Arrayed MIMO Radar. Harry Commin. MPhil/PhD Transfer Report October Supervisor: Prof. A. Manikas

Transcription

1 D E E E C A P G Arrayed MIMO Radar Harry Commin MPhil/PhD Transfer Report October 00 Supervisor: Prof. A. Manikas

2 Abstract MIMO radar is an emerging technology that is attracting the attention of both researchers and practitioners, which employs multiple transmit signals and has the ability to jointly process signals received at multiple receive antennas. In this report, the various MIMO radar configurations are defined and discussed. Motivated by its suitability for the direct application of powerful modern digital array signal processing techniques, the collocated arrays configuration is chosen as the focus of this report. From this basis, a detailed discussion of performance bounds in array processing is developed. The parameter C is introduced as a figure of merit for comparing the performances of practical direction-finding (DF) algorithms in terms of their superresolution capabilities. C takes values between 0 and, with higher values indicating better resolving capability and C = denotes an algorithm with the theoretically ideal resolution performance. Analytical expressions for C can be derived for a number of DF algorithms. In this report, three such expressions are derived for MUSIC, optimal Beamspace MUSIC and Minimum Norm. It is found that optimal beamspace MUSIC yields the smallest resolution separation, which can approach the ideal when incident signals have equal powers. Some preliminary work related to an investigation into the role of complex correlation coeffi - cient is presented. In particular, the -emitter CRB is derived for arbitrary correlation, and an accessible discussion of the spatial smoothing technique is presented. Finally, a variety of important topics for significant future research are identified, emphasising the wide-ranging potential for useful ongoing research contribution in the field of MIMO radar.

3 Contents Abstract Contents 3 Notation 5 Introduction 6. Array Configurations in MIMO Radar Parameter Estimation SIMO Array Signal Model MIMO Radar Signal Model The "Virtual Array" Concept in MIMO Radar Performance Bounds in Array Processing 4. Uncertainty Hyperspheres and the Parameter C Theoretical Detection Bounds Theoretical Resolution Bounds Theoretical Estimation Error Bounds Cramer-Rao Bounds Other Estimation Error Bounds Algorithm Comparison using the Figure of Merit C 4 3. Main Results Discussion Correlated Signals and Spatial Smoothing Conclusions and Future Work 9 4. Conclusions Future Work Appendix A Expressing Ideal Resolution Performance in Terms of ξ music ( s) 3 3

4 CONTENTS 4 Appendix B Lee and Wengrovitz s Angular Separation Measure 36 Appendix C Ideal Resolution Performance for ULA 38 Appendix D Algorithm-specific Resolution Threshold Expressions 39 Appendix E -Source CRB (correlated signals) 4 Appendix F An Accessible Analysis of Spatial Smoothing 44 F. Relating Source Covariance Matrices F. Effective Source Signal Correlation F.3 Quantifying the Decorrelating Effect F.4 Backward Spatial Smoothing References 50

5 Notation a, A Scalar a, A Column Vector A, A Matrix (A) ij (i, j) th element of A ( ) T Transpose ( ) H Conjugate transpose 0 N (N ) vector of zeros I N (N N) identity matrix a = vec(a) a is formed by stacking columns of A L [A] Linear subspace spanned by the columns of A L [A] Complement subspace to L [A] P A A ( A H A ) A H Projection operator onto L [A] P A I P A Projection operator onto L [A] E { } Expectation operator N Number of array antennas L Number of data snapshots M Number of signal sources ρ Complex correlation coeffi cient R ab Covariance matrix of a and b diag (a) Matrix whose diagonal elements are given by a exp(a) Element-by-element exponential 5

6 Chapter Introduction Radar has received great research interest for many decades []. Its basic goal is to provide the user with information about targets by estimating various parameters of interest (such as target bearing, range and velocity). In early radar systems, this was achieved by mechanically steering a directional transmit/receive antenna across the whole space and processing the (electromagnetic) signals reflected back to the receiver. However, by instead employing an array of multiple antennas at the transmitter and/or receiver, a number of important breakthroughs in radar theory have since been made. Firstly, using the concept of beamforming, an antenna array is able to coherently combine signals to synthesise a concentrated directional beampattern. This beampattern can be steered electronically, removing the need for any mechanical steering of the radar platform. Furthermore, with the development of powerful modern digital signal processing techniques, radar theory has seen dramatic improvement in recent years. In particular, adaptive techniques such as optimal beamforming and superresolution direction-finding (DF) have attracted enormous research interest []. The most sophisticated arrayed radar configuration is Multiple Input Multiple Output (MIMO), whereby the system employs multiple transmit waveforms and has the ability to jointly process signals received at multiple receive antennas. Due to the general nature of this definition, many traditional radar configurations can be viewed as (restricted) special cases of the MIMO paradigm. It is well documented in the literature that harnessing the full potential of MIMO could offer numerous significant improvements in radar performance, compared to traditional methods. 6

