Atomic-decomposition-based parameter estimation of multiple overlapped signals J. Liu, H. Meng, Y. Liu and X. Wang A novel algorithm is presented for atomic decomposition (AD)-based parameter estimation of multiple signals that are partially overlapped in the time domain. By using a subspace orthogonal matching pursuit technique, the proposed algorithm can overcome resolution problems and obtain higher parameter estimation accuracy than other commonly employed AD algorithms, making it quite suitable for the detection and estimation of multiple signals. Introduction: Atomic decomposition (AD) [1] refers to a class of adap- tive approximation techniques that provide a sparse, flexible and phys- ically meaningful representation of a signal with a set of well- localised time–frequency functions called atoms. Matching pursuit (MP) [1], the most commonly used AD method, iteratively extracts atoms that are the most strongly correlated with the signal. Unfortunately, MP is inherently a greedy technique that has poor res- olution when multiple signals partially overlap in time [2, 3]. In this case, the features of the extracted atoms can hardly describe signal com- ponents. To overcome this limitation, the greediness detector (GD) algorithm [2, 3] is employed to check the correctness of every atom extracted by MP. GD is able to resolve close time-spaced signals, but it is only applicable at high ENR (signal-energy-to-noise-power-ratio). Hence, a novel method, referred to as subspace orthogonal matching pursuit (SOMP), is presented in this Letter. SOMP is based on a sub- space orthogonal projection technique and can distinguish multiple close signals even at low ENR. Signal model: Let D ¼ {h g (n)} be a highly redundant dictionary of unit- energy atoms and x(n) be the signal under analysis, where n ¼ 1, ..., N, N is the length of the samples. AD can represent the signal by the expan- sion of atoms x(n)= ∑ p b p h g p (n)+ r(n) p = 1, 2, ... (1) Coefficient b p is usually a complex number, and r(n) is zero-mean complex white Gaussian noise of power s 2 . The dictionary is composed of chirplets, because linear frequency modulation is very common for radar signals, and chirplets exhibit good time–frequency concentration [2]. The basic chirplet function is parameterised through the four- component vector g =[a, b, T , f ] T [ {G = R + × R 3 } as h g (n)= a p   1/4 e −a/2(n−T ) 2 e j[2pf (n−T )+pb(n−T ) 2 (2) where a is inversely related to the signal duration d, d =  2/a √ . T, f and b are the chirplet’s mean time, mean frequency and chirp rate, respectively. The atom parameters are discretised to obtain K atoms, as in [1]. SOMP algorithm: When several signals are close in time, MP always extracts a large-scale (i.e. long duration) atom that absorbs the most energy. This large-scale atom is defined as a greedy atom [2, 3], and this phenomenon is called ‘overfitting’. Hence, it is natural to consider that, if we can construct a subspace spanned by both the greedy atom (large-scale) and the true atoms (small-scale) corresponding to the signal components, the projection of signals on the subspace obtained by the least squares (LS) method can tend towards the true atoms but not concentrate on the greedy atom. Based on this consideration, a sub- space extraction method is first proposed, in which the selected atoms to span a subspace at each iteration should satisfy the following conditions. First, in order to guarantee the performance of projection, once an atom exists in a subspace, other atoms with similar scales and strong corre- lation to it should be excluded from the subspace. Moreover, besides the similar scale atoms, the subspace should also include different scale atoms, while the correlation between them has no limitation. The subspace extraction method can be summarised as follows: 1. x p (n) represents the pth signal residual, and x 0 (n) ¼ x(n). Set q ¼ 1 and the correlation threshold Th, scale similarity p, subspace atom number m. Calculate the inner product module between x p (n) and every atom as r k =|kx p (n), h g k (n)l| k = 1, 2, ... , K (3) Inner product k . , . l is defined as kx(n), y(n)l = ∑ n x(n)y ∗ (n). 2. Choose the best correlated atom with x p (n) as k q = arg max k {r k }. 3. Eliminate atoms with similar scales and strong correlation to h g kq : r k = 0, if 1 r a kq ≤ a k , ra kq and |kh g kq (n), h g k (n)l| ≥ Th k = 1, 2, ... , K (4) 4. Set q ¼ q + 1, repeat steps 2–4 to determine k 1 , k 2 , ..., k m and construct subspace as S =[h g k 1 h g k 2 ... h g km ]. The description above explains how to extract a subspace, and the SOMP algorithm for close signal representation will now be elaborated: 1. Set p ¼ 1, support set V 0 ¼ f and parameter 1 to end the iteration. 2. Construct subspace S p =[h g k 1 h g k 2 ... h g km for the signal residual x p−1 (n) by the method stated above. 3. Calculate the orthogonal projection of x p−l (n) on S p by LS: (c 1 , c 2 , ... , c m ) T =(S H p S p ) −1 S H p x p−1 (5) 4. Choose the atom h g gp with the maximal projection coefficient module as g p = arg max k {|c k |, k = 1, 2, ... , m} (6) and merge it into the support set V p =[V p−1 , h g gp ]. 5. Define P Vp = V p (V H p V p ) −1 V H p as the projection matrix. Calculate x p (n) by projecting x(n) onto the orthogonal complement of V p as x p =(I − P Vp )x = x − V p (V H p V p ) −1 V H p x (7) 6. Determine the terminal condition. If ‖x p (n)‖ 2 2 ≤ 1‖x(n)‖ 2 2 , the algorithm ends with the extracted atoms {h g g 1 , h g g 2 , ... , h g gp }; other- wise, set p = p + 1 and repeat steps 2–6 until the terminal condition is achieved. Simulation results: In this Section, the same two signals as in [2, 3] are considered, which consist of two chirplets of unit energy, with the same scale and frequency, no chirp rate and close in time, as shown in Fig. 1. The first atom extracted by MP (i.e. the greedy atom) is shown as well. Figs. 2 and 3 illustrate the root-mean-square-errors (RMSEs) of the parameter estimates, such as a 2 and T 1 , against ENR, for 1000 Monte-Carlo trials. ENR (¼|b| 2 /s 2 ) [2, 3] is usually used in the relevant literature, because it does not depend on signal parameters. The Cramer- Rao bound, calculated as in [4], is also depicted as a benchmark. In GD, r = 100, Th = 5, m = 10 −4 and 1 = 0.05 [2, 3]. In SOMP, m = 3, r = 5, Th = 0.2 and 1 = 0.05. Simulation results show that MP always extracts the greedy atom first, and this results in a rather poor estimation performance. Both GD and SOMP can obtain good parameter estimates at high ENR. However, at low ENR (,25 dB), the estimation accuracy of GD is low, because the first extracted atom by GD is the greedy atom and the determination of its greediness, in this case, is wrong. It further emphasises that GD is only effective at high ENR. Nevertheless, for SOMP, the estimates are quite robust in noisy environments and the greedy atom is not extracted, even at low ENR (.13 dB). The effectiveness of SOMP can be explained as follows. LS can give rise to the subspace projection that minimises the distance to the signal as in (5), whereas the inner product only reflects the correlation between the signal and the atoms. By overcoming the overfitting problem caused mainly by the simple use of the inner product, the subspace projection by LS can reflect the real structure of the signal components. As noise increases, the projections of noise on different atoms in the subspace are approximately equivalent, so that noise will not destroy the relative amplitude of the projection coeffi- cients. Therefore, the SOMP algorithm is effective in noisy environments. ELECTRONICS LETTERS 8th July 2010 Vol. 46 No. 14