and communication deal with very similar problems [4]. Over the years, radar systems have developed considerably. These developments can be attributed to the increase in computation power and advances in hardware design. While early radar systems utilized a directional antenna, today’s array radar systems can synthesize beams and simultaneously scan the whole space [5]. In array radars, the system is composed of many closely spaced omni-directional antennas that either transmit or receive signals. Nonetheless, in some radar systems, only one antenna transmits energy. Such systems are known as active radars [6], [7]. In active radar systems, the transmitting antenna transmits a known waveform. This waveform is reflected back from the target toward the receiving array. The radar system’s tasks are to detect the existence of the target and to estimate its unknown parameters, e.g., range, speed, and direction. In active array applied to radar systems, solutions for the detection and parameter estimation problems are well documented in the literature (see, the survey in [8]). These solutions can be divided into two groups. The first group of solutions is based on high resolution techniques, e.g., MUSIC or maximum likelihood (ML) [8]. In the second group of solutions, the array of sensors is used to steer a beam toward a certain direction in space, in a manner similar to a conventional radar with a directional antenna [5]. The advantages of array radars are well known (see, for example [9], [5], [8], [10], [11]). The lack of any April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 3 mechanical elements in the system and the ability to steer multiple beams at once are two examples of such advantages. Systems with more than one transmitting element have been proposed and built as well [9]. In these systems, an array of transmitting elements is used. In the conventional approach, an array of closely spaced transmitting elements is used to cohere a beam toward a certain direction in space, a process called beamforming [5]. The main advantage of these systems is the lack of any mechanical elements, which reduces the system’s complexity and improves its performance. Recently a new and interesting concept in array radar has been introduced by the synthetic impulse and aperture radar (SIAR) [12], [13], [14]. In SIAR systems, the elements of the transmitting array emit orthogonal waveforms. Through synthetic pulse formation, SIAR achieves the advantages of wideband radar (improved range resolution) while individual antennas transmit narrowband waveforms. Unlike conventional beamformers, SIAR features isotropic radiation (an advantage in terms of the probability of intercept of the radar waveform by a third party). But, like beamformers, SIAR exploits the full correlation of signals transmitted and received at the array elements. Radars utilizing multiple transmitters and orthogonal waveforms have also been proposed in [15], [16], where they have been referred to as MIMO radars. The performance of radar systems is limited by target scintillations [17]. Targets are complex bodies composed of many scatterers. The range to, and the orientation of, the target determines the amount of energy reflected from these scatterers, and small changes in range or orientation can result in a large increase or decrease in the amount of energy reflected from the target [17], [4]. Both experimental measurements and theoretical results demonstrate that scintillations of 10 dB or more in the reflected energy can occur by changing the target aspect by as little as one milliradian [2]. These scintillations are responsible for signal fading, which can cause a large degradation in the system’s performance. Most, if not all, radar processing ignores target scintillations and considers target scintillations as an unavoidable loss. Target radar cross section (RCS) fading can reduce the received signal energy to a level that does not allow reliable detection. One way to mitigate the effect of target fading is to maximize the received energy from the target. This in turn implies that one should maximizes the system’s coherent processing gain. Since processing gain can be realized with closely spaced antennas, most systems use an array of closely spaced antennas. Motivated by recent developments in communication theory [18], we question the conventional wisdom of maximizing the coherent processing gain. In this paper, we demonstrate that there are alternative optimization criteria that outperform beamforming and leads to a new radar architecture. In particular, we demonstrate that the utility of an array of sensors in which the spacing between the elements of the array is very large, and that orthogonal waveforms should be transmitted from the elements of the transmitting array. The key point in the radar architecture is that sensors both at the transmitter and the April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 4 receiver of the radar are separated such that they experience an angular spread1 . As will be seen in the sequel, such a system forms a multistatic radar. The advantage of this system is that the average received energy (averaged across all the independent radars) is approximately constant, i.e., it does not fade as in conventional systems. In essence, we will show that spatial diversity gain outweights the coherent processing gain. We will demonstrate that an additional advantage of such systems is that the sensor outputs are combined non-coherently. Communication systems that use the same principle for improving their performance are called Multiple Input Multiple Output (MIMO) systems. Since our proposed system is inspired by MIMO communications, and it exploits that statistical properties of the target RCS, we refer to it as statistical MIMO radar. The reader should note that the whole notion of statistical MIMO radar is new, and the main purpose of this paper is to introduce this concept and to demonstrate its advantages. For clarity and mathematical tractability we use a simple model which ignores Doppler effects and clutter. More realistic models are left to subsequent work. The rest of the paper is organized as follows: Section II develops a signal model that generalizes current signal models. This model is then used to classify different array radars and in particular to describe our proposed statistical MIMO radar. Section III analyzes the performance of the proposed radar, while Section IV examines the optimal invariance detector. Section V provides the summary and concluding remarks. II. Signal Model A. General Theory The point source assumption dominates the models used in radar theory [2]. For closely spaced sensors and large range between the target and the array, the point source assumption is an excellent approximation. However, as has been noted in the array signal processing community, a more accurate model is the distributed source model [19], [20], [21]. Unlike the common point source model, the distributed source model accounts for the spatial characteristics of the target. In statistical MIMO radar, the spacing between the array elements is large. Due to the target’s complex shape and the distance between the array elements, every element observes a different aspect of the target. Therefore, the point source model is not adequate for describing the received signal in statistical MIMO radar, and a more detailed model must be developed. In what follows we assume that the target is located at some point (x0 , y0 ) in space, and it is stationary during the observation time, i.e, Swerling I model [2]. The target has a rectangular shape whose dimensions are ∆x × ∆y meters. In addition, we assume that our system is composed of M transmitters and N receivers. With reference to Figure 1, denote by (txk , tyk ), k = 1, . . . , M , the locations (in the two dimensional space) of the M transmitters, and by (rxl , ryl ), l = 1, . . . , N the locations of the N receivers. 1 Angular spread is the target’s radar cross section variability as a function of aspect ratio. