Master Thesis Separation of Non-Stationary Sources

Transcription

1 Master Thesis Separation of Non-Stationary Sources Lisbeth Andersson Perth, March 21 Australian Telecommunications Research Institute Curtin University of Technology Australia

2 i Preface This master thesis project is part of the Master of Science degree, in Electrical Engineering, at the Royal Institute of Technology (Kungliga Tekniska Högskolan, KTH) in Stockholm, Sweden. The work has been conducted for the Australian Telecommunications Research Institute in Perth, Australia, under the supervision of Prof. A. Zoubir. The project will be examined by Assoc. Prof. S. Parkvall and, Ph.D. Student B. Völcker, who is also an assistant supervisor.

3 ii Acknowledgment I would like to thank my supervisor Prof. A. Zoubir for his valuable support and guidance throughout the project. I would also like to thank Prof. S. Nordholm, Dr. P. Pelin, and Assoc. Prof. M. Skoglund, for helping me make my dream of doing my master thesis project in beautiful Australia come true, and Ph.D. Student B. Völcker, for his valuable support. Further, I wish to express my gratitude to all my colleagues and friends at the CSP group, ATRI and ATcrc, for the comfortable atmosphere they all have contributed to creating. Lisbeth Andersson Perth, February 21

4 iii Abstract This master thesis deals with the problem of source separation of linear instantaneous mixtures of non-stationary, polynomial-phased signals received by a uniform linear antenna array. The main goal with source separation is to recover the source signals and to estimate the mixing matrix, which is the transfer function between the source signals and the signals received by the array sensors. This can then be used to estimate the direction of arrival of the source signals. Up to date, mostly non-parametric attempts have been made. The difficulty with nonparametric methods is that they either need to know if a selected time-frequency point corresponds to a signal autoterm or crossterm, and/or they are simply too computationally expensive. This thesis suggests a parametric approach using the discrete polynomial-phase transform (DPPT) to solve the problem. The discrete polynomial-phase transform is nonlinear, and also a computationally efficient, technique. The parametric method avoids the selection of time-frequency points, and no time-frequency distributions need to be calculated.

5 Contents 1 Introduction 1 2 Data model 2 3 Non-stationary signals and the Wigner-Ville distribution 4 4 Non-parametric source separation Spatial time-frequency distributions The blind sources separation algorithm (JD) Spatial averaging of time-frequency distributions Combined joint diagonalization and joint anti-diagonalization (JAD) Parametric source separation An example using the discrete polynomial-phase operator The source separation algorithm using the discrete polynomial-phase transform Model order selection Estimating the mixing matrix and the lowest phase coefficient of the source signals Simulations and results Performance Index Comparison between the different source separation techniques Two linear FM signals Computational Complexity iv

6 v 6.4 If the polynomial order, Q = max{q i }, is not known If the polynomial orders, Q i, of the source signals are not equal Polynomial order selection Randomly chosen polynomial coefficients Conclusions and future work 52

7 Chapter 1 Introduction Source separation is important in a variety of applications, such as radar, sonar array systems, antenna array processing, tracking, and jamming. The problem is to separate the unknown individual source signals given linear mixtures of them. Up to date mostly non-parametric techniques have been used for source separation of non-stationary signals, i.e. a signal whose spectral contents varies with time. The non-parametric techniques exploit the differences in the time-frequency signatures of the sources, and a combined set of spatial time-frequency distributions being diagonalized. The difficulty with non-parametric methods is that they either need to know if a selected time-frequency point corresponds to a signal autoterm or crossterm, and/or they are simply too computationally expensive. This thesis suggests a parametric approach using the discrete polynomial-phase transform (DPPT) to solve the problem. The discrete polynomial-phase transform is nonlinear, and also a computationally efficient, technique. The purpose of this thesis is to show that efficient source separation can be made using a parametric technique, and to compare it with the non-parametric techniques. 1

