Maximum Likelihood Signal Parameter Estimation Via Track Before Detect

Maximum Likelihood Signal Parameter Estimation via Track Before Detect

Murat Uney¨ and Bernard Mulgrew Daniel Clark Institute for Digital Communications, School of Engineering, School of Eng.& Physical Sciences, The University of Edinburgh, Heriot-Watt University EH9 3JL, Edinburgh, UK EH14 4AS, Edinburgh, UK Emails: {M.Uney, B.Mulgrew }@ed.ac.uk Email: [email protected]

Abstract—In this work, we consider the front-end processing ciated with different range-bearing bins. These estimates are for an active sensor. We are interested in estimating signal then used as design parameters in constant-false-alarm rate amplitude and noise power based on the outputs from filters detection algorithms [3]. However, this approach is prone to that match transmitted waveforms at different ranges and bearing angles. These parameters identify the distributions in, for errors due to incorrect identification of bins that contain only example, likelihood ratio tests used by detection algorithms and noise or signal-and-noise as a result of that this identification characterise the probability of detection and false alarm rates. task requires the temporal information in the received signal Because they are observed through measurements induced by a which is ignored. Estimation of the signal energy E requires (hidden) target process, the associated parameter likelihood has a collection of temporal samples, as well. One way of doing this time recursive structure which involves estimation of the target state based on the filter outputs. We use a track-before-detect is to replace the MF bank in the basedband processing chain scheme for maintaining a Bernoulli target model and updating with iterative processing algorithms preferably working with the parameter likelihood. We use a maximum likelihood strategy high sampling rates (e.g., [4], [5]). This places requirements and demonstrate the efficacy of the proposed approach with an on the hardware architecture that are hard to satisfy in practice. example. In this work, we use multiple snapshots from the MF bank I.INTRODUCTION collected across a time interval and perform spatio-temporal Active sensors send energy packets towards a surveillance processing to jointly estimate E and β2. We treat the problem region in order to locate objects within from the reflections. as a parameter estimation problem in state space models. This For example, radars transmit radio frequency (RF) electro- allows us to integrate all the information in the measurements magnetic (EM) pulses and locate reflectors by searching for in a single likelihood function for both E and β2. Such likeli- the pulse waveform in the spatio-temporal energy content of hood functions require the state distribution of the underlying the received signal. This search is often performed using the (target) process given the measurement history, which can matched filtering technique in which the received signal is be found by the prediction stage of Bayesian filtering (or, projected onto versions of the transmitted waveform shifted tracking) recursions [6]. As the measurements are the MF so as to encode the desired reflector locations [1]. The more outputs (as opposed to the outputs from a detection algorithm the energy a projection has, the more likely that it is due to as in widely studied tracking scenarios), the corresponding the presence of a reflector. recursions describe a track-before-detect algorithm. In this work, we are interested in estimating parameters In this framework, we derive explicit formulae for the related to the signal at the output of the matched filters (MF) parameter likelihood and its score function, i.e., the log- of a radar. This signal is composed of a distorted version likelihood gradient. We use a maximum likelihood (ML) of the waveform auto-correlation function in the presence of approach to design an unbiased and minimum variance esti- a reflector and additive thermal noise. Detection algorithms mator. In particular, we maximise the log-likelihood by using aim to decide on the existence of an object based on these a coordinate ascent algorithm [7] in which we select the outputs sampled at selected time instances so as to give directions of increase based on the gradient and perform the energy of the aforementioned projected signal [2]. These (golden section) line search along these directions. decisions are characterised by a probability of detection and This article is outlined as follows: We give the ML problem a probability of false alarm