7 . Introduction 7 Particularly, since MIMO radars can jointly employ transmit and receive degrees of freedom after the signal is received, the total number of degrees of freedom available to the system can be greatly increased. This yields advantages including: improved resolution performance [3] and the ability to simultaneously detect a higher number of targets (i.e. greater parameter identifiability ) [4]. "Collocated arrays" is a typical configuration in MIMO radar, whereby transmit and receive antenna arrays are situated close together (such that the directions to the targets are the same for both arrays). This configuration will be the main focus of this report. In Section., all the various MIMO configurations found in the literature will be reviewed and their relative strengths and challenges discussed. It will be shown that the collocated configuration is particularly wellsuited to the direct application of adaptive array processing techniques and so the remainder of this report will focus on such techniques. In Section., the concept of superresolution directionfinding (DF) will be introduced. The fundamental bounds on DF performance of an array system will then be defined and explored. The parameter C is introduced in order to characterise the impact of practical DF algorithms on resolution performance. C is first derived in general terms, then specific analytical expressions for C are derived and discussed for the MUSIC, optimal Beamspace MUSIC and Minimum Norm algorithms for uncorrelated signals. Various topics of ongoing and future work are discussed in Section 4 and finally the report is concluded in Section 4... Array Configurations in MIMO Radar When designing the general physical layout of the antenna arrays in MIMO radar, there are two main approaches to consider:. Widely-separated Antennas [5] (a) Radar targets are complex bodies composed of many scatterers and so the power of the signal reflected back to the receiver can vary dramatically even for very small variations in illumination/observation angle. This can cause severe performance degradation in radar systems [6]. By distributing (transmit and/or receive) antennas far apart in space, multiple independent aspects are obtained, significantly reducing the effect of these fluctuations ("scintillations").

8 . Introduction 8 (b) Another useful consequence of the highly diverse signal paths (transmitters to scatterers to receivers) yielded by this configuration is that the fading properties of each path are not fully correlated and so a signal decorrelation effect is observed. (c) Finally, since the targets are illuminated/observed from multiple angles, their velocities will be different relative to each widely-spaced antenna. This yields a range of different Doppler frequencies (a Doppler spread ) which can be exploited to improve moving target detection (MTD) performance. Specifically, targets with low radial velocity with respect to one antenna will tend to have a larger radial velocity with respect to another widely-spaced antenna. Thus, MTD performance for targets which are diffi cult to distinguish from background clutter is improved.. Closely-spaced antennas [7] (a) Assuming a narrowband signal model, if transmit/receive elements are situated within the "local region" of a common phase origin, then relative antenna displacements give rise to consistent, predictable signal phase offsets. This allows array response vectors to be defined as a function of the array (electrical properties and geometry) and signal bearings. The array manifold is then defined as the locus of the array response vectors across the whole parameter space. From this coherent basis, a family of extremely powerful adaptive array processing techniques can be applied. (These will be discussed in greater detail in Section.). In the literature, we find that these two approaches give rise to three possible MIMO radar configurations:. Distributed MIMO Radar: Both transmitters and receivers are widely-separated.. Transmit Diversity MIMO Radar: Transmitters are widely-separated and receivers are in a closely-spaced array. 3. Collocated MIMO Radar: Both transmitters and receivers are in closely-spaced arrays. Furthermore, these arrays are situated close together (such that target bearings are the same with respect to both arrays).

9 . Introduction 9 Clearly, each of these approaches has their respective strengths and challenges. Transmit Diversity MIMO Radar has received significant research interest, since it can be considered as a hybrid approach that seeks to capitalise on the advantages offered by both closely-spaced and widely-separated antennas [8, 9]. However, this report will be concerned with collocated MIMO radar since it is the most suited to employing advanced adaptive array processing techniques. A further major advantage of the collocated arrays configuration is that properties of the transmitted vector signal are always known to the receiver and, similarly, parameters estimated at the receiver can be fed back to the transmitter. This provides a basis for the development of novel collaborative techniques whereby various transmitter characteristics (such as array geometry and waveform design) can be adaptively optimised depending on parameters estimated at the receiver. Moreover, the collocated arrays configuration has further practical advantages in that all processing can be done in situ, without requiring a wireless communication link to a central processor.. Parameter Estimation A large part of this report will focus on parameter estimation. In this report, targets are assumed to be stationary with respect to the radar platform, allowing a strong focus to be placed on the problem of estimating the directional parameters of targets (direction-finding). Early approaches to direction-finding with sensor arrays involved simply steering (electronically) a directional beam around in space and constructing a DF spectrum from the received signal power across the parameter space. To do this, a complex weight vector (generally designed to be optimal in some sense) is used to linearly combine coherent signals into a beam. Since antenna characteristics are independent of the direction of energy flow, a given weight vector provides the same beampattern when applied to either transmitting or receiving arrays. Many optimal beamformers can be found in the literature. The Conventional Beamformer, due to Bartlett [0], simply uses the manifold vector as the weight vector, which acts to maximise expected output power in a data-independent sense. However, significant improvement in DF performance is offered by beamformers that utilise the received array data (adaptive beamformers). A representative early example is Capon s Maximum Likelihood Method [], which For historical reasons, the Capon Beamformer is termed maximum likelihood, but it generally does not provide a maximum likelihood estimate in a DF context. It is more correctly referred to as the Minimum Variance Distortionless Response (MVDR) beamformer.