8 Chapter 2 Data model The source signals are received by a uniform linear antenna array, and therefore, only instantaneous linear mixtures of a set of signals are observed. The data model used is x(t) = y(t) + n(t) = As(t) + n(t) t R (2.1) where x(t) is the noisy linear mixture of source signals received by the antenna array sensors. The sampling scheme is uniform, i.e. t n = n, n = 1,..., T, where denotes the sampling interval and T the total number of samples. The matrix A is the mixing matrix, assumed to be complex and of full column rank. This implies that the source signals must come from different directions of arrival. The largest distance between any two sensors in the array is orders of magnitude smaller than the distance from the source to the sensor array, i.e. the so-called far field case. The vector source signal, s(t)=[s 1 (t), s 2 (t),..., s I (t)] T, is a collection of non-stationary signals, i.e. signals whose spectral contents vary with time. All s i (t) are assumed to have constant amplitude and to be statistically independent. The source signals have polynomial-phase s i (t) = b i e j(a,i+a 1,i t+a 2,i t a Qi,it Q i) i = 1, 2,..., I (2.2) where the number of signals, I, are known. The polynomial orders, Q i, of the source signals can be different and unknown. The sources signals (wavefields) arise 2

9 3 from a point source and have constant velocity of propagation (no Doppler). The additive noise, n(t), is modulated as a stationary, temporally white, zeromean, complex random process, independent of the source signals. The noise is also spatially white, i.e., E[n(t + τ)n H (t)] = σ 2 nδ(τ)i (2.3) where σ 2 n denotes the variance of the noise, n H (t) denotes the conjugate transpose of n(t), I the identity matrix, E[.] is the statistical expectation operator, and δ(τ) is the Dirac delta. The mixture of source signals received by the antenna array sensors is a linear mixture if the mixing matrix, A, is not time varying. A is not time varying if all source envelope signals remains essentially unchanged over any time interval that is smaller than the time required for the wavefield to travel across the array, i.e. the maximum time delay must be much less than the inverse of the bandwidth (narrow band assumption). For non-stationary signals this may not be the case. It depends on the antenna aperture and signal bandwidth. Narrowbanded signals are assumed in most existing multi channel models. All the source separation techniques presented in this thesis assume that the mixture of the source signals is a linear mixture.

10 Chapter 3 Non-stationary signals and the Wigner-Ville distribution Non-stationary signals are signals whose spectral characteristics are varying with time, e.g. polynomial phased signals s(t) = be j(a +a 1 t+a 2 t a Q t Q ) (3.1) For example, s(t) = e.5π+.3πt+.2πt2 its Fourier transform can be seen in Figure 3.1. is a non-stationary signal. The signal and The instantaneous frequency, f(t), is defined as the slope (derivative) of the phase curve divided by 2π. f(t) = 1 dφ(t) (3.2) 2π dt In this particular case f(t) = t, which is linear in t and its slope is equal to.2. When the phase of a signal is quadratic, then its frequency changes linearly with time and the signal is called a linear chirp, or a linearly swept frequency. However it is not so easy to recognize the signal in Figure 3.1 as being a chirp. The frequency variation produced by the time-varying phase is called frequency modulation (FM). The name chirp comes from the audible sound similar to a siren or a chirp, which the linear variation of the frequency can produce. The Fourier transform shows where a signal exists in the frequency domain, but it does not show how its frequency characteristics change over time. The well known 4

11 5 Real part of s(t) Time in samples Absolut value of the Fourier transform of s(t) Frequency Figure 3.1: The signal, s(t) = e.5π+.3πt+.2πt2, and its Fourier transform, using 128 samples. Short Time Fourier Transform (STFT) divide the signal into small enough segments, where these segments of the signal can be assumed to be stationary and calculates the fourier transform for each segment. For this purpose, a window function is chosen. The width of this window must be equal to the segment of the signal where its stationarity is valid. The STFT will give us an idea of how the signals frequency characteristics change over time. To be able to observe how the signal frequency characteristics change over time time-frequency distributions are used. They describe the energy density of a signal simultaneously in both time and frequency [2]. This can be used to analyze signals which are non-stationary. The discrete-time form of Cohen s class of time-frequency distributions [2], for a signal s(t) is D ss (t, f) = φ(m, l)s(t + m + l)s H (t + m l)e j4πfl (3.3) l= m= The kernel φ(m, l) characterizes the time-frequency distribution and is a function