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING t1 5 r1 r2 t2 ∆X ∆Y x_0 y_0 Fig. 1. Spatial Configuration In the figure, t1 and t2 represent transmitters, while r1 and r2 represent receivers. We assume that the target is composed of an infinite number of random, isotropic and independent scatterers, uniformly distributed over [x0 − ∆x 2 , x0 + ∆x 2 ] × [y0 − ∆y 2 , y0 + ∈ [− ∆x 2 , complex gain of the scatterer located at (x + x0 , y + y0 ), for (x, y)  Σ(x, y) as a zero mean, white, complex random variable, and E |Σ(x, y)|2 ∆y 2 ]. Denote by Σ(x, y) the ∆y ∆y ∆x 2 ] × [− 2 , 2 ]. We model 1 . The last assumption = ∆x∆y is responsible for normalizing to one the average energy returned from the target. △ In order to describe the received signal model, we need the following definitions: Denote by d(x, y, x′ , y ′ ) = p ′ ′ △ ,y ) the time (x − x′ )2 + (y − y ′ )2 be the distance between (x, y) and (x′ , y ′ ). Let τ (x, y, x′ , y ′ ) = d(x,y,x c it takes a signal transmitted from (x, y) to reach (x′ , y ′ ), where c is the speed of light. Assume that a q E narrow-band signal, denoted by M sk (t), is transmitted from the kth transmitter, where ||sk (t)||2 = 1 and E is the total average received energy. The received signal at the lth receiver is the superposition of the signals reflected from all the scatterers. Denote by rlk (t) the received signal at the lth receiver due to the signal transmitted from the kth transmitter, where rlk (t) obeys rlk (t) = April 26, 2004 Z x0 + ∆x 2 x0 − ∆x 2 Z y0 + ∆y 2 y0 − ∆y 2 r E sk (t − τ (txk , tyk , γ, β) − τ (rxl , ryl , γ, β)) Σ(γ − x0 , β − y0 )dγdβ M DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 6 A change of variables β + x0 → β and γ + y0 → γ results in the following expression for rlk (t): rlk (t) = − Z ∆x 2 − ∆x 2 Z ∆y 2 − ∆y 2 r E sk (t − τ (txk , tyk , x0 , y0 ) − (τ (txk , txk , x0 + γ, y0 + β) − τ (txk , tyk , x0 , y0 )) M τ (rxl , ryl , x0 , y0 ) − (τ (rxl , ryl , x0 + γ, y0 + β) − τ (rxl , ryl , x0 , y0 )))Σ(γ, β)dγdβ Invoking the narrowband assumption: r Z ∆y Z ∆x 2 2 E sk (t − τ (txk , tyk , x0 , y0 ) − τ (rxl , ryl , x0 , y0 )) rlk (t) = M − ∆x − ∆y 2 2 ·e−j2πfc (τ (txk ,tyk ,x0 +γ,y0 +β)−τ (txk ,txk ,x0 ,y0 )+τ (rxl,ryl ,x0 +γ,y0 +β)−τ (rxl,ryl ,x0 ,y0 )) Σ(γ, β)dγdβ (1) where fc is the carrier frequency. This model can be further simplified by using the following approximations, τ (txk , tyk , x0 + γ, y0 + β) − τ (txk , tyk , x0 , y0 ) p p (txk − (x0 + γ))2 + (tyk − (y0 + β))2 − (txk − x0 )2 + (tyk − y0 )2 = c p p (txk − x0 )2 + γ 2 + 2γ(txk − x0 ) + (tyk − y0 )2 + β 2 + 2β(tyk − y0 ) − (txk − x0 )2 + (tyk − y0 )2 = c p p (txk − x0 )2 + 2γ(txk − x0 ) + (tyk − y0 )2 + 2β(tyk − y0 ) − (txk − x0 )2 + (tyk − y0 )2 ≈ c p p β(txk −x0 )+γ(yxk −y0 ) (txk − x0 )2 + (tyk − y0 )2 + √ − (txk − x0 )2 + (tyk − y0 )2 (txk −x0 )2 +(tyk −y0 )2 ≈ c β(txk − x0 ) + γ(tyk − y0 ) β(txk − x0 ) + γ(tyk − y0 ) , (2) = p k= c · d(txk , tyk , x0 , y0 ) c (txk − x0 )2 + (tyk − y0 )2 where the first approximation is due to the fact that γ 2 + β 2 << (txk − x0 )2 + (tyk − y0 )2 , and the second √ √ is due to x + ǫ ≈ x + 2√ǫ x for ǫ << x. By combining (1), and (2), and using the narrowband signal assumption, the received signal can be modeled as follows: rlk (t) = · = r E sk (t − τ (txk , tyk , x0 , y0 ) − τ (rxl , ryl , x0 , y0 )) M Z ∆y Z ∆x β(txk −x0 )+γ(tyk −y0 ) β(rxl −x0 )+γ(ryl −y0 ) 2 2 −2jπfc ( c·d(tx + c·d(rx k ,tyk ,x0 ,y0 ) l ,ryl ,x0 ,y0 ) e Σ(γ, β)dγdβ − ∆y 2 − ∆x 2 αlk r E sk (t − τ (txk , tyk , x0 , y0 ) − τ (rxl , ryl , x0 , y0 )), M (3) where △ αlk = Z ∆x 2 − ∆x 2 Z ∆y 2 − ∆y 2 e −2jπfc β(txk −x0 )+γ(tyk −y0 ) β(rxl −x0 )+γ(ryl −y0 ) + c·d(rx c·d(txk ,tyk ,x0 ,y0 ) l ,ryl ,x0 ,y0 )
Σ(γ, β)dγdβ. (4) Since Σ(x, y) is a random field, αlk is a random variable. The exact distribution of αlk depends on the exact distribution of Σ(x, y). However, due to the central limit theorem, αlk is approximately a complex April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 7 normal random variable. The mean and variance of αlk are given as follows: E {αlk } = Z ∆x 2 − ∆x 2  E |αlk |2 = Z ∆y Z Z ∆x 2 2 − ∆x 2 − ∆y 2 ∆y 2 Z ∆x 2 − ∆x 2 e − ∆y 2 Z ∆y 2 − ∆y 2 β(txk −x0 )+γ(tyk −y0 ) β(rxl −x0 )+γ(ryl −y0 ) + c·d(rx c·d(txk ,tyk ,x0 ,y0 ) l ,ryl ,x0 ,y0 ) −2jπfc e −j2πfc
E {Σ(γ, β)} dγdβ = 0, (5) β(txk −x0 )+γ(tyk −y0 ) β(rxl −x0 )+γ(ryl −y0 ) η(txk −x0 )+ζ(tyk −y0 ) η(rxl −x0 )+ζ(ryl −y0 ) + c·d(rx − c·d(tx − c·d(rx c·d(txk ,tyk ,x0 ,y0 ) l ,ryl ,x0 ,y0 ) k ,tyk ,x0 ,y0 ) l ,ryl ,x0 ,y0 ) E {Σ(γ, β)Σ∗ (η, ζ)} dγdβdηdζ = Z ∆x 2 − ∆x 2 Z ∆y 2 − ∆y 2 1 dβdγ = 1, ∆x∆y where we used the fact that E {Σ(γ, β)Σ∗ (η, ζ)} = 1 ∆x∆y δ(γ (6) − η)δ(β − ζ). Consequently, the distribution of αlk can be approximated as αlk ∼ CN (0, 1). In order to write the received signal in compact-matrix form, we need additional notation. Let ψk = 2πfc (τ (txk , tyk , x0 , y0 ) − τ (tx1 , ty1 , x0 , y0 )) (7) ϕl = 2πfc (τ (rxl , ryl , x0 , y0 ) − τ (rx1 , ry1 , x0 , y0 )). (8) Since the signal is narrowband, s(t−τ (txk , tyk , x0 , y0 )−τ (rxl , ryl , x0 , y0 )) = e−jψk −jϕl s(t−τ (tx1 , ty1 , x0 , y0 )− τ (rx1 , ry1 , x0 , y0 )); and the received signal model, (3), can be further simplified as follows: r E −jψk −jϕl sk (t − τ (tx1 , tx1 , x0 , y0 ) − τ (rx1 , ry1 , x0 , y0 )). rlk (t) = αlk e M (9) The received signal at the lth receiver is the superposition of all the signals originating from the various transmitters plus the additive noise. Denote by rl (t) the received signal and by nl (t) the additive noise at the lth receiver, then rl (t) is given by the following model rl (t) = r M E X αlk e−jψk −jϕl sk (t − τ ) + nl (t), M (10) k=1 △ where τ = τ (tx1 , tx1 , x0 , y0 ) − τ (rx1 , ry1 , x0 , y0 ). Denote by r(t) = [r1 , . . . , rN (t)]T the collection of the received signals at the various receiving elements, and by s(t) = [s1 (t), . . . , sM (t)]T the collection of the transmitted signals from the various transmitting elements. The received signal vector can be described by the following simple model, r E diag (a(x0 , y0 )) H diag (b(x0 , y0 )) s(t − τ ) + n(t) r(t) = M (11) where H is an N ×M matrix, referred to as the channel matrix, such that [H]ji = αji ; diag (v) is a diagonal matrix with v on its diagonal; a(x0 , y0 ) = [1, eiϕ2 , . . . , eiϕN ]T is a N × 1 vector, which is a function of the target location, usually referred to as the receiver steering vector; b(x0 , y0 ) = [1, eiψ2 , . . . , eψM ]T is an M × 1 vector, which is a function of the target location, usually referred to as the transmitter steering vector; and n(t) = [n1 (t), . . . , nN (t)]T is an N × 1 vector representing the additive noise. We assume that April 26, 2004
DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 8 n(t) is a white, zero mean, complex normal random process with correlation matrix σn2 IN , where IN is the N × N identity matrix. In order to complete the received signal model, (11), we need to characterize the distribution of the channel matrix H. As will be seen in the sequel, the exact distribution of the channel matrix depends on the distances between the array elements and the target. Different array-target configurations give rise to different channel matrix distributions. The MIMO2 concept is related to one configuration; whereas the conventional model is related to another. Before we discuss these different configurations, we analyze the distribution of the channel matrix further. We have already demonstrated that every element of the channel matrix H is a zero mean, unit variance complex normal random variable. Denote by α = [α11 , . . . , α1M , α21 , . . . , αN M ]T the vector that contains all the elements of the matrix H. It is clear from our previous discussion that α is complex normal random vector. Since the mean of each element of α is zero, α ∼ CN (0MN , Rα ), where 0MN is the M N × 1  all-zero vector, and Rα = E ααH , (the superscript denotes conjugate transpose,) is the correlation matrix of the vector α. The matrix Rα , depends on the exact arrays-target configuration. However, there exists several cases in which we can approximate Rα using very simple expressions. Consider αjk and αjl . The correlation between αjk and αjl is given by,   (Z ∆x Z ∆y Z ∆x Z ∆y β(rxj −x0 )+γ(ryj −y0 ) β(txk −x0 )+γ(tyk −y0 ) 2 2 2 2 + −2jπfc  c·d(txk ,tyk ,x0 ,y0 ) d·d(rxj ,ryj ,x0 ,y0 ) e Σ(γ, β) E αjk α∗jl = E − ∆y 2 − ∆x 2 ·e = 2π jλ c Z  − ∆x 2 − ∆y 2 η(rxj −x0 )+ξ(ryj −y0 ) η(txl −x0 )+ξ(tyl −y0 ) + d(rx d(txl ,tyl ,x0 ,y0 ) j ,ryj ,x0 ,y0 ) ∆x 2 − ∆x 2 Z ∆y 2 − ∆y 2 Z ∆x 2 − ∆x 2 Z ∆y 2 − ∆y 2 e 2π −j λ c   Σ(η, ξ)dγdβdηdξ ) β(rxj −x0 )+γ(ryj −y0 ) η(rxj −x0 )+ξ(ryj −y0 ) β(txk −x0 )+γ(tyk −y0 ) η(txl −x0 )+ξ(tyl −y0 ) + − d(tx − d(rx d(txk ,tyk ,x0 ,y0 ) d(rxj ,ryj ,x0 ,y0 ) j ,ryj ,x0 ,y0 ) l ,tyl ,x0 ,y0 )  ·E {Σ(γ, β)Σ(η, ξ)} dγdβdηdξ   Z ∆x Z ∆y β(rxj −x0 )+γ(ryj −y0 ) β(rxj −x0 )+γ(ryj −y0 ) β(txl −x0 )+γ(tyl −y0 ) β(txk −x0 )+γ(tyk −y0 ) 2π 2 2 + − − −j λ 1 d(tx ,ty ,x ,y ) d(rx ,ry ,x ,y ) d(tx ,ty ,x ,y ) d(rx ,ry ,x ,y ) c j j 0 0 j j 0 0 k k 0 0 l l 0 0 dγdβ = e ∆y ∆x∆y − ∆x − 2 2 Z ∆x Z ∆y