10 . Introduction 0 minimises array output power (variance) subject to the linear constraint that the signal arriving from the direction of interest is undistorted. While adaptive beamformers have much better resolution performance and interference-rejection capability than data-independent methods, they are also much more sensitive to modelling errors. Therefore, much research effort has been directed towards robust adaptive beamforming in recent decades []. More recently, a new family of "subspace-based" methods (based on eigenanalysis of the observed signal covariance matrix) have been developed. In general, they take a geometric approach to the DF problem by seeking to estimate the (orthogonal) signal and noise subspaces (which together make up the N-dimensional complex space in which the array manifold resides). Manifold vectors can then be orthogonally projected onto the estimated noise subspace and the norm of the resulting vector provides a measure of proximity between the two. Repeating this process for manifold vectors computed across the entire parameter space then provides a DF null spectrum. (Alternatively, projections onto the signal subspace provide the inverse null spectrum, or "pseudospectrum"). Representative examples are the MUSIC [3] and Minimum Norm [4] algorithms. As discussed later in this section, the subspace-based approach is capable of yielding extremely powerful DF performance and so will be a major topic of this report. In addition to beamformers and subspace-based techniques, several other families of parameter estimation algorithms can be found in the literature, such as Maximum Likelihood type [5, Chapter 8] and Parallel Factor Analysis type [6] (of which ESPRIT [7] is a widely-used example). However, the specific details of the many DF algorithms developed during recent decades are not central to this report. Such algorithms are reviewed extensively in the literature (see, for example, [], [8] and [5, Chapters 7-9]). In this report, a more general theoretical framework will instead be developed, before later returning to consider specific practical DF algorithms within that framework..3 SIMO Array Signal Model In order to aid later discussion regarding direction-finding, a general SIMO (single input, multiple output) array signal model will be derived first, before expanding to MIMO radar. Consider M narrow-band plane wave signals impinging on an array of N sensors. The (N )

11 . Introduction received signal vector at the array output (in the presence of noise) can be modelled as follows: x(t) = S(θ, φ)m(t) + n(t) (.) where m(t) is the M-vector of the baseband source signals, n(t) is additive noise and S(θ, φ) is the manifold matrix, having the following structure: S(θ, φ) = [S(θ, φ ), S(θ, φ ),..., S(θ M, φ M )] (.) with parameter vectors θ = [θ, θ,..., θ M ] T and φ = [φ, φ,..., φ M ] T denoting the directional parameters associated with the M sources (e.g. azimuth and elevation, where azimuth is measured anti-clockwise from the positive x-axis). The (N ) complex vector S(θ, φ) is the array manifold vector (array response vector): S(θ, φ) exp ( jr k(θ, φ)) (.3) of the source signal impinging from (θ, φ). In Equation.3, the array geometry is represented by the (N 3) real matrix: r [r x, r y, r z ] = [r, r,..., r N ] T R N 3 (.4) and k(θ, φ) is the wavenumber vector: k(θ, φ) π λ u(θ, φ) (.5) where λ is the signal wavelength and u(θ, φ) is the (3 ) real unit vector pointing from (θ, φ) towards the origin..4 MIMO Radar Signal Model In its simplest form, a collocated MIMO radar system can be viewed as a SIMO receiver whose multiple scalar inputs are given by the transmit array s MISO outputs reflecting from multiple targets. Denoting transmitter quantities with ( ) and assuming targets to be stationary with respect to the radar platform, the equivalent baseband scalar signal at the k th target can therefore be expressed as: z k (t) = S H (θ k, φ k ) m(t τ k )

12 . Introduction where m(t) is the ( N ) vector of transmit waveforms and τ k is the time taken for the signal to propagate from the transmitter to the k th target. The (K ) vector signal at the K targets, z(t) [z, z,..., z K ] T, can then be considered to be somewhat analagous to m(t) in the SIMO array signal model. Thus, assuming collocated arrays, the received signal vector at the array output is given by: x(t) = K β k S(θ k, φ k ) S H (θ k, φ k ) m(t τ k ) + n(t) (.6) k= where β k is the complex fading coeffi cient associated with the k th target, which includes path losses and the target s radar cross section (RCS). An important point to note is that, unlike in the general SIMO model, the cross-correlations of the signals at the targets increase as targets move close together, even if transmit waveforms are orthogonal..5 The "Virtual Array" Concept in MIMO Radar A problem with the MIMO radar signal model given in Section.4 is that the degrees of freedom are inconventiently distributed amongst the transmitter and receiver. Since the transmit and receive arrays share a common phase origin, it would be desirable to devise a scheme whereby the whole system operated coherently. Specifically, it would be constructive to translate all modelled parameters at the transmitter across to the receiver (or vice versa), so their degrees of freedom could be combined in a collaborative manner. In the literature, we find that an attempt has been made to achieve this using the concept of the virtual array [9, 0]. However, we will see that this approach has crucial flaws that prevent it from being directly applicable to our theoretical framework at present. The MIMO radar virtual array concept relies on N equipower, orthogonal transmit waveforms that can be separated by N matched filters at the outputs of the N receiver antennas. Furthermore, it requires targets to be in the same range bin. In this case the received signal vector at the output of the array can be written as: x(t) = Sdiag(β) S H m(t) + n(t) (.7) where β [β, β,..., β K ] T. The columns of S and S are given by the K source position vectors

13 . Introduction 3 associated with the transmit and receive arrays, respectively. Thus, matched filtering yields: y(t) = E{vec[x(t) m H (t)]} = E { vec [( Sdiag(β) S H m(t) + n(t) ) m H (t) ]} (.8) Since transmit waveforms are orthogonal and equipowered, E{ m(t) m H (t)} = I N. Similarly, since noise is white, E{n(t) m H (t)} = 0 N N and so: { [ K ]} y(t) = E vec β k S(θ k, φ k ) S H (θ k, φ k ) (.9) k= This can be written in the form: { K } y(t) = E S v (θ k, φ k )β k k= (.0) where the manifold vector of the virtual array has been defined as: [ S v (θ k, φ k ) vec S(θ k, φ k ) S ] H (θ k, φ k ) (.) While the virtual array geometry described by the virtual manifold vector appears promising at first, the virtual array system as a whole has diffi culties. In particular, it is evident from Equation.0 that the virtual SIMO input signals are in fact given by the complex fading coeffi cients, estimated during the matched filtering process. So, immediately the assumption must be made that β will exhibit some degree of statistical variance from pulse to pulse (i.e. in slow time), otherwise no virtual signal power can be observed by the virtual array. However, even if this assumption is made (i.e. the Swerling II target model is assumed), analysis of the system is still not straightforward. For example, the finite sampling effect seems diffi cult to define in this context, since each estimate of β requires multiple snapshots, then each estimate of the covariance matrix of the virtual array output, R yy, requires multiple β estimates. Clearly, there are unresolved issues regarding the virtual array. However, since it offers a potentially huge increase in the number of degrees of freedom, gaining an in-depth understanding of the virtual array (or developing a more useful alternative) should be considered an important topic for future research.