12 6 of both the time and lag variables. The time-frequency distribution used in this thesis is called the Wigner-Ville distribution, which is a quadratic joint time-frequency representation, whose kernel is equal to 1. It is the Fourier transform of the correlation function with respect to the delay parameter. The Wigner-Ville distribution is defined as V s1,s 2 (t, f) = s 1 (t + τ 2 )sh 2 (t τ 2 )e j2πft (3.4) τ= If s 1 = s 2 it is called the auto-wigner-ville distribution, otherwise it is called the cross-wigner-ville distribution. The auto-wigner-ville distribution is real valued. The Wigner-Ville distribution of the signal s(t) = e.5π+.3πt+.2πt2 can be seen in Figure 3.2. In this figure it is easy to see that the instananeous frequency varies linearly with time. WVD Frequency Time in samples Figure 3.2: The Wigner-Ville distribution of s(t) = e.5π+.3πt+.2πt2. The auto-wigner-ville distribution of a signal, when the signal is the sum of two

13 7 or more signals, is not the sum of the Wigner-Ville distribution of the individual signals, but contains an additional term. This additional term is usually referred to as a crossterm, and the other terms as autoterms. To make this clear, let us look at an example. The auto-wigner-ville distribution of x(t) = s 1 (t) + s 2 (t) is V x,x (t, f) = V s1,s 1 (t, f) + V s2,s 2 (t, f) + 2Re{V s1,s 2 (t, f)} (3.5) Figure 3.3 is an example of the Wigner-Ville distribution of the sum of two linear FM signals. The crossterms can be seen between the two autoterms. It should be noted that the crossterms can be much larger than the autoterms frequency time in samples Figure 3.3: The Wigner-Ville distribution of the sum of two linear FM signals. The spatial time-frequency distribution matrix for the discrete-time form of Cohen s class of time-frequency distributions, for a vector signal s(t) = [s 1 (t) s 2 (t) s 3 (t)... ] T is Dss(t, f) = l= m= φ(m, l)s(t + m + l)s H (t + m l)e j4πfl (3.6)

14 Chapter 4 Non-parametric source separation First the time-frequency separation algorithm presented in [1] is being used to estimate the source signals and the mixing matrix. This blind source separation technique can estimate the mixing matrix and recover the source signals up to a fixed permutation and some complex factors. This means that the original labeling of the sources can never be recovered, and that a satisfactory estimation of the sources is ŝ i (t) = α iˆbi e j(â,i+â 1,i t+â 2,i t â Qi,it Q i) i = 1, 2,..., I (4.1) where α i is an arbitrary, unknown and complex factor. This fixed scalar factor between the source and the corresponding column of the mixing matrix can be expressed as x(t) = y(t) + n(t) = where a i denotes the ith column of A. I i=1 a i α i α i s i (t) + n(t) (4.2) An advantage of this ambiguity is that the source signals can be treated as if they had unit power, i.e., 1 R ss () = lim T T T s(t)s H (t) = I (4.3) t=1 where T is the number of samples. 8

15 9 Since the sources are assumed to be uncorrelated and have unit power 1 R yy () = lim T T T t=1 y(t)y H 1 (t) = lim T T T As(t){As(t)} H = AA H (4.4) t=1 1 R xx () = lim T T T x(t)x H (t) = R yy () + σni 2 (4.5) t=1 Accordingly, AA H = R xx () σ 2 ni (4.6) Equation (4.6) will be useful later. 4.1 Spatial time-frequency distributions Under the linear data model of Equation (2.1), the spatial time-frequency matrix of the signals received by the antenna array, without the additive noise, can be expressed as V x1,x 1 (t, f) V x1,x 2 (t, f)... V x1,x M (t, f) V x2,x D xx (t, f) = 1 (t, f) V x2,x 2 (t, f)... V x2,x M (t, f) = AD ss(t, f)a H V xm,x 1 (t, f) V xm,x 2 (t, f)... V xm,x M (t, f) where, V s1,s 1 (t, f) V s1,s 2 (t, f)... V s1,s I (t, f) V s2,s D ss (t, f) = 1 (t, f) V s2,s 2 (t, f)... V s2,s I (t, f) (4.7) V si,s 1 (t, f) V si,s 2 (t, f)... V si,s I (t, f) D ss (t, f) is of dimension I I and D xx (t, f) of dimension M M, where M is the number of antenna array sensors and I the number of source signals. The off-diagonal elements of D ss (t, f) are the crossterms and the diagonal elements are the autoterms. For each time-frequency point which corresponds to a