β(txk −x0 ) γ(tyk −y0 ) β(txl −x0 ) γ(tyl −y0 ) 2 2 2π 1 1 −j λ −j 2π − − d(tx ,ty ,x ,y ) d(tx ,ty ,x ,y ) c k k 0 0 l l 0 0 = e e λc d(txk ,tyk ,x0 ,y0 ) d(txl ,tyl ,x0 ,y0 ) dγ, dβ (12) ∆x − ∆x ∆y − ∆y 2 2 1 where we used the fact that E {Σ(x, y)Σ(u, v)} = ∆x∆y δ(x − u)δ(y − v). o o n n ∗ In order to get further insight into E αjk αjl , we develop conditions under which E αjk α∗jl is △ either approximately zero or approximately one. Define d = d(txk ,tyk ,x0 ,y0 )−d(txl ,tyl ,x0 ,y0 ) , 2 1 ∆x Z ∆x 2 − ∆x 2 e 2π −j λ c d(txk ,tyk ,x0 ,y0 )+d(txl ,tyl ,x0 ,y0 ) , 2 △ and d′ = and consider, β(txk −x0 ) β(tx −x ) − τ (tx ,tyl ,x 0,y ) τ (txk ,tyk ,x0 ,y0 ) l l 0 0
1 dβ = ∆x Z ∆x 2 e 2π −j λ c (d−d′ )(txk −x0 )−(d+d′ )(txl −x0 ) β (d−d′ )(d+d′ ) dβ, (13) − ∆x 2 where λc is the carrier wavelength. It is easily seen that a sufficient condition for the above expression to be much less than one is that the integrand completes at least one cycle or equivalently 2 For ′ ′ 2π (d−d )(txk −x0 )−(d+d )(txl −x0 ) ∆x λc (d−d′ )(d+d′ ) 2 readability, in the sequel, we shorten “statistical MIMO” to plain “MIMO”. April 26, 2004 DRAFT > SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 9 π. Using some algebraic manipulations this condition can be reduced to (txk − txl )(d − d′ ) + 2d′ x0 > λc (d + d′ )(d − d′ )/∆x. (14) Without loss of generality, we can choose d′ to have the same sign as x0 , hence 2d′ x0 > 0, and we can strengthen the inequality by requiring that d′ = (txk − txl ) > λc (d + d′ )/∆x . (15) Condition (15) has a simple physical interpretation. The target is iluminated by the transmiter and it reflects the energy back. The target can be regarded as an antenna with aperture ∆x and beamwidth λc /∆x. If two transmiters are not within the same receive beamwidth of the target, then they see different “aspects” of the target with uncorrelated RCSs. Our condition examines whether the two transmiters are separated by more than the angular spread of the target or not. This simple condition, (15), allows us to determine whether the elements of the channel matrix H are considered correlated or not. Consider, for example, the jkth and ilth elements of the channel matrix H. If at least one of the following four conditions (1) rxj − rxi > d(rxj , ryj , x0 , y0 )λc /∆x, (2) txk − txl > d(txk , tyk , x0 , y0 )λc /∆x, (3) ryj − ryi > d(rxj , ryj , x0 , y0 )λc /∆y, (4) tyk − tyl > d(txk , tyk , x0 , y0 )λc /∆y (16) holds, the jkth and ilth elements of the channel matrix are uncorrelated. Note that (16) is a sufficient condition. In contrast to (16), we would like to develop a similar condition under which two elements of the channel matrix H are deemed correlated. Based on the discussion so far, if the integrands in (13) do not complete more than, say, 10% of a cycle, the correlation between αjk and αil is close to one. It can be easily seen from (16) that if the following four conditions hold jointly (1) rxj − rxi << d(rxj , ryj , x0 , y0 )λc /∆x, (2) txk − txl << d(txk , tyl , x0 , y0 )λc /∆x, (3) ryj − ryi << d(rxj , ryj , x0 , y0 )λc /∆y, (4) tyk − tyl << d(txk , tyk , x0 , y0 )λc /∆y (17) then the jkth and ilth elements of the channel matrix are approximately fully correlated. B. Model Classification Our received signal model, (11), can be used to describe many different systems. These systems differ in two aspects: the correlation between the elements of the channel matrix H and the design of the April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 10 transmitted signals. In this paper, we consider four canonical systems, which we believe represent the four extreme cases. These four systems are the conventional phased array radar, the MIMO radar, and the multiple input single output (MISO) and the single input multiple output (SIMO) radars. Each of these four canonical systems represents a large number of similar systems. Consequently, it is sufficient to examine these systems in order to compare which one is preferable and where. B.1 Phased Array Radar Phased array radars are probably the most common radar systems that use an array of sensors. Phased array radars utilizes on an array of closely spaced sensors. That is, every pair of elements obeys condition (17). In addition, in phased array radar the transmitted signal is given by s(t) = b̃s(t), where b̃ is usually referred to as the transmitter steering vector. When every pair of elements obeys condition (17), all the elements of the channel matrix are fully correlated. Therefore, the channel matrix is given by α1N M , where α ∼ CN (0, 1), and 1N M is the N × M all-ones matrix. Particularizing the received signal model, (11), to this case, results in the following received signal, r(t) = r E αa(x0 , y0 )b(x0 , y0 )H b̃s(t − τ ) + n(t). M (18) In phase array radar systems, both a(x0 , y0 ) and b(x0 , y0 ) are functions of the angle between the array and the target. It is common to denote these steering vectors by a(θ) and b(θ′ ) respectively, where θ (θ′ ) is the angle between the receiving (transmitting) array and the target. If, in addition, the receiver uses a beamformer to steer towards directions θ̃, and the transmitter steering vector b̃ = b(θ′ ), then the output of the beamformer is r E αa(θ̃)H a(θ)b(θ′ )H b(θ̃′ )s(t − τ ) + n(t). (19) M   This model represents a bistatic radar where b(θ′ )H b θ˜′ plays the role of the transmit antenna pattern,   and a(θ)H a θ̃ is the receive antenna pattern. Since n(t) = a(θ̃)H n(t) is a linear transformation of   n(t), n(t) is a zero mean, complex normal random process with correlation function ||a θ̃ ||2 σn2 δ(τ ) = H r(t) = a(θ̃) r(t) = N σn2 δ(τ ). In phased array radar, a processing gain of N M can be realized by taking θ = θ̃ and θ′ = θ̃′ , a(θ̃)H a(θ) = N , and b(θ̃)H b(θ) = M . This processing gain does not come free of cost. Phased array radars are sensitive to the distribution of the fading coefficient α. If it happens that α is small, there is no way to overcome this fade, and the target will not be detected, or the estimation error will be large. MIMO radars are capable of overcoming this exact problem by exploiting spatial diversity. The radars proposed in [15] has been refered to by the authors as MIMO radars. In these radars each transmitter transmits one of M orthogonal waveforms. It was shown in [15] that with proper processing, phased array radar that use this type of signaling can realize a coherent processing gain of N . However, April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 11 since the transmitted signals are orthogonal, this radar has a low probability of intercept (LPI). According to our classification these radars belong to the phased array category. B.2 MIMO Radar While in phased array radar, the inter-element spacing between every pair of elements obeys condition (17), in MIMO radar the inter-element spacings between each pair of elements obeys condition (16). Therefore, all the elements of the channel matrix are uncorrelated, that is α ∼ CN (0, IMN ). Particularizing the received signal model, (11), to this case results in the following model, r E Hs(t − τ ) + n(t). r(t) = M (20) With MIMO radar, each transmitter-receiver pair sees a different aspect of the target. In particular, by proper selection of the transmitted signals from the various transmitters, one can generate the equivalent of M N radar systems. Assume that each of the M transmitters transmits one of M orthogonal signals, and denote these signals by si (t), i = 1, . . . , M . By matched filtering each of the elements of r(t) with each of the transmitted q E αji si (t − τ ) + nji (t), where nji (t) is a zero mean, complex normal signals, we can reconstruct rji (t) = M random process with correlation function Rn (τ ) = σn2 δ(τ ). While in conventional radar the processing gain is M N and the radar sees only single aspect of the target, in MIMO radar, there is no coherent processing gain, but we synthesize M N independent radars. As a result, MIMO radar overcomes deep fades. B.3 MISO and SIMO radar In conventional radar, all the inter-element spacings are small; whereas in MIMO radar, elements are widely spaced. It is sometime desirable to mix these two approaches. MISO and SIMO radars are a compromise between phased array radars and MIMO radars. In MISO radar, the inter-element spacings of the transmitting elements obey (16), while the inter-element spacings of the receiving elements obey (17). In contrast, in SIMO radar, the inter-element spacings of the transmitting elements obey (17), while the inter-element spacings of the receiving elements obey (16). In what follows we concentrate