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 bear- ing 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 = ∅.