14 Chapter Performance Bounds in Array Processing In general, the resolution performance of an array system is a function of array aperture and number of sensors, N. In practice, these resources are limited and so it is desirable to achieve high resolution performance without increasing the size of the array. Subspace-based techniques are particularly powerful in this respect, since they are capable of exhibiting asymptotically infinite resolving capability as the number of data snapshots, L, becomes large. In other words, they belong to the family of "superresolution" DF techniques. Since the number of snapshots available in practice is finite, the estimated statistics of the noise-contaminated received signal are imperfect. This finite sampling effect therefore imposes limits on system performance, even when the array is assumed to be calibrated. Specifically, theoretical bounds on three key aspects of DF performance arise:. Detection Performance: the capability of a system to correctly estimate the number of signals, M, impinging on the array.. Resolution Performance: the capability of a system to yield M separate, distinct directional parameter estimates corresponding to the M impinging signals. 3. Estimation Accuracy: the mean square error of the directional parameter estimates (which can only be obtained following successful detection and resolution), with respect to true target directions. 4

15 . Performance Bounds in Array Processing 5 In the case of detection and resolution, overall success depends particularly on the two most closely-spaced sources. Detection and resolution performance can therefore each be characterised by a different threshold separation, which must be satisfied in order for detection/resolution to be achieved with high probability. These thresholds are dependent upon various system parameters such as: signal-to-noise ratio (SNR), L, N, array geometry, source bearings, relative source powers, signal correlation and the specific practical DF algorithm employed. In this report, the roles of all these parameters will be explored in a general sense, except signal correlation; signals are assumed to be uncorrelated. However, this is considered an extremely important topic for future research and a preliminary discussion will be developed in Section 3.3. In order to explore performance bounds in this chapter, the concept of uncertainty hyperspheres will first be introduced as a means to characterise the uncertainty in the system due to the finite sampling effect. While Cramer-Rao bounds are used in the definition of uncertainty hyperspheres, these will not be discussed in detail until later (in order to preserve the logical order of: detection, then resolution, then estimation).. Uncertainty Hyperspheres and the Parameter C For a given array, the array manifold is defined as the locus of the manifold vectors for all (θ, φ) across the whole parameter space. In the presence of finite sampling effects, the uncertainty remaining in the system (corresponding to a given point on the manifold) after L snapshots can be represented using an N-dimensional hypersphere: It has been proven in [, Ch. 8, p. 99] N dim complex observation space Figure.: Visualisation of an uncertainty hypersphere in an N-dimensional complex space.

16 . Performance Bounds in Array Processing 6 that if the directional parameters, (θ, φ), are expressed as a function of the arc length of the manifold curve, then the radius, σ e, of the uncertainty hypersphere is given by the square root of the single-source Cramer-Rao Lower Bound expressed in terms of the arc length of the manifold: σ e = (SNR L) (.) This hypersphere therefore represents the smallest achievable uncertainty due to the presence of noise after L snapshots. In other words, this performance would be achieved by the theoretically ideal DOA estimation algorithm, whereby any inter-dependency between the various parameters of the multiple received signals (such as cross-correlation) have been somehow eliminated and no additional uncertainty has been introduced. For any non-ideal practical algorithm, this radius will be larger. To model this effect, the parameter C (where 0 C > ) is introduced, which acts to scale the hyperspheres accordingly: σ e = (SNR L) C (.) Clearly, if analytical expressions can be obtained for C for different practical algorithms, then C can be used as a useful figure-of-merit parameter to compare their performances. This could provide important insight in a number of ways. Firstly, it will give a clear indication of which algorithm is the superior for a given scenario (higher value of C denotes superiority). Secondly, if C is found to be close to, then it can immediately be concluded that the algorithm is near-ideal for that scenario. In other words, if system performance is still unsatisfactory, then there is no point in considering the use of a more complex algorithm; more favourable scenario parameters must be sought (for example, by increasing signal powers or the array aperture). Finally, since C contains all the non-idealities of a given algorithm (and only its non-idealities), the analytical form of C may provide some insight regarding the cause of these imperfections (and therefore how to eliminate them). In Chapter 3, several algorithm-specific expressions for C will be derived and discussed.