on the MISO radars, and we omit the discussion on SIMO radars due to space limitations. Since every pair of transmitting elements obeys (16), αji and αkl are uncorrelated for i 6= l. However, since every pair of receiving elements obeys (17), αji and αki are fully correlated. Therefore, the channel matrix, H, equals 1N αH , where α ∼ CN (0, IM ), and 1N is the all ones vector of length N . Particularizing the received signal model, (11) to this case results in the following model, r E a(θ)αH s(t − τ ) + n(t). r(t) = M (21) MISO radar can realize a coherent processing gain of ||a(θ)||2 = N and create M independent radars. To see this, suppose that, as in MIMO radar, each of the M transmitters transmits one of M orthogonal April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 12 signals, si (t) (i = 1, . . . , M ). The receiver first steers toward θ, and then uses a beamformer to combine the received signal vectors. By matched filtering the beamformed signal we can construct the following q E αi si (t − τ ) + ni (t), where ni (t) is a zero mean, complex normal random process M signals, ri (t) = N M with correlation function Rn (τ ) = N σn2 δ(τ ). A complete analysis of MISO radars and their application for direction finding can be found in [22]. III. Basic Performance Comparisons The radar detection problem can be formulated as follows [17]: H0 : Target does not exist at delay τ H1 : Target exists at delay τ . (22) Many variants of this basic detection problem have been investigated and analyzed in the past. These variants differ by the assumed signal model, the unknown parameters, etc. In this section, we investigate the best achievable performance with phased array, MIMO, and MISO radars. We then compare the various systems and determine the optimal one. Since we are interested in the inherent limitations of the various systems, in this section, we assume that θ, θ′ , and the noise level, σn2 , are all known in advance. The optimal, in the Neyman-Pearson sense, detector is the Likelihood Ratio Test (LRT), which is given by [23], T = log f (r(t)|H1 ) > H1 < δ, f (r(t)|H0 ) H0 (23) where f (r(t)|H0 ) and f (r(t)|H1 ) are the probability density functions (pdf) of the observation vector given the null and alternate hypotheses, respectively, and δ is a threshold, set by the desired probability of false alarm. A. MIMO Radar The following Lemma determines the structure of the MIMO radar’s LRT detector. △ R Lemma 1: Denote by x the N M × 1 vector such that [x]iN +j = ri (t)sj (t − τ )dt, that is, x is the output of a bank of matched filters. The optimal detector is given by > H1 T = ||x||2 <H0 δ, (24) where δ is set to ensure the required probability of false alarm. Proof of Lemma 1: See Appendix A. Notice the non-coherent nature of the MIMO radars detector. It is easy to verify that x is distributed as follows:   n H0 q , x= E  M α + n H1 April 26, 2004 (25) DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING where n ∼ CN 0, σn2 IMN 13
and α ∼ CN (0, IN M ). Therefore, x is a zero mean complex random vari
E + σn2 IMN under the alternate able with correlation matrix σn2 IMN under the null hypothesis, and M hypothesis. This leads to the following distribution of the test statistic,  2  σn χ2 H0 2 (2MN2)  , ||x||2 ∼ σ E n  χ2(2MN ) H1 2M + 2 (26) where χ2(d) denotes a chi-square random variable with d degrees of freedom. It is well known that the probability of false alarm, the probability of detection, and the threshold are all connected by a series of one-to-one relations. These relations can be written with the aid of (26). The probability of false alarm can be expressed PrF A = Pr (T > δ|H0 ) = Pr  σn2 2 χ >δ 2 2MN    2δ = Pr χ22MN > 2 . σn (27) It follows that δ is set using the following formula, δ= where Fχ−1 2 σn2 −1 (1 − PrF A ) , F 2 2 χ2M N (28) denotes the inverse cumulative distribution function of a chi-square random variable with 2M N 2M N degrees of freedom. The probability of detection is given by, PrD = = Pr (T > δ|H1 ) = Pr 1 − Fχ22M N E M  σ2 E + n 2M 2  χ22MN >δ  = 1 − Fχ22M ! σn2 (1 − PrF A ) . F −1 2 + σn2 χ2M N E M 2δ + σn2 ! (29) It is interesting to note that both the test statistic, (24), and the threshold, (28), are independent of the transmitted energy E. Therefore, the optimal detector, even in the case of unknown signal energy, is given by (24) and (28). This establishes the fact that (24) is a Uniformly Most Powerful (UMP) detector for which only the noise level needs to be known [24]. B. Phased Array Radar Consider a phased array radar system. The following lemma describes the structure of the phased array LRT Detector, Lemma 2: Let x = R rH (t)a(θ)s(t − τ )dt be the output of the spatial-temporal matched filter. The optimal detector is given by > H1 T = |x|2 <H0 δ (30) where δ is set to ensure the required probability of false alarm is achived. Proof of Lemma 2: See Appendix B. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 14 The optimal detector for the phased array system turns out to be the beamformer steering to direction θ followed by a simple linear filter (also known as the Echardt filter) [25]. It is interesting to note that the optimal detector, whether target fading exists or not, is the same. The optimal detector is independent of the transmitter steering vector, b̃, used. Optimizing this vector will result in the optimal phased array system. It is easy to see that x is described by the following model,   n H0 q , (31) x= E  ||a(θ)||2 b̃H b(θ′ )α + n H1 M
where α ∼ CN (0, 1) and n ∼ CN 0, σn2 ||a(θ)||2 . This gives rise to the following distribution of the test statistic, T ∼    2 σn ||a(θ)||2 2 χ(2) 2  2 σn ||a(θ)||2 E||a(θ)||4|b̃H b(θ ′ )|2 χ2(2) + 2 2M H0  H1 . (32) The probability of false alarm, the probability of detection, and the threshold can be easily derived using the same derivation leading respectively, to (27), (28), and (29). The probability of false alarm, the probability of detection, and the threshold are, respectively, given by,   2δ 2 , PrF A = Pr χ(2) > 2 σn ||a(θ)||2 σ 2 ||a(θ)||2 −1 δ= n Fχ2 (1 − PrF A ) , 2 2  PrD = 1 − Fχ22   σn2 σn2 + E||a(θ)||2|b̃H b(θ ′ )|2 M (33) (34)  .  Fχ−1 2 (1 − PrF A ) 2 (35) Since the cumulative distribution function Fχ22 (·) is a monotonic increasing function, the probability of detection is maximized when the argument of (35) is minimized. This in turn happens when |b̃H b(θ′ )|2 is maximized. Using the Cauchy-Schwartz inequality, |b̃H b(θ′ )|2 is maximized when b̃ = b(θ′ ). Substituting the optimal b̃ into (35) results in the following expression for the probability of detection,     2 σn2 σn −1 −1   2 F (1 − Pr ) = 1 − F F (1 − Pr ) , (36) PrD = 1 − Fχ22   2 FA FA χ2 2 ′ 4 χ22 (σn2 + EN M ) χ2 σn2 + E||a(θ)||M||b(θ )|| where we used the fact that ||a(θ)||2 = N and ||b(θ′ )||2 = M . C. MISO Radar The last system we consider is the MISO radar. It is very easy to verify that MISO radar is a hybrid of both the MIMO and the phased array systems. Therefore, it should not be a surprise that the optimal detector is a combination of the corresponding optimal detectors. The following lemma describes the MISO radar’s optimal detector. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING △ Lemma 3: Denote by x an M × 1 vector such that [x]i = R 15 rH (t)a(θ)si (t − τ )dt, that is, x is the output of a bank of spatial-temproal matched filters. The optimal detector is then given by > H1 T = ||x||2 <H0 δ, (37) where δ is set to ensure the required probability of false alarm. The proof of Lemma 3 is a simple modification of the proofs of Lemmas 1 and 2, and hence it is omitted. The receiving array is composed of closely spaced sensors. Consequently, the receiver can cohere a beam toward the target and coherently combine the signals received at the receiving array. However, the spatially combined signal contains several orthogonal signals sent from different antennas. The optimal detector de-spreads these orthogonal signals. Since these signals arrive with unknown gains and phases, the detector combines these signals non-coherently, the same way the MIMO radar does. The probability of false alarm and the probability of detection can be computed from (27), (29), (33), and (36), and they are given respectively by  PrF A = Pr χ2(2M) > 2δ σn2 ||a(θ)||2  , (38) σn2 ||a(θ)||2 −1 Fχ2 (1 − PrF A ) , 2M 2 ! σn2 −1