16 1 signals autoterm, the D ss (t, f) matrix will almost be a diagonal matrix, since all the elements in the matrix, but one on the diagonal, will be close to zero. 4.2 The blind sources separation algorithm (JD) The non-parametric time-frequency separation algorithms only work if there are more antenna array sensors than source signals, i.e. M > I. The separation algorithm is divided into two steps. The first step is to whiten the signals received by the sensors. This can be achieved by multiplying the received signal with a whitening matrix, W. This whitening procedure transforms the mixing matrix A into a unitary matrix. z(t) = Wx(t) = W(As(t) + n(t)) = Us(t) + Wn(t) (4.8) where U = WA is a unitary matrix, i.e. UU H = U H U = I. U is an I I matrix. The whitening matrix W satisfies lim T 1 T T Wy(t)y H (t)w H = WR yy ()W H = I (4.9) t=1 The matrix which includes the snapshots of the signals received by the linear antenna array is an M T matrix. The matrix containing the whitened signals is an I T matrix, where T is the number of samples and I the number of source signals. This whitening procedure reduces the determination of the M I mixture matrix A to that of a unitary I I matrix U. z(t) = Us(t) is now a unitary mixture of the source signals. This means that the autocorrelation matrix of the sources s(t) (treated as if they had unit power, see Equation (4.3)) is equal to the autocorrelation matrix of the whitened signals z(t), and if U were to change to another unitary matrix, the autocorrelation matrix of z(t) would not change.

17 11 By using Equation (4.6) and (4.9), the whitening matrix can be obtained from [ ] H 1 1 Ŵ = h 1,..., h I (4.1) λ1 ˆσ n 2 λi ˆσ n 2 where λ 1,...,λ I are the I largest eigenvalues and h 1,...,h I the corresponding eigenvectors of ˆRxx (). An estimate of σn 2 can be obtained from the average of the remaining eigenvalues of ˆR xx (). Equation (4.7) can now be extended to D zz (t, f) = WD xx (t, f)w H = (4.11) UD ss (t, f)u H (4.12) Since U is a unitary matrix and D ss (t, f) is a diagonal matrix (for t-f points corresponding to signals autoterms) the eigenvalues of D zz (t, f) are equal to the diagonal entries of D ss (t, f). Since U = WA, the mixing matrix can be obtained from  = Ŵ# Û (4.13) where Ŵ # = (Ŵ H Ŵ) 1 Ŵ H denotes the pseudoinverse of Ŵ. The second step is to find this missing unitary matrix U. U can be obtained as a unitary diagonalization matrix of D zz (t, f) for some t-f points corresponding to signal autoterms. This joint diagonalization algorithm to obtain U is described in [3] and can be described as the problem of the diagonalization of M k, k = 1, 2,..., K, normal matrices. This is defined as the minimization of the JD criterion K I b H i M k b i 2 (4.14) k=1 i=1