17 . Performance Bounds in Array Processing 7. Theoretical Detection Bounds Detection fails when the estimated number of signals impinging on the array falls below M. For a subspace-based method, this occurs when the dimensionality of the signal subspace falls below M. Detection threshold therefore occurs when two uncertainty hyperspheres just overlap (such that those two sources tend to contribute just one signal eigenvector). This geometrical scenario is shown in Figure.. Figure.: Detection threshold occurs when the uncertainty hyperspheres just touch. It is shown in [, Ch. detection: 8], that this geometrical model leads to the square root law for p det thr = ṡ( p +p ) (σ e + σ e ) (.3) where p represents a directional parameter, such as θ, φ or cone angles []. ṡ (p) is the rate of change of manifold arc length at point p and κ is the manifold s principal curvature (where ˆκ also takes into account the inclination angle of the manifold). For a parameterisation in terms of θ, these properties of the manifold are related to familiar system parameters as follows [3]: ṡ (θ) = π cos (φ) R θ (.4) κ (θ) R θ (.5) ˆκ (θ) = κ (θ) T R 3 N θ (.6)

18 (SNRxL) det,thr (db). Performance Bounds in Array Processing 8 where R θ ( r y cos (θ) r x sin (θ) ) and R θ R θ R θ. This allows us to compare ideal detection capability for a variety of array geometries and scenario parameters, as shown in Figure ULA Non uniform Linear Circular Y shaped Azimuth (degrees) Figure.3: Example comparing the ideal detection capabilities of various 5-element uniformlyspaced arrays (d r = λ/) as a function of azimuth. φ 0 = 30, P /P = 0.6, θ =..3 Theoretical Resolution Bounds Resolution fails when the number of directional parameter estimates falls below M. For a subspace-based method, this occurs when the number of intersections between the estimated signal subspace and the array manifold falls below M. The geometry of the resolution threshold scenario is shown in Figure.4, where the signal subspace associated with two sources first fails to form two distinct intersections with the array manifold. It is shown in [, Ch. resolution: 8], that this geometrical model leads to the fourth root law for p res thr = ṡ( p 4 4 +p ) ) (ˆκ ( σe + ) σ e N where κ is the manifold s principal curvature and ˆκ also takes into account the inclination angle of the manifold. Again, using the expressions given in [3], a comparison of various ideal resolution (.7)

19 . Performance Bounds in Array Processing 9 Figure.4: Resolution threshold occurs when the worst-case estimated signal subspace just touches the array manifold. capabilities for a variety of array geometries and scenario parameters is shown in Figure.5..4 Theoretical Estimation Error Bounds In the literature, a number of approaches can be found that seek to describe lower bounds on estimation accuracy. These generally involve a discussion of the estimated parameter vector s error covariance matrix (since, for unbiased estimators, mean square error and error variance are equal). Specifically, they seek to set a lower bound on the error covariance matrix of any estimate, ˆp, of the true parameter vector p [p, p,..., p M ] T. In our discussion, a deterministic signal model [4, 5] is assumed..4. Cramer-Rao Bounds The most popular estimation error bound in array processing is the Cramer-Rao Bound (CRB). It is a statistical result, based on the inversion of the appropriate Fischer information matrix with dimension equal to the total number of unknown parameters. Since, in the deterministic model, the unknown parameters consist of both parameters of interest (DOAs) and nuisance parameters (e.g. noise variance and complex signal amplitudes), we are actually only interested in a relatively small submatrix of the inverse Fischer information matrix. An explicit formulation of the relevant

20 (SNRxL) res,thr (db). Performance Bounds in Array Processing ULA Non uniform Linear Circular Y shaped Azimuth (degrees) Figure.5: Example comparing the ideal resolving capabilities of various 5-element uniformlyspaced arrays (d r = λ/) as a function of azimuth. φ 0 = 30, P /P = 0.6, θ =. submatrix was first introduced in [6] for the single parameter case (e.g. azimuth only), based on the following assumptions []: A: N > M and the manifold vectors are independent. A: Noise is a zero-mean, temporally white Gaussian process. A3: Noise is spatially white (from sensor to sensor). A4: All parameters other than p are known a priori. The exact Cramer-Rao lower bound on the covariance matrix of the unbiased estimate ˆp of parameter vector p is given as: CRB(p) = σ n ( L Re [ M H (t) HM (t) ]) t= where M (t) diag (m (t)) and H ṠH P S Ṡ, with P S defined as in the Notation section and Ṡ the matrix of manifold vectors differentiated with respect to p. However, a significantly more easily evaluated (and therefore popular) result is the asymptotic CRB for large L. That is : Note also that a simple extension to the multiple-parameter case (e.g. azimuth, elevation, range) was presented by Yau and Bresler [7] (also see [8, p. 53]) (.8)

21 . Performance Bounds in Array Processing ( [ CRB(p) = σ n Re H L ˆR mm]) T for suffi ciently large L (.9) Since these expressions involve a projection of Ṡ (i.e. the sensitivity of the manifold vectors to variations in p) onto the noise subspace, it can be observed that ultimate estimation accuracy will therefore increase as the Ṡ gradient vectors approach orthogonality to the signal subspace. This degree of orthogonality is determined by how steeply the array manifold varies for small changes in p about the direction of interest, p k. Thus, the shape of the array manifold is profoundly and fundamentally important in determining an array system s ultimate estimation accuracy. (This was clearly also the case for detection and resolution). In practice, we find (see e.g. [, Fig. 8.4]) that, as angular separation become suffi ciently large, the CRB in the multiple-emitter scenario approaches the equivalent single-emitter value. In other words, a wide range of scenarios can be characterised using just two expressions (given here in terms of geometric properties of the array manifold):. The single-emitter CRB: CRB [p ] = (SNR L)ṡ(p ) (.0). The -emitter CRB (for closely-spaced emitters): CRB [p A] = (SNR L) ṡ(p ) s (ˆκ ) (.) N where A is the array manifold. While most of this report considers only uncorrelated sources and leaves arbitrary complex correlation coeffi cient as a topic for future reseach, a new derivation will be presented here. A partial derivation of the -emitter CRB for correlated sources is given in [, p. 84]. In Appendix E, this proof is completed by considering the circular approximation to the array manifold. This yields the final result: CRB [p ] = (SNR L) ṡ (p ) 4 ( s) (ˆκ ( p) N ) ( ) (.) Re [ρ] P P