Fχ2 (1 − PrF A ) . PrD = 1 − Fχ22M 2M σn2 + EN M δ= D. Discussion (39) (40) The most useful performance measure of any detector is its probability of detection. Therefore, we should compare the probability of detection of the various systems in order to determine the best. We will do this comparison based on numerical examples in the next section. Another approach, which is analytically tractable, is to compare the various systems based on a single scalar performance measure. Such a performance measure should summarize the detector performance in a single number instead of a function. One such performance measure is the detector’s signal-to-noise ratio (SNR) (also referred to as divergance in [26]). Let T be some test statistic (detector). The detector’s SNR, denoted by β, is defined as follows: |E (T |H0 ) − E (T |H1 ) |2 . (41) [Var (T |H0 ) + Var (T |H1 )] √ If T |Hi , i = 0, 1 is normally distributed, β represents the normalized distance between the means of β= 1 2 the distribution of the detector’s test statistics under the null and alternate hypotheses. This in turn represents our ability to distinguish between the two hypotheses. Although, in our problem, T |Hi is not normally distributed, we find this measure very useful in capturing the detector’s performance in one simple number. Consider a MIMO radar system. According to (26), E (T |H0 ) = M N σn2 , and E (T |H1 ) = M N σn2 +N E, and hence |E (T |H0 ) − E (T |H1 ) |2 = N 2 E 2 . April 26, 2004 Also, according to (26), Var (T |H0 ) = M N σn4 , and DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 16

Var (T |H1 ) = M N E 2 /M 2 + 2Eσn2 + σn4 , and hence [Var (T |H0 ) + Var (T |H1 )] = M N E 2 /M 2 + 2Eσn2 /M + 2σn4 . Combining these results, we obtain the following detector SNR βMIMO = MN E2N 2 E2N = . 4 2 2 2 + σn + Eσn /M ) M (E /2M + σn4 + Eσn2 /M ) (42) (E 2 /2M 2 Define the SNR, denoted by ρ, as the ratio between the total transmitted energy and the noise level △ per receive element, that is ρ = E/σn2 . The detector SNR can be further simplified and is given by the following expression, βMIMO = ρ2 N . M (1 + ρ2 /2M 2 + ρ/M ) (43) Now consider a phased array radar system. According to (32), E (T |H0 ) = N σn2 , and E (T |H1 ) = N σn2 + N 2 M E, and hence |E (T |H0 ) − E (T |H1 ) |2 = N 4 M 2 E 2 . Also, according to (32), Var (T |H0 ) = N 2 σn4 , and Var (T |H1 ) = N 2 σn4 + E 2 N 4 M 2 + 2σn2 EN 3 M , and hence [Var (T |H0 ) + Var (T |H1 )] = 2N 2 σn4 + E 2 N 4 M 2 + 2σn2 EN 3 M . The SNR of the optimal detector for the phased array radar system is given by the following expression, βphased array = N 4 M 2E 2 = 1 2 4 2 4 2 2 3 2 [2N σn + E N M + 2σn EN M ] = 1+ N 2M 2E2 1 4 2 2 2 2 2 [2σn + E N M + 2σn EM N ] N 2 M 2 ρ2 . + N ρM (44) M 2 N 2 ρ2 /2 Consider now a MISO system. The SNR of the optimal detector for a MISO is given by the following expression (a complete derivation is omitted due to space limitations), βMISO = E2N 4  1 4 2 3 2 2 2σn N M + 2N Eσn + E2N 4 M = ρ2 N 2 ρ2 N 2 = (45) M + ρ2 N 2 /M + N ρ M (1 + ρ2 N 2 /2M 2 + N ρ/M ) Figures 2 and 3 depict the optimal detectors’ SNR for the phased array, MIMO, and MISO systems. In Figure 2, we assume M = 1 and N = 4, while in Figure 3, we assume that M = N = 4. It is evident from the figures that, while for high SNR, the MIMO system is optimal, for low SNR, the phased array system is optimal. Assume that ρ >> 1, that is high SNR. The optimal detectors’ SNR can then be approximated as follows: βMIMO ≈ 2N M, βphased array ≈ 2, βMISO ≈ 2M. (46) It is easily seen that in terms of its ability to distinguish between the two hypotheses, the MIMO radar is the best system, whereas the phased array system is the worst system. The received signal in a phased array system enjoys a coherent processing gain. However, the random gain due to target fluctuations, α, is Rayleigh distributed, and hence there is a non-negligible probability that α is either small or large. When α is small, although the transmitted power is high, the received power is low. Therefore, the received SNR is low with non-negligible probability. The MIMO radar system, on the other hand, is equivalent to April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 17 1 10 0 β 10 −1 10 MIMO phased array MISO −2 10 −10 −5 0 5 SNR [dB] 10 15 20 Fig. 2. The optimal detector’s SNR in various systems. M = 1, N = 4. Note that the MISO and phased array curve fall . on top of each other N M independent radar systems, each of which is subject to Rayleigh distributed random gain. Since the number of independent radars is large, some of them are subject to deep fades, while others experience strong gains. However, the average received power does not vary considerably. Therefore, the MIMO radar is not subject to the deep fades that affect the phased array radar. Assume that ρ << 1, that is low SNR. The optimal detectors’ SNR can be approximated as follows: βMIMO ≈ ρ2 N/M, βphased array ≈ ρ2 N 2 M 2 , βMISO ≈ 2M. (47) For low SNR, the phased array radar system’s disadvantage becomes its advantage. Since with nonnegligible probability the random gain, α, is large, in those instances, the instantaneous SNR is high compared with the average SNR. Consequently, the target is detected by the phased array radar. On the other hand, the MIMO radar system does not enjoy this benefit. The received SNR can not deviate considerably from the average received SNR (recall that we have M N independent radars). Consequently, the probability of detection of the MIMO radar system is lower than that of the phased array system. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 18 2 10 1 10 0 β 10 −1 10 −2 10 MIMO phased array MISO −3 10 −10 −5 0 5 SNR [dB] 10 15 20 Fig. 3. The optimal detector’s SNR in various systems. M = N = 4. . IV. The Optimal Invariant Detector In the previous section, we examined the performance of the optimal detector when all the parameters are known. This assumption is an unrealistic one, and usually both the noise level, σn2 , and the transmitted power, E, are unknown. In real life systems, the noise level, σn2 , depends on the clutter, and the transmitted power, E, includes implicitly the unknown target average radar cross section (RCS). Consequently, it is unrealistic to assume that either E or σn2 are known in advance [4]. In radar theory there exist detectors that do not require explicit knowledge of either E or σn2 . These detectors are usually constant-false-alarm-rate detectors (CFAR), and some of them are optimal in a very important sense [17]. In what follows, we derive the optimal detector for MIMO and MISO systems when both the noise level, σn2 , and the transmitted energy, E, are unknown. We demonstrate that a very intuitive extension of the Mean Level Detector (MLD) is the optimal detector for MIMO radar systems in the invariant sense [27], [28]. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 19 When unknown parameters exist, many hypothesis testing problems do not possess an UMP detector. However, if we restrict our attention to a class of detectors having some special properties, e.g., CFAR, sometimes an optimal detector can be found. Usually these special properties are so natural that it is hard to imagine using a detector that does not possess these properties. The mathematical theory governing these special detectors is the theory of invariant decision rules. The theory of invariant decision rules is beyond the scope of this paper. (We refer the reader to [29], [24], [28] for an introduction to invariant hypothesis testing theory.) In order to enhance an intuitive understanding of the subject, we describe the mathematical theory of invariant decision rules in the context of our detection problem. We first formulate the exact detection problem when both the noise level and the average received power are unknown and then develop the optimal detector in the invariance sense. A. Problem Formulation In all the systems discussed so far, the output of the matched filter sampled at t = τ was a sufficient statistic. However, if both the noise level, σn2 , and the average received power, E, are unknown, the information contained at the output of the matched filter sampled at t = τ is not sufficient for constructing a detector with bounded probability of false alarm. Take for example (25). Assume the null hypothesis, H0 , and that σn2 = 2. The output of the matched filter sampled at t = τ is a zero mean, complex normal random vector with correlation matrix 2I. Assume, on the other hand, the alternate hypothesis, H1 , and that σn2 = 1 and E = M . Again, the output of the matched filter sampled at t = τ is a zero mean, complex normal random vector with correlation matrix 2I. This simple example demonstrates that once both E and σn2 are unknown, we can find one combination of the parameters that belongs to the null hypothesis, and another that belongs to the alternate hypothesis, that yield the same output of the matched filter sampled at t = τ , i.e., the output is unidentifiable [30]. In order to overcome this problem, the common practice is to obtain additional samples of the noise process. This can be done, for example, by sampling the output of the matched filter in the vicinity of t = τ . For a discussion on how to obtain these noise samples we refer the reader to [17]. Here we assume that L samples of the noise process were obtained by the receiver, and denote by y = [y1 , . . . , yL ]T the