18 12 over the set of unitary matrices B = [b 1,..., b I ] Therefore some t-f points need to be selected, which belong to the signals autoterms. The algorithm works even if only one t-f point is selected, but to reduce the impact of a t-f point with high noise contamination, we should choose more than one. The selection of a greater number of t-f points increases the robustness with respect to the noise while localizing the source energy in the t-f domain. One interesting thing discovered while running simulations is that it does not seem to matter if all the selected autoterm points belongs to one signal or different signals. If one source signal was always known to the receiver, autoterm points could easily be selected belonging to the known source signal. The source signals can now be estimated from ŝ(t) = ÛH Ŵx(t) = ÛH z(t) (4.15) Spatial averaging of time-frequency distributions The main problem with the proposed blind source separation algorithm, based on time-frequency signal representations in [1], is the difficulty in choosing only timefrequency points which corresponds to signal autoterms. Even if only one crossterm is selected, and several autoterms, the result can be devastating. To reduce the influence of crossterms, spatial averaging can be performed on the signals received by the linear antenna array [4]. The assumption that the signals are mutually uncorrelated is then no longer necessary. To be able to do spatial averaging the linear antenna array configuration must be known. The antenna array must also be extended to include more antenna array sensors. The original M sensors (Figure 4.1) becomes M + (M 1) = 2M 1 sensors (Figure 4.2).

19 13 Figure 4.1: Original antenna array configuration. Figure 4.2: The extended antenna array configuration.

20 14 The signals are received by an equispaced linear antenna array with the interelement spacing d =.5λ, where λ is the RF wavelength. The first antenna array element will receive each source signal τ i, i = 1,..., I seconds before the second element. This time delay will correspond to a shift in the phase of each source signal when observed by each antenna due to the narrow band assumption. This phase shift can be described as φ i = 2πd λ sin θ i (4.16) where θ i is the direction of arrival of the source signals. Under the assumption that the distance, d, is much smaller than the distance from the source to the antenna element, the centre array sensor can be taken as a reference and the mixing matrix can be assumed to be a matrix of steering vectors, i.e. A = e jω 1(M 1)d e jω 2(M 1)d... e jω I(M 1)d e jω 12d e jω 22d... e jω I2d e jω 1d e jω 2d... e jω Id e jω 1d e jω 2d... e jω Id e jω 12d e jω 22d... e jω I2d e jω 1(M 1)d e jω 2(M 1)d... e jω I(M 1)d (4.17) where ω i = 2π λ sin θ i If we consider only two signal sources the received signal at the mth antenna array sensor, without noise, is x m (t) = s 1 (t)e jmdω 1 + s 2 (t)e jmdω 2 m = 1 M, 2 M,..., 1,, 1,..., M 1 (4.18) The Wigner-Ville distribution of the signal received at the centre array sensor, x

21 15 and x m, without noise, is V x,x m (t, f) = V s1,s 1 (t, f)e jdω 1 e jmdω 1 + V s2,s 1 (t, f)e jdω 2 e jmdω 1 + V s1,s 2 (t, f)e jdω 1 e jmdω 2 + V s2,s 2 (t, f)e jdω 2 e jmdω 2 = [V s1,s 1 (t, f) + V s2,s 1 (t, f)] e jmdω 1 + (4.19) [V s1,s 2 (t, f) + V s2,s 2 (t, f)] e jmdω 2 Spatial averaging of the received signals, s =, 1,..., M 1, is defined as V (s) xx (t, f) = V x,xs (t,f)+v x,x s (t,f) 2 = [V s1,s 1 (t, f) + Re{V s2,s 1 (t, f)}] e jsdω 1 + (4.2) [V s2,s 2 (t, f) + Re{V s1,s 2 (t, f)}] e jsdω 2 Note that the terms in the brackets in Equation (4.2) are real. The spatial average time-frequency matrix can now be expressed as D xx (t, f) = xx (t, f) xx (t, f) x,x (t, f)... V xx (M 3) (t, f)..... xx (t, f) (4.21) V () xx (t, f) V (1) xx (t, f) V (2) xx (t, f)... V (M 1) V (1) xx (t, f) V () xx (t, f) V (1) x,x (t, f)... V (M 2) V (2) xx (t, f) V (1) xx (t, f) V ().. V xx (M 1) (t, f) V x,x (M 2) (t, f) V x,x (M 3) (t, f)... V () D xx (t, f) is Toeplitz, i.e. elements on the diagonal (and on the subdiagonals) are equal r i,j = r i+1,j+1, and Hermitian, i.e. r i,