22 . Performance Bounds in Array Processing It is very interesting to note that this expression comprises three separate contributions: CRB [p ] = CRB [p ] G S whereby CRB [p ] is the single-source CRB. The real scalar G is a geometry term, reliant upon the shape of the array manifold in the neighbourhood of s. Finally, S is a signal term, dependent on the statistical properties of the source signals. Clearly, only the real part of the correlation coeffi cient effects the CRB, but this effect can be reduced by increasing signal powers (which is consistent with our intuition, when we consider the distribution of data snapshots on the signal subspace as a function of P, P and ρ)..4. Other Estimation Error Bounds Despite its widespread use in the literature, the CRB can be found to be somewhat inadequate in providing a reliable, tight bound. This is for two main reasons:. There exists some threshold (SNR L), below which the estimation accuracy deviates from its linear behaviour. The CRB fails to model this large error region (see Figure.6).. The "estimation error threshold" at which this occurs is not straightforward to predict, so it is diffi cult to be certain as to exactly what range of (SNR L) values the CRB is valid for. For this reason, estimation error bounds are generally divided into two classes that deal with each region separately: small-error bounds and large-error bounds (where the CRB is an example of a small-error bound). Other small-error bounds include the Bhattacharyya inequality (see [9] and references therein). The non-linear, high-error region is significantly more computationally complex to model. Indeed, the original Barankin Lower Bound has therefore seen a number of simplifications, such as the Chapman and Robbins, Hammersley and Kiefer approaches (and hybrids thereof). These bounds are discussed in more detail in [5, pp ] and references therein.

23 . Performance Bounds in Array Processing 3 Resolution Threshold MSE CRB Estimation Error Threshold No Information Region Large Error Region Asymptotic Region SNR x L (db) Figure.6: Typical MSE behaviour of a direction finding estimator as a function of (SNR L), compared with Cramer-Rao bound. (Adapted from: [9]). In a multiple-source scenario, the "No Information Region" refers to the sub-resolution-threshold region.

24 Chapter 3 Algorithm Comparison using the Figure of Merit C In Section., the parameter C was introduced as a figure of merit for comparing superresolution DF algorithm performance. In this chapter, various algorithm-specific analytical expressions for C will be derived by studying the resolution capabilities of each algorithm in context to the theoretically ideal. 3. Main Results By substituting the algorithm-specific resolution threshold expressions given in Appendix D into the expressions for ideal resolution performance derived in Appendices A and C and using the relationship derived in Appendix B, the main results of this chapter are now presented. Specifically, the C parameters for MUSIC, Minimum Norm and optimum Beamspace MUSIC are given in Equations : 4

25 3. Algorithm Comparison using the Figure of Merit C 5 C music = C min_norm C music = C opt_beamspace = (N ) [ + + L(N+)(πdr cos θ cos θ ) 60 ] (ULA with P = P ) (3.) 5 (N + ) (ULA with P = P ) (3.) ) 4 ( + 4 P P ( ) [ 8 3 P ( ) ] P (N ) + P L(ˆκ P ( p) N )( s) (3.3) ) 4 ( + 4 P P ( ) 8 3 P P [ ( 4 3 P P )(N ) ( ) + P L(ˆκ P ( p) N )( s) ( ) 4 3 P P ] (3.4) In these expressions, P P is the ratio of the two closely-spaced sources powers, d r is sensor spacing in units of λ, θ θ +θ is the midpoint between the two sources and s source separation in terms of manifold arc length. Note that Equations 3. and 3. apply only to a uniform linear array (ULA) and equipower sources, while Equations 3.3 and 3.4 are valid for unequal source powers and arbitrary array geometry. The approximate expression of Equation 3. is only valid for suffi ciently small separations (such that Equation 3. is dominated by the / (N ) term). 3. Discussion As discussed in [30], optimal Beamspace MUSIC is defined by the beamforming preprocessor, B opt, which maximises resolution performance. Applying the matrix B opt to the received data snapshots before any eigenspace-based technique (such as MUSIC or Minimum Norm) yields the same optimal resolution performance. Since B opt is independent of P P, it follows that the effect of on resolution performance will be the same for all eigenspace-based algorithms. Specifically, P P it is evident from Equations 3.3 and 3.4 that P P causes performance degradation relative to the Proof of these conditions and a detailed discussion of how to obtain B opt are given in [30].

26 3. Algorithm Comparison using the Figure of Merit C 6 equipower case, approximately given by: C eigenspace ) 4 ( + 4 P P ( ) C eigenspace 8 3 P given P P P = (3.5) which can be approximated by: C eigenspace ( ) 4 + P P C eigenspace 5 given P = P for P P From Equations , it can be seen that MUSIC (with arbitrary array geometry) and Minimum Norm (ULA geometry) can both exhibit near-ideal performance for 3-element arrays when P P =. However, optimal Beamspace MUSIC can also achieve near-ideal resolution performance for larger arrays (N > 3), when P P = and the following condition holds: L (ˆκ ( p) N 4 ) ( s) (3.6) Since, ( s) ṡ( θ) θ θ (which grows rapidly with increasing N), this condition generally holds for small L, N and θ θ. In Figure 3., the C parameters for the three algorithms are compared for ULAs with increasing numbers of sensors. In Figure 3., the variation of C as a function of azimuth is shown (where the shape of the plot depends on the array geometry).optimum Beamspace MUSIC clearly exhibits the best resolution performance, but its superiority is less outstanding for larger source separations. The same effect can also be observed for larger numbers of snapshots. A general trend is that these algorithms perform closer to the ideal when N, L and θ θ are restricted, but there may be considerable scope for improvement by future algorithms that can better utilise the greater resolving capacity of the system as these quantities increase. 3.3 Correlated Signals and Spatial Smoothing Our discussion so far has considered only uncorrelated signals. In practice, signals are often correlated (particularly as radar targets become closely-spaced) and it is well-known that this can