L × 1 vector that contains these samples, that is y ∼ CN 0, σn2 IL . Denote by x the sufficient statistic when all the parameters are known, and by l(x) = |x| the length of △ x. In addition, denote by z̃ = [yT , xT ]T the concatenation of y and x. It is easy to verify that independent of the exact system, e.g., phased array, MIMO, MISO, used,

z̃ ∼ CN 0, diag [σn2 1L×1 , (σn2 + µ)1l(x)×1 ] , (48) where µ depends on the system considered and whether or not a target exists. It can be easily shown that z = [||y||2 , ||x||2 ] is a sufficient statistic for z̃. Therefore, we will restrict our attention to z instead April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING of z̃. The sufficient statistic vector z is distributed as follows,   2 σn2 + µ 2 σn 2 , χ χ(2l(x)) . z∼ 2 (2L) 2 20 (49) The detection problem can be formulated H0 : µ = 0, σn2 ∈ R+ H1 : µ > 0, σn2 ∈ R+ , (50) where R+ is the set of positive, real numbers. This detection problem does not possess an UMP detector. However, it is only natural to restrict our attention to detectors that are invariant to scaling, that is, detectors, the performance of which depends only on the SNR and not on the actual noise level and average received signal energy. Moreover, if the structure of this detector is independent of the exact SNR, then the resulting detector is what is known as an UMP invariant detector. B. Invariant Decision Rules and the Optimal Detector In this section we provide a short introduction to invariant decision rules. We describe only the essential mathematical concepts without any specific proofs. After each step, we demonstrate the mathematical concept with our detection problem, (50). Let Z be a random vector whose pdf, denoted by fZ (z|θ), θ ∈ Θ, is known up to some unknown parameter vector, denoted by θ, belonging to a parameter space denoted by Θ. We denote by z the observation vector, and by Z the observation space. Assume that Θ = Θ0 ∪ Θ1 and Θ0 ∩ Θ1 = ∅. We associate Θ0 with the null hypothesis, and Θ1 with the alternate hypothesis, that is, if H0 is true, θ ∈ Θ0 , and if H1 is true, θ ∈ Θ1 . In our detection problem θ = [σn2 , α] and Θ = R+ 2 . In addition, Θ0 = (R+ , 0), and Θ1 = (R+ , R+ − {0}). Let G denote a group of transformations from z to z. In addition, let the group operation be composition, △ that is, if g1 , g2 ∈ G, then g = g2 g1 is the transformation that takes z to g(z) = g2 (g1 (z)). Definition 1: The family of distributions fZ (z|θ) is said to be invariant under the group G, if for every g ∈ G and θ ∈ Θ there exists a unique θ ′ ∈ Θ such that g(z) is distributed according to f (z|θ ′ ). Denote this θ ′ by ḡ(θ). The interpretation is that g has left the pdf invariant in form, but the parameter θ has changed to ḡ(θ) Our main objective is to determine the optimal detector invariant to scaling, that is, the detector outcome when presented with both z and az should be the same. Using this in our problem, leads to G = {g(z)|g(z) = az, a 6= 0}. It is easy to verify that if g(z) = az, then ḡ(θ) = aθ. Definition 2: A group of transformations G leaves a hypothesis-testing problem invariant if G leaves both families of distributions {f (z|θ), θ ∈ Θ0 } and {f (z|θ), θ ∈ Θ1 } invariant. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 21 It is easy to show that a hypothesis testing problem remains invariant if and only if ḡ(Θ0 ) = Θ0 and ḡ(Θ1 ) = Θ1 . This condition expresses the requirement that the original dichotomy of Θ is maintained for any transformation g. The detection problem of (50) is an invariant hypothesis test problem. To see this, note that for every θ = (σn2 , 0) ∈ Θ0 , ḡ(θ) = (a2 σn2 , 0) ∈ Θ0 . Further, ḡ(Θ0 ) ⊆ Θ0 . In addition, for every θ = (σn2 , 0) ∈ Θ0 , θ = ḡ(θ/a2 ), hence Θ0 ⊆ ḡ(Θ0 ). Consequently, ḡ(Θ1 ) = Θ1 . A decision rule is a mapping of the observation space, z, to the decision space {H0 , H1 }. We say that a decision rule φ : z → {H0 , H1 } is G invariant if for all z and g ∈ G, φ(g(z)) = φ(z). (51) This condition reflects the requirement that the decision should be invariant to the group of transformations. For example, in our detection problem we require that whether we observe z or az, the decision should be the same. This requirement is manifested in mathematical terms by equation (51). The definition of the invariant decision rule leads to the following definition: Definition 3: A function T (z) is said to be maximal invariant with respect to G if (invariance) T (g(z)) = T (z), ∀z, and ∀g ∈ G, and (maximality) T (z1 ) = T (z2 ) implies z1 = g(z2 ) for some g ∈ G. The interpretation of maximal invariant statistic is beyond the scope of this paper, and the interested reader is referred to [29]. It is easy to verify that a maximal invariant statistic for our detection problem is given by, M (z) = M ([||y||2 , ||x||2 ]) = ||x||2 . ||y||2 (52) Theorem 1: A decision rule is G-invariant if and only if it is a function of a maximal invariant statistic [24]. This theorem asserts that in searching for the optimal detector we can limit our search only to functions of M (z) of (52). The following theorem describes the optimal scale invariant detector for our detection problem: Theorem 2: Consider the detection problem of (50). An UMP scale invariant (UMPI) detector exists and it given by ||x||2 > H1 (53) < δ. ||y||2 H0 Proof of Theorem 2: The proof of the theorem is divided into two parts. In the first part we will derive T = the optimal detector for two simple hypotheses and in the second we will demonstrate that the detector obtained in the first part is UMP. Consider the following two simple hypotheses. Under the null hypothesis, the SNR is zero, while under the alternate hypothesis the SNR is a fixed value greater than zero. We are looking for the optimal invariant detector for deciding between these two hypothesis. According to Theorem 1 and the NeymanPearson Lemma, the optimal detector for this simple hypothesis testing problem is given by, T = April 26, 2004 f (M (z)|H1 ) > H1 < δ. f (M (z)|H0 ) H0 (54) DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 22 The maximal invariant test statistic, M (z), is the result of dividing two independent chi-squared random variables. Consequently, χ2(2l(x)) 2 l(x) χ(2l(x)) /2l(x) l(x) = F2l(x),2L 2 2 χ(2L) L χ(2L) /2L L   2 µ l(x) σ 2 + µ χ(2l(x)) = 1 + F2l(x),2L , M (z)|H1 ∼ n 2 σn χ2(2L) σn2 L M (z)|H0 ∼ = (55) (56) where and F2l(x),2L denotes an F-distributed random variable with 2l(x) and 2L degrees of freedom. Now,   M (z)L f (M (z)|H0 ) = fF2l(x),2L (57) l(x)   M (z)L . (58) f (M (z)|H1 ) = fF2l(x),2L l(x)(1 + SNR) where fF2l(x),2L (·) is the pdf of an F2l(x),2L random variable. Since the pdf of the F distribution is monotonically decreasing, f (M (z)|H1 , SNR)/f (M (z)|H0 ) is monotonically increasing. Consequently, (54) is equivalent to the following detector > H0 T = M (z) <H1 δ. (59) Since both the optimal test statistic T and δ are independent of the unknown parameters, (59) is a uniformly most powerful scale invariant detector [24]. ✷ C. Discussion Theorem 2 describes the optimal detector for the general detection problem in (50). In this subsection, we discuss this general problem in the context of the phased array, MIMO, and MISO radar systems. We compare the various systems based on their optimal scale invariant detector’s SNR. Denote by ρ = E/σn2 the average received SNR. By using the same method leading to (43), (44), and (45), respectively, we have (2L − 4) (2M N + 2L − 2) 2L − 4 βphased array-UMPI = βphased array 2L 2L − 4 βMISO−UMP I = βMISO 2L + 2M − 2 βMIMO−UMP I = βMIMO (60) (61) (62) From the expressions it is evident that the number of noise samples has great influence on the UMPI detectors’ SNR. In each case, the SNR is equal to the optimal detector’s SNR multiplied by a system dependent factor smaller than one. This factor represents the degradation in performance due to the unknown parameters, and it approaches one as the number of noise samples approaches infinity. This should not come as a surprise because, when the number of noise samples is high, the noise level can be estimated with very small error. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 23 The number of independent noise samples we can obtain depends on the number of transmitters and receivers, the system complexity, and the model used. In clutter limited systems the only way to obtain samples of the noise process is to sample the output of the filters which are matched to the transmitted signals at locations (delays) that do not contain targets. In a typical radar system, between 8 and 16 such locations are used for estimating the noise level. Consider one such location [17]. In our proposed MIMO radar system, we can sample each of the M N matched filters at that delay and obtain M N independent samples of the noise process. Similar reasoning applies for the MISO system as well. In the phased array system, at each such location only N independent samples can be obtained because only one signal is transmitted by the transmitting array. Therefore, in practical radar systems we can expect about 10M N independent samples of the noise process if MIMO or MISO systems are used, and about 10N samples if a phased array radar systems are used. Figures 4 and 5 depict the performance of the phased array, MIMO, and MISO radar systems’ UMPI detectors. We consider two scenarios, in the first M = 1 and N = 4, while in the second M = N = 4. In addition, the number of independent samples of the noise process was set to either 8M N or M N . Figures 4 and 5 demonstrate the validity of our theoretical results. We can see that when the number of noise samples is large, the degradation in SNR is negligible. However, if the number of noise samples is very small, the degradation in the SNR of the MIMO and MISO systems is larger than that of the phased array system. V. Numerical Examples In this section we consider several numerical examples which compare the performance of the various systems. These examples also validate the analysis conducted in the previous sections. In our first example we consider systems having 4 receiving antennas and 1 transmitting antenna, that is N = 4, M = 1. In this example, we consider both the MIMO radar system and the phased array radar system. Note that for M = 1, the MISO radar system is equivalent to the phased array radar system. Figure 6 depicts the probability of miss of the optimal detectors for known and unknown noise level, as a function of the SNR. The probability of false alarm was fixed at PF A = 10−6 . The figure contains graphs for both known and unknown noise levels. In the case of unknown noise level, L = 4 and L = 32 independent samples of the noise process are used by the processor. We can see that the results validate our analysis. At low SNR (for which the probability of detection is less than 0.5) the phased array system outperforms the MIMO system, while at high SNR the MIMO system outperforms the phased array system. Moreover, when the number of independent samples of the noise process is small, both the MIMO system and the phased array system exhibit large performance degradation, and the performance degradation of the MIMO system is much larger than the performance degradation of the phased array system. However, even for a moderate number of samples of the noise process, e.g., 32 samples, the performance degradation due to the unknown noise level is only about 1.5 April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 24 1 10 0 β 10 −1 10 MIMO, L=8MN phased array, L=8MN MISO, L=8MN MIMO, L=MN phased array, L=MN MISO, L=MN −2 10 −10 −5 0 5 SNR [dB] 10 15 20 Fig. 4. The optimal detector’s SNR for various systems. M = 1, N = 4. . dB. Figure 7 depicts the receiver operating curves of the MIMO and phased array radar systems under known/unknown noise level conditions. The average received SNR was set at ρ = 10 dB. These results establish the advantage of the MIMO radar system over the phased array radar system. For known noise level, the MIMO radar system outperforms the phased array radar system even for very low probabilities of false alarm. For unknown noise level and 32 independent samples of the noise process, a MIMO radar system outperforms the phased array radar system whenever the probability of false alarm is greater than 10−8 . In our second example we consider systems having 4 receiving antennas and 2 transmitting antennas, that is N = 4, M = 2. In this example, we consider the MIMO, MISO, and phased array systems. Figure 8 depicts the probability of miss of the optimal detectors for both known and unknown noise levels as a function of the SNR. The probability of false alarm was fixed at PF A = 10−6 . We assume that when the April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 25 2 10 1 10 0 β 10 −1 10 −2 10 MIMO, L=8MN phased array, L=8MN MISO, L=8MN MIMO, L=MN phased array, L=MN MISO, L=MN −3 10 −10 −5 0 5 SNR [dB] 10 15 20 Fig. 5. The optimal detector’s SNR for various systems. M = N = 4. . noise level is unknown the receiver obtains 64 independent samples of the noise process. At low SNR the phased array system outperforms the MIMO system, while at high SNR the MIMO system outperforms the phased array system. However, we can see that the MIMO radar system outperforms the phased array system when the probability of miss is about 20%. We can also see from the results that the MISO system is, in a sense, a compromise between the MIMO and phased array systems. Therefore, at low SNR, the MISO system outperforms the MIMO system, while at high SNR the MISO system outperforms the phased array system. Figure 9 depicts the receiver operating curves of the MIMO, MISO, and phased array radar systems under known noise level and unknown noise level conditions. The average received SNR was set at ρ = 10 dB. This figure establishes the advantage of the MIMO radar system over the phased array radar system. For known noise level, the MIMO system outperforms the phased array system at probability of false alarm greater than 10−8 ; whereas, for unknown noise level, the MIMO system outperforms the April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 26 0 10 −1 PMD 10 −2 10 MIMO N= 4 Phased Array N= 4 MIMO N=4 L=4 Phased Array N= 4 L=4 MIMO N=4 L=32 Phased Array N= 4 L=32 −3 10 Fig. 6. 0 5 10 15 SNR 20 25 30 Probability of detection as the function of the SNR, N = 4, M = 1. Note that the curves corresponding to the MISO and phased array radars fall on top of each other. phased array system at a probability of false alarm greater than 10−6 . Again, the MISO system offers a compromise between the two systems. VI. Summary and Concluding Remarks In this paper, we introduced the concept of MIMO radar. This concept differs substantially from current regimes in which closely spaced antenna arrays are used. With closely spaced antenna elements it is possible to cohere a beam towards a direction in space and to realize a coherent processing gain. However, these systems are prone to severe target fading, and hence they may suffer considerable performance degradation. In contrast, MIMO radar can not cohere a beam toward a certain direction in space, and hence it can not realize any processing gain. However, MIMO radar exploits that target angular spread to combat target fading. MIMO radar is composed of many independent radars, each of which sees a different aspect of the target, enabling the MIMO radar to exploit spatial diversity to overcome target fading. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 27 1 0.9 0.8 PD 0.7 0.6 0.5 0.4 0.3 −10 10 MIMO N= 4 Phased Array N= 4 MIMO N=4 L=32 Phased Array N= 4 L=32 −8 10 −6 −4 10 10 −2 10 0 10 Pfa Fig. 7. Receiver’s operating curves, N = 4, M = 1. . In addition, we extended the current models used in radar theory to account for both range and angular spreads, and sparse antenna arrays. While, for closely spaced arrays, this model is equivalent to the current existing models, for sparse antenna arrays, this model provides for the first time a simple method for modeling the signals received from complex targets. In addition, this model offers insights into the advantages and disadvantages of each system considered in this paper. From the model we can learn how MIMO radar can exchange coherent processing gain for diversity gain. More importantly, we investigated and compared the inherent performance limitations of both conventional phased array radars and the newly proposed radars. We derived the optimal detectors for both the conventional and the newly proposed systems when the parameters are either known or unknown. We demonstrated that the MIMO radar outperforms the conventional phased array radar whenever the probability of detection is at a reasonable level, say above 80%. So far, we have ignored the fact that, in practice, a phased array radar system needs to scan the whole April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 28 0 10 MIMO N= 4 Phased Array N= 4 MISO N= 4 MIMO N=4 L=64 Phased Array N= 4 L=64 MISO N= 4 L=64 −1 PMD 10 −2 10 −3 10 0 5 10 15 SNR 20 25 30 Fig. 8. Probability of detection as a function of the SNR, N = 4, M = 2. . space. Once a phased array radar system realizes a processing gain of M , it needs M times more time to scan the whole space. If we compare the MIMO system and the phased array system under the constraint that both systems need to scan the whole space and use the same total energy for accomplishing this task, phased array radar systems incur an SNR loss of 10log10M dB compared with the SNR of the MIMO systems. Therefore, in practical systems, the advantage of MIMO system is even greater than the one reported here. It is the current state of understanding that one should always try to increase the SNR by coherent combining. In this paper, we suggest to deviate from this regime and to balance between coherent processing and diversity. This paper provides a partial answer to the problem we refer to as “detection with unit energy.” This problem is defined as follows: given a fixed amount of energy and a scintillating target, should we cohere a beam toward the target and see only one aspect of the target or observe several aspects of the target with reduced power. Another aspect of the same problem is the need to decide April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 29 1 0.9 0.8 PD 0.7 0.6 0.5 MIMO N= 4 Phased Array N= 4 MISO N= 4 MIMO N=4 L=64 Phased Array N= 4 L=64 MISO N= 4 L=64 0.4 0.3 −10 10 −8 10 −6 −4 10 10 −2 10 0 10 Pfa Fig. 9. Receiver’s operating curves, N = 4, M = 2. . whether to transmit one strong pulse or to divide our energy between several pulses? The performance improvement achieved by MIMO radar is not limited to detection. In subsequent papers, we will provide a detailed analysis which demonstrates the superiority of MIMO radar in many aspects over the conventional phased array radar. We show that MIMO radar enjoys lower range, location, angle of arrival, and Doppler estimation errors. A preliminary analysis that demonstrates these results can be found in [22], [31]. References [1] W. M. Siebert, “A radar detection philosophy,” IEEE Trans. on Information Theory, vol. 2, pp. 204–221, Sep. 1956. [2] M. Skolnik, Introduction to Radar Systems. McGraw-Hill, 3rd ed., 2002. [3] P. M. Woodward, Probability and Information Theory with Application to Radar. Artech House Books, MA, 1953. [4] H. L. V. Trees, Detection, Estimation, and Modulation Theory, vol. III. John Wiley & Sons, NY, 1968. [5] S. Haykin, J. Litva, and T. J. Shepherd, Radar Array Processing. New York: Springer - Verlag, 1st ed., 1993. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 30 [6] A. Dogandzic and A. Nehorai, “Cramer-Rao bounds for estimating range, velocity, and direction with an active array,” [7] S. Pasupathy and A. N. Venetsanopoulos, “Optimum active array processing structure and space-time factorability,” [8] L. Swindelehurst and P. Stoica, “Maximum likelihood methods in radar array signal processing,” Proc. of the IEEE, IEEE Trans. on Signal Processing, vol. 49, pp. 1122–1137, June 2001. IEEE Trans. Aerospace and Electronic Systems, vol. 10, pp. 770–778, 1974. vol. 86, pp. 421–441, Feb. 1998. [9] A. Farina, Antenna Based Signal Processing Techniques for Radar Systems. Norwood, MA: Artech House, 1992. [10] H. Wang and L. 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Robey, “Performance bounds for adaptive coherence of sparse array radar,” in Proc. of the 11th conf. on Adaptive Sensors Array Processing, Mar. 2003. [16] D. Rabideau, “Ubiquitous MIMO digital array radar,” in in Proc. of the Asilomar Conference on Signals, Systems, and Computers, November 2003. [17] N. Levanon, Radar Principles. John Wiley & Sons, 1st ed., 1988. [18] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multiple antennas,” Bell Labs Technical Journal, vol. 1, no. 2, pp. 41–59, 1996. [19] F. Friedmann, R. Raich, J. Goldberg, and H. Messer, “Bearing estimation for a distributed source of nonconstant modulus ounds and analysis,” IEEE Trans. on Signal Processing, vol. 51, pp. 3027–3035, Dec. 2003. [20] Y. U. Lee, J. Choi, I. Song, and S. R. Lee, “Distributed source modeling and direction-of-arrival estimation techniques,” IEEE Trans. on Signal Processing, vol. 45, pp. 960–969, Apr. 1997. [21] R. Raich, J. Goldberg, and H. Messer, “Bearing estimation for a distributed source: Modeling, inherent accuracy limitations and algorithms,” IEEE Trans. on Signal Processing, vol. 48, pp. 429–441, Feb. 2000. [22] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “MIMO radar: An idea whose time has come,” in Proc. of the IEEE Int. Conf. on Radar, April Philadelphia, PA, 2004. [23] H. L. V. Trees, Detection, Estimation, and Modulation Theory, vol. I. John Wiley & Sons, NY, 1968. [24] E. L. Lehmann, Testing Statistical Hypotheses. John Wiley & Sons, 2nd ed., 1986. [25] J. B. Lewis and P. M. Schultheiss, “Optimum and conventional detection using a linear array,” The Journal of the Acoustical Society of America, vol. 49, pp. 1083–1091, Apr. 1971. [26] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, vol. 2nd. Pearson Education, 2002. [27] P. Grieve, “The optimum constant false alarm probability detector for relatively coherent multichannel signals in gaussian noise of unknown power,” IEEE Trans. on Information Theory, vol. 23, pp. 708–721, Nov. 1977. [28] L. Scharf and D. Lytle, “Signal detection in gaussian noise of unknown level: An invariance application,” IEEE Trans. on Inf. Theory, vol. 17, pp. 404–411, July 1971. [29] T. S. Ferguson, Mathematical Statistics: A decision Theoretic Approach. Probability and Mathematical Statistics, Academic Press, NY, 1967. [30] S. Kay, Fundementals of Statistical Signal Processing: Part II. Prentice Hall, 1998. [31] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Statistical MIMO radar,” in The 12th conf. on Adaptive Sensors Array Processing, March 2004. April 26, 2004 DRAFT SUBMITTED TO THE IEEE TRANSACTIONS ON SIGNAL PROCESSING 31 Appendices I. Proof of Lemma 1 In this section we compute the LRT detector for MIMO radar systems. We start by deriving the pdf of the received measurements under both the null and alternative hypotheses. Consider first the alternative hypothesis. Z Z − R 1 2 σn √E ||r(t)−H s(t−τ )||2 dt M f (r(t)|H1 ) = f (r(t)|H1 , α)f (α)dα = ce √ E PM P R Z |ri (t)− M − 12 αij sj (t−τ )|2 dt i σn j=1 = ce f (α)dα R R R √ E PM √ P P Z M E α∗ r(t)s∗ αij |ri (t)|2 dt− M − 1 (a) j (t−τ )dt− M j=1 ij j=1 = ce σn2 i R √E H √E H √ E 2
||r(t)||2 dt Z 2 − 12 − M − α x− M x α+|| M α|| 2 ′ σn σn e =ce e−||α|| dα ′ =ce − R ||r(t)||2 dt 2 σn E ||x||2 + 2M 2 E ) σn (σn + M Z e − 1 2 σn √E √ 2 x M 2+ E σn M 2 + E α− √ σn M △ ′′ dα = c e − f (α)dα E r ∗ (t)sj (t−τ )dt+ M R ||r(t)||2 dt 2 σn + PM j=1 |αij |2 f (α)dα E ||x||2 M 2 (σ2 + E ) σn n M (63) R ri (t)sj (t − τ )dt, α is the N M × 1 vector that contains R the fading coefficients, and in (a) we use the fact that sj (t)si (t)dt = δij . where x is an N M × 1 vector such that [x]iN +j = The pdf of the received measurements under the null hypothesis is given by, R 2 ||r(t)|| dt f (r(t)|H0 ) = ce − 2 σn (64) Combining (63), (64), and (23) results in the following detector, > H1 T = log f (r(t)|H1 ) − log f (r(t)|H0 ) = ||x||2 <H0 δ. (65) II. Proof of Lemma 2 In order to prove Lemma 2, we need to derive the pdf of the received measurements under both the null and alternative hypotheses. We first consider the alternative hypothesis. R √E Z Z − 1 αa(θ)bH (θ ′ )b̃s(t−τ )||2 dt ||r(t)− M f (r(t)|H1 ) = f (r(t)|H1 , α)f (α)dα = ce σn2 f (α)dα R
E 2 H
R √E ||r(t)||2 dt Z ℜ αbH (θ ′ )b̃ rH (t)a(θ)s(t−τ )dt + M |α| ||b̃ b(θ)a(θ)||2 +|α|2 − − 1 −2 M 2 σn dα = ce e σn2 R R R ′ 2 H 2 2 2 E|b(θ )b̃| || ||r(t)|| dt ′ =ce where x = R − 2 σn − r (t)a(θ)s(t−τ )dt|| ( 2 1+ E |b(θ′ )b̃|2 M σn M ) ||r(t)|| dt ′ =ce − 2 σn − E|b(θ′ )b̃|2 |x|2 2 1+ E |b(θ′ )b̃|2 M σn M ( ), (66) rH (t)a(θ)s(t − τ )dt. The pdf of the received measurements under the null hypothesis is given by (64). Combining (66), (64), and (23) results in the following detector, T = log f (r(t)|H1 ) − log f (r(t)|H0 ) = April 26, 2004 Z > H1 rH (t)a(θ)s(t − τ )dt <H0 δ. (67) DRAFT