which can be found given the definition in Sec. II and detail the parameter likelihood in energy of the reflected pulse at the receiver front-end E, and, Sec. III. In Sec. IV, we derive the gradient of the objective the noise power β2 (or, the standard deviation of the noise function and detail an iterative optimisation procedure. We process). These parameters also determine the signal-to-noise demonstrate the proposed approach with an example in Sec. V. ratio which can be used to characterise the expected accuracy II. PROBLEM DEFINITION in further levels of processing [1]. Often only the noise power β2 is estimated using spatial We consider a pulse transmitted towards a surveillance windows over one snapshot of the outputs from MFs asso- region which gets reflected if it interacts with an object at state T T T x = [xl , xv ] where xl is the location, xv =x ˙ l is the velocity time step 1 through k. Therefore, the ML estimation problem of the object and (.)T denotes the transpose of a vector. These can be formulated as reflections are sought in the the spatio-temporal energy content 2 2 (E,ˆ βˆ ) = argmax log l(z1, · · · , zk|E,β ), (5) of the received signal by matched filtering [1]. Typically, the E,β2 filter output is sampled with a period of the pulse length so as the computation of which will be described next. to compute the correlation of the transmitted waveform with the received signal corresponding to the ith range bin of width III. THE SIGNAL PARAMETER LIKELIHOOD th ∆r and j bearing bin of width ∆φ. Therefore, at time step Let us represent with a random set Xk, the events that there k, the filter output for the bin (i, j) is given by exists a reflector with state xk, and, none, in which cases X = {x } and X = ∅, respectively. X is referred to r (i, j)=< m(i, j), r > (1) k k k k k as a Bernoulli random finite set (RFS) [8]. The likelihoods 2 where r is the (complex) received signal and m(i, j) represents for zk(i, j) given the signal parameters to be estimated E,β th the transmitted waveform shifted to the (i, j) bin. for the cases that Xk = {xk} and Xk = ∅ are well known Let Et represent the energy of the transmitted pulse, i.e., results in the literature: After the uniformly distributed phase H H Et = m m where (.) denotes the Hermitian transpose. In is marginalised-out, the modulus zk(i, j) is distributed with a the presence of a reflecting object at state xk, the inner product Rician distribution, i.e., above leads to 2 l1(zk(i, j)|xk; E,β ) jθk 2 rk(i, j)= Ete hi,j (xk)+ nk (2) , p(zk(i, j)|Xk = {xk}; E,β ) 2 2 2 where nk ∼ N (.;0,β ) is circularly symmetric complex 2zk(i, j) zk(i, j) + E 2zk(i, j)E 2 = exp − I0 ,(6) Gaussian noise with complex power β , hi,j (xk) specifies the β2 β2 β2 ratio of E that has been reflected from the (i, j)th bin with if X = {x } and x ∈ C−1(i, j), where I is the zero order (an unknown) phase θ ∼U(0, 2π]. k k k 0 k modified Bessel function of the first kind. Otherwise, z (i, j) In this work, we assume a sensor resolution such that an k follows a Rayleigh law given by [1, Chp.6] object in the surveillance region affects only a single range- 2 2 bearing bin, i.e., l0(zk(i, j)|β ) , p(zk(i, j)|Xk = ∅; E,β ) 2 2zk(i, j) zk(i, j) hi,j (xk)= HδC(xk),(i,j) (3) = exp − . (7) β2 β2 where δ is the Kronecker’s delta function, C : X → M × N Let us define the intensity map z related quantities: maps object states to range-bearing bins and H is the reflection k coefficient. 2 2 Λ(zk|xk,E,β ) , l1(zk(C(xk))|xk; E,β ) We consider (2) and (3), and, are interested in estimating 2 2 l0(zk(i, j)|β ), (8) the received signal energy E , EtH, and, the noise power β . These signal parameters determine the signal-to-noise ratio at (i,j)Y∈C(xk) the matched filter by where C(xk) denotes the set of range-bearing bins comple- 2 E menting C(xk), and, l0 and l1 are given in (6) and (7), SNR = 10log , (4) 10 β2 respectively. Similarly 2 2 and are also required to compute false alarm rates and object Λ(zk|β ) , l0(zk(i, j)|β ) (9) detection probabilities of threshold rules [1]. i,j Y We treat these parameters as (non-random) unknown con- where the product is over all the range-bearing bins. stants and consider an ML solution. We now specify the We assume that the noise processes for different bins and arguments of the likelihood function: The reflection phase θk time steps are independent given the state of the object process in (2) models the ambiguity related to the exact position of Xk. Hence, the reflector within the (i, j)th bin which cannot be mitigated. 