27 3. Algorithm Comparison using the Figure of Merit C 7 Figure 3.: C opt_beamspace, C min_norm and C music as a function of the number of sensors, N, for a ULA. In each case, θ = 34, θ = 35, P P = and L = 00. opt. Beamspace MUSIC Figure 3.: C opt_beamspace and C music for various source separations (for a 5-element uniform X-shaped array) with P P = and L = 00. Since C music is relatively insensitive to changes in θ θ, the separate traces cannot be distinguished.

28 3. Algorithm Comparison using the Figure of Merit C 8 severely degrade resolution performance. A popular method of decorrelating signals is spatial smoothing [3]. In [3], it is shown that, for two equipowered, fully-correlated (coherent) sources impinging on a ULA, the resolution performance of the forward-backward spatially-smoothed MUSIC algorithm is worse by a factor of approximately 4 (Nπd r cos θ cos θ ), compared to the standard, uncorrelated case. However, this result does not provide a great deal of insight into the effect of arbitrary complex correlation coeffi cient, ρ; like many results found in the literature, only the fully coherent signals problem is considered (motivated by the multipath propagation problem in mobile communications, where the primary concern is simply to restore the correct rank to the resulting covariance matrix). A major topic for future research related to this project will be to gain a comprehensive understanding of the impact of ρ on DF performance. As a part of this, it would be constructive to also fully understand the impact of employing a given decorrelating technique in a correlated signal environment. In the case of spatial smoothing, we find that there is a trade-off in that a decorrelation effect can be obtained at the expense of a reduced effective array aperture and with restrictions imposed on the usable array geometry. However, discussions in the literature on this topic seem to be particularly inaccessible. Specifically, there seems to be some confusion regarding the nature of valid subarray geometries and little attention seems to be paid to the analysis of non-coherent (partially correlated) signals. Therefore, an accessible basic analysis of conventional and backward spatial smoothing is presented in Appendix F in order to clarify a number of such points.

29 Chapter 4 Conclusions and Future Work 4. Conclusions In this report, the various MIMO radar configurations have been defined and compared. Using the collocated arrays configuration as a basis, a general discussion of parameter estimation in array processing was then developed. The theoretical performance bounds imposed by the finite sampling effect on an array system were defined and discussed in detail. The parameter C was proposed as a figure of merit for comparing superresolution DF algorithms. Representative example analytical expressions were derived for the MUSIC, optimal Beamspace MUSIC and Minimum Norm algorithms (for uncorrelated signals). It was found that, when sources have equal powers, all these algorithms can offer near-ideal resolution performance for 3-element arrays. However, only optimum Beamspace MUSIC can achieve this for larger arrays. Some preliminary work related to the analysis of complex correlation coeffi cient was presented. In particular, the -emitter CRB was derived for arbitrary ρ, and an accessible discussion of the spatial smoothing technique was presented. 4. Future Work In Section.5, a MIMO radar virtual array model was discussed and found to be somewhat flawed. However, the general concept of reformulating the MIMO signal model into an equivalent SIMO or MISO system remains a powerful one and should therefore be afforded considerable research 9

30 4. Conclusions and Future Work 30 effort. Only a very limited proportion of this report was related to the study of correlated sources. However, as discussed in Section 3.3, understanding the impact of signal correlation of DF performance is of paramount importance. One possible way to expand our theoretical framework to allow further insight into this problem could be to slacken the constraint on the definition of the uncertainty hypersphere s radius. That is, it should seek to characterise the performance of the algorithm which is ideal in every sense except with respect to non-zero correlation. There are several important scenario considerations (in addition to signal correlation) that have been neglected entirely in this report. Perhaps the most important is non-zero target velocity. Estimating target velocities by Doppler processing is often a key requirement in radar and so the system model will need to be changed in order to account for (and ultimately estimate) these frequency shifts. Another important consideration is clutter. In radar, signals don t just reflect back off targets, but also off clouds, the ground, or any number of objects that we are not interested in. Distinguishing clutter from targets and rejecting the clutter (whilst preserving as many degrees of freedom as possible for parameter estimation) is an important challenge. A further assumption that was made in this report was the narrowband signal assumption. However, wideband and ultra-wideband MIMO radar are attracting increasing research interest. This would require a somewhat radical reconsideration of the MIMO system model, since signal amplitudes and phases will tend to vary across the aperture of the array under the wideband assumption. Finally, perhaps the most promising topic for future research will be the development of novel collaborative techniques, whereby various transmitter characteristics (such as array geometry and waveform design) can be adaptively optimised depending on parameters estimated at the receiver. This process of feeding information back from the receiver to the transmitter could be a particularly powerful utilisation of the collocated MIMO radar configuration.