2 Λ(zk|xk,E,β ), if Xk = {xk} The modulus of (2), i.e., p(z |X ,E,β2)= (10) k k z 2 (Λ( k|β ), if Xk = ∅. zk(i, j) , |rk(i, j)| We would like to compute the parameter likelihood based neglects the phase and is a sufficient statistic when testing on all measurements covering time step 1 through k which whether rk(i, j) is induced by a reflector at state xk or noise can be found as alone. Therefore, we treat the intensity map given by zk(i, j)s k as measurements based on which the ML estimation will be 2 2 l(z , · · · , zk|E,β )= p(zt|z t ,E,β ) relying upon. 1 1: −1 t=1 Let us denote with z the concatenation of z (i, j)s for all Y k k = l(z , · · · , z |E,β2) range-bearing bins, i.e., the intensity map. We would like to 1 k−1 z z 2 base the parameter likelihood on all measurements covering ×p( k| 1:k−1,E,β ) after using the chain rule of probabilities in the first line. The an object that existed at k − 1 continuing to exist at time k is recursive structure revealed in the second line is typical to given by PS . A Bernoulli model at time k−1 characterised by parameter estimation problems in state space models [9], [10], (rk−1,sk−1(xk−1)), then, leads to the following prediction: in which the update term is found by marginalising-out the rk|k−1 = Pb(1 − rk−1)+ rk−1PS underlying process Xk, i.e., Pb(1 − rk−1) z z 2 sk|k−1(x)= b(x) p( k| 1:k−1,E,β )= rk|k−1 2 2 rk−1PS p(zk|Xk,E,β )p(Xk|z1:k−1,E,β )δXk (11) + π (x|x )s (x )dx (16) r k|k−1 k−1 k−1 k−1 k−1 Z k|k−1 Z where the right hand side is a set integral [8, Chp.11] as Xk where πk|k−1 is the state transition density [11]. is a set random variable. The first term inside the integral is Upon receiving zk, the posterior model is given by [11] the likelihood at k given by (10), and, the second term is the 2 prediction of Xk based on the previous measurements. rk|k−1 gk(xk|E,β )dxk rk = 2 The objective function for the ML solution in (5), hence, is 1 − rk|k−1 + rRk|k−1 gk(xk|E,β )dxk 2 given by gk(xk|E,β ) sk(xk)= R 2 2 2 2 J(E,β ; z1:k)= J(E,β ; z1:k−1)+log p(zk|z1:k−1,E,β ). gk(xk|E,β )dxk (12) l (z (C(x ))|x ; E,β2) 2 , R1 k k k (17) gk(xk|E,β ) 2 sk|k−1(xk). Let us now consider the likelihood update term in (11). l0(zk(C(xk))|β ) The computation of the predictive term inside the integral As a result, the predictive term in (13) required for the is detailed later in Section III-A. For our discussion on the parameter likelihood is found by iterating prediction and likelihood update term, suppose that it is given by update cycles using (16) and (17). r s (x ), if X = {x } The object birth model we use is selected so as to have z 2 k|k−1 k|k−1 k k k p(Xk| 1:k−1,E,β )= a uniform distribution in the location component which is (1 − rk|k−1, if Xk = ∅. (13) nonzero in the sensor field of view, and, a uniform distribution Then, (11) expands using the set integration rule in [8, Chp.11] in the velocity component which is nonzero if the speed is with a Bernoulli RFS characterised by (13) and the likelihood between selected minimum and maximum values: in (10) as T T T b([xl , xv ] )= UF OV (xl)Uvmin≤|xv|≤vmax (xv ). (18) z z 2 z 2 z 2 p( k| 1:k−1,E,β )= p( k|∅,E,β )p(∅| 1:k−1,E,β ) IV. MAXIMUM LIKELIHOOD SIGNAL AMPLITUDE AND 2 2 NOISE POWER ESTIMATION + p(zk|Xk = {xk},E,β )p(Xk = {xk}|z1:k−1,E,β )dxk The objective function of the ML solution in (5) is not Z z 2 = (1 − rk|k−1)Λ( k|β ) straightforward to maximise partly because, in practice, only a 2 noisy approximation of it can be obtained via particle methods, +rk k Λ(zk|xk,E,β )sk k (xk)dxk. (14) | −1 | −1 and, the term due to E in the right hand side (RHS) of (15) is Z 2 2 dominated by the influence of β through the first term (see, After dividing both parts of the equation above by Λ(zk|β ), it can easily be shown that e.g., the example in Section V). Our ML realisation strategy is to apply coordinate ascent 2 2 log p(zk|z1:k−1,E,β ) = logΛ(zk|β )+ along the projections of the log-likelihood gradient. This can be viewed as a subgradient approach with the difference that log 1 − r + k|k−1 we do not explicitly select step sizes to move along the l (z (C(x ))|x ; E,β2) r 1 k k k s (x )dx (15) selected subgradient direction. Instead, we perform (golden k|k−1 l (z (C(x ))|β2) k|k−1 k k Z 0 k k ratio) line search along the subgradient direction. The gradient Therefore, the ML objective can be recursively evaluated using equals the sum of the gradients of the log update term in (15) (15) in (12). Next, we discuss the computation of the predictive over time. Note that, the first term in the RHS of (15) is state distribution in (13) using Bernoulli track before detect. independent of E, hence,