31 Appendix A Expressing Ideal Resolution Performance in Terms of ξ music ( s) In this appendix, numerous results are used from [, Ch. 8]. For notational compatibility, manifold vectors will therefore be denoted using a (s) here. The MUSIC [3] null spectrum is given by: ξ music (s) = a H (s) P A a (s) We wish to explore the value in the MUSIC null spectrum associated with the arc length s between the two sources at s and s. That is: ξ music ( s) = a H ( s) P Aa ( s) The geometric layout is shown in Figures A.-A.: = a H ( s) (I N P A ) a ( s) = N a H ( s) P A a ( s) Using these figures, expressions for the required inner product, a H ( s) P Aa ( s), will now be developed. In order to do this, a number of related geometric quantities, shown in Figure A., must first be derived: By first considering the entire (blue) right-angled triangle of Figure A., it is clear that: 3

32 A. Expressing Ideal Resolution Performance in Terms of ξ music ( s) 3 Centre of Curvature Origin Figure A.: Circular approximation of the array manifold, in the neighbourhood of s. Centre of Curvature Origin Figure A.: "Side-on" view, showing a ( s) and its orthogonal projection onto the signal subspace L ([a, a ]). cos (γ ) = ˆκ N Using this result, the cosine rule can now be applied to Figure A.3 to determine the angle γ between a ( s) and P A a ( s). First the length b is evaluated: b = ˆκ N ( cos ψ) + N ( cos ψ) cos (θ ) ˆκ = ˆκ ( χ χ ) + N

33 A. Expressing Ideal Resolution Performance in Terms of ξ music ( s) 33 Figure A.3: Geometric quantities required in the derivation of a H ( s) P A a ( s). where, for notational convenience, χ ( cos ψ). Now, applying the cosine rule again: ˆκ χ = ˆκ ( χ χ ) + N N ˆκ (χ χ) + N cos (γ ) ( ) N χ κ cos (γ ) = N ˆκ (χ χ) + N The inner product a H ( s) P A a ( s) can now be evaluated by considering the appropriate right angled triangle (including the yellow shaded region in Figure A.). That is, a H ( s) P A a ( s) = a H ( s) PA a ( s) cos γ = N cos γ and so: a H ( s) P Aa ( s) = a H ( s) (I N P A ) a ( s) = N N cos γ ( ) N χ κ = N ( ) (χ χ) + N By separately evaluating the numerator and the denominator, it is relatively straightforward to show: [( ) N + ˆκ N ˆκ 8 (cos ψ ) a ] a H ( s) P Aa ( s) = N N a 4 Following a series of straightforward but tedious manipulations, this reduces to: a H ( s) P Aa ( s) = ( ˆκ N ˆκ ) N ˆκ 4 (cos ψ ) ) (N a 4

34 A. Expressing Ideal Resolution Performance in Terms of ξ music ( s) 34 At this stage, the following approximations are used: Approximation : cos ψ = 4 a ˆκ ( ) 8 a ˆκ = 8 a ˆκ Assumption used: 64 a 4 ˆκ 4 4 a ˆκ a ˆκ 6 Note : Since κ = r, where r = [, Table 8., p. 86], it follows that κ. Note : Since ˆκ = κ sin ζ [, Eq. 8.], ˆκ κ ). Approximation : Assumption used: ( N a 4 ) N N a 4 Therefore, a H ( s) P Aa ( s) = = ( ˆκ ) ( N ˆκ 4 8 a ˆκ ) N N ( ˆκ ) ( ) N 8 a ) (ˆκ N ( s) 4 64 (A.)

35 A. Expressing Ideal Resolution Performance in Terms of ξ music ( s) 35 Now, substituting into Equation D. yields: which concludes the proof. (SNR L) res = ( + 4 P P ) 4 3ξ music ( θ) C (A.)

36 Appendix B Lee and Wengrovitz s Angular Separation Measure In this appendix, it is proven that the angular separation measure,, given by Lee and Wengrovitz in [30] is, in fact, a scaled small-angle approximation to the change in manifold arc length, s. The parameter is given as: k N N [ r T i (u(θ ) u(θ )) ] i= = N r (k(θ ) k(θ )) We know k(θ ) k(θ ) = π [cos φ(cos θ cos θ ), cos φ(sin θ sin θ ), 0] T and so: = π N cos φ N [r xi (cos θ cos θ ) + r yi (sin θ sin θ )] i= = π N cos φ[ r x (cos θ cos θ ) + ry (sin θ sin θ ) For suffi ciently small θ θ, we have: cos θ cos θ sin(θ m ) θ θ sin θ sin θ cos(θ m ) θ θ 36 + r T x r y (cos θ cos θ )(sin θ sin θ )

37 B. Lee and Wengrovitz s Angular Separation Measure 37 Therefore: π N cos φ[ r x (sin(θ m ) θ θ ) + ry (cos(θm ) θ θ ) r T x r y (sin(θ m ) θ θ )(cos(θ m ) θ θ )] [ = π N cos φ θ θ r x sin(θ m ) + ] ry cos(θm ) r T x r y sin(θ m ) cos(θ m ) θ θ ṡ(θ m ) N N s for suffi ciently small θ θ which concludes the proof.

38 Appendix C Ideal Resolution Performance for ULA For a ULA with its phase centre taken to be its centroid (with antenna spacing d r ), the sensor locations are given by: r x = d r [ (N ) (N ), +,..., Using the method of finite differences, it is easily shown that: ] T (N ) (C.) r x = Then, using this result, we similarly find: d r (N 3 N) κ = r x r x r x r x = 3(3N 7) 5(N 3 N) (C.) (C.3) For symmetric linear arrays, ˆκ = κ. Substituting Equations C. and C.3 into Equation D. and setting P P = then yields: (SNR L) res = 5760 (πd r cos θ cos θ ) 4 N (N ) (N 4) C (C.4) 38