2 ∂ log l1 l1 A. Bernoulli track before detect ∂ log p(z |z ,E,β ) rk|k−1 sk|k−1 k 1:k−1 = ∂E l0 ∂E 1 − r + r l1 s Recursive updating of a Bernoulli object model using k|k−1R k|k−1 l0 k|k−1 measurements with likelihood models in the form of (10) ∂ log l1 rk|k−1 gkR can be carried out using the Bayesian recursive filtering = ∂E (19) 1 − r + r g principles [11], [12]. An important component of filtering with k|k−1R k|k−1 k RFS models is the object appearance, or, birth model. The where the first line follows after differentiating theR term inside probability of object appearance at time k is given by Pb and the integral and using the identity ∂l1/∂E = l1 ×∂ log l1/∂E. the state of this object is distributed as b(x). The probability of The second line is obtained through the definition of gk in (17). 6 The partial derivative of log l with respect to E can easily be x 10 1 1 found [13] as 1 1 ) 2

β 0 0.8 0.8 ∂ log l 2E 2z (C(x )) I (2z (C(x ))E/β2) |E, 1 k k 1 k k 1:k = − + (20) 0.6 0.6 2 2 2 −1 ∂E β β I0(2zk(C(xk))E/β ) log l(z 0.4 0.4

−2 0.2 where I0 and I1 are the modified Bessel functions of the first 1.5 0.2 2 0.2 0.1 0.15 2.5 0.05 0 0.05 0.1 0.15 0.2 0.25 kind of order zero and one, respectively. After substituting (20) 2 1.5 2 2.5 2 E β E β in (19),we get (a) (b) (c) 2 Fig. 1. (a) Log-likelihood surface for the example scenario evaluated over ∂ log p(zk|z1:k−1,E,β ) 2Erk the grid . ≤ E ≤ . and . ≤ β2 . with step sizes of . and = − 2 + 1 5 2 5 0 01 0 25 0 05 ∂E β 0.01, respectively. (b) The normalised profile of the surface along E-axis for 2 2 zkI1 varying β . (c) The normalised profile along β for varying E. 2rk|k−1 2 gk β I0 , (21) 1 − rk|k−1 +R rk|k−1 gk using golden section search [14]. where rk and gk are given in (17). R The iterations terminate if two consecutive points are closer Next, we consider the partial derivative of the log update than a selected tolerance value. In other words, the solution 2 ˆ ˆ2 2 term with respect to β : to (5) is declared as (E, β ) = (Em+1,βm+1) if 2 2 T 2 T ∂ log p(zk|z1:k−1,E,β ) [Em+1,βm+1] − [Em,βm] < δ. 2 = ∂β The computations are carried out using particle methods:

∂ log l1 ∂ log l0 We sample from the predictive distribution in (13) using a ∂ log l rk|k−1 ∂β2 − ∂β2 gk 0 + (22) Sequential Monte Carlo realisation [11] of the recursions given ∂β2 1 − r + r g i,j R k|k−1 k|Rk−1 k by (16) and (17). In order to sample from the birth model X −1 The first partial derivative inside the integralR above can be in (18), we find a grid of L samples for each bin C (i, j) found [13] as and concatenate with velocity components generated from 2 2 2 Uvmin≤|xv|≤vmax . The integrals in (19)–(23) are estimated ∂ log l1 1 z + E 2zkE I1(2zkE/β ) = − 1 − k + , using samples generated from gk during Bernoulli track before ∂β2 β2 β2 β2 I (2z E/β2) 0 k detect within the Monte Carlo method [15, Chp.3]. and the derivative of log-noise term with respect to the noise V. EXAMPLE power is given by Let us consider an example scenario consisting of an object ∂ log l 1 z2 T 0 = − + k . with initial state x0 = [−503.5, 4974.6, 5, 0] moving in ∂β2 β2 β4 accordance with a constant velocity motion model with a small Therefore, the first term inside the integral in (22) is given by process noise term and a sensor located at the origin with range and bearing resolutions of 10m and 1◦, respectively. ∂ log l ∂ log l E2 2Ez I 1 0 k 1 (23) The sensor field of view is selected as the region bounded 2 − 2 = 4 − 4 . ∂β ∂β β β I0 by 4800m and 5200m in range and ±10◦ around the y-axis As a result, the gradient of the log-likelihood function in order o restrict the size of the intensity map and hence the in (12) can be computed using (19)–(23) in the recursive form volume of computations needed. As a result, a 40×20 intensity given by map is observed for k = 200 time steps with signal amplitude 2 2 ∂ log p(zk |z1:k−1,E,β ) E = 2 and noise power β = 0.1. The target signal, hence, 2 2 ∂E z z 2 has SNR in the corresponding bin. ∇J(E,β ; 1:k)= ∇J(E,β ; 1:k−1)+ ∂ log p(zk |z1:k−1,E,β ) ∼ 16dB " ∂β2 # In Fig. 1(a), we present the ML objective surface evaluated (24) for a given realisation of the measurement history z1:200 and a In order to maximise J, we adopt a coordinate ascent grid of (E,β2) values using the parameter likelihood detailed 2 approach: Starting from an initial point (E0,β0 ), at each in Section III together with MC computations. For Bernoulli iteration m, we find the gradient vector (24) and select a line track before detect, we use L = 225 uniform grid points for search direction as follows: each of the 800 state bins given by C−1(i, j) to represent b(x) 2 T ∇J(E,β ; z1:k) em in (16). Here, π is selected as a constant velocity motion model (25) dm = 2 T k∇J(E,β ; z1:k) emk with a small additive process noise term. We maintain 10000 [0, 1]T , if m =0, 2, 4, ... samples generated from sk in (17) after resampling weighted em = samples from gk. The probability of birth Pb =0.01 whereas [1, 0]T , if m =1, 3, 4, ... ( the PS =0.99. Then, we solve the following one dimensional problem There are much less target associated measurements con- tributing to (15) compared to noise assoicated terms. This 2 2 T z (Em+1,βm+1) = arg min J([Em,βm] + λdm; 1:k) λ∈[0,λmax] manifests itself in Fig. 1(a) as the significantly smaller cur- (26) vature of the surface along E compared to that along β2. 0.25 There is a tradeoff between the observation length and the accuracy in which E can be estimated (see, e.g., [17]). 0.2 The relations between the Hessian, Fisher information and the 0.15 2 associated Cramer-Rao lower bound can be explored in order β

0.1 to investigate this tradeoff. Another possible extension of this work is to accommodate multi-Bernoulli models in order to 0.05 handle multiple moving objects via track-before-detect [18].

1 1.5 2 2.5 Esimation of parameters for other Swerling target types [1] E within the proposed ML framework is also left as future work. Fig. 2. Iterative maximisation of the parameter likelihood for 200 Monte Carlo runs: Initial point (blue circle), and converged estimates (red crosses) ACKNOWLEDGMENT joined by estimates in the intermediate steps (green crosses and dashed lines). Diamond is the correct value of (E,β2). This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) grants EP/J015180/1 and Correspondingly, the Cramer-Rao lower bound (CRLB) for EP/K014277/1, and the MOD University Defence Research E is higher and estimates of E will be less accurate [16, Collaboration in Signal Processing. App. 8A]. For a closer look to the log-likelihood objective function, we provide the normalised profiles of the surface REFERENCES 2 along E and β axis in Fig. 1(b) and (c), respectively. Note [1] M. A. Richards, Fundamentals of Radar Signal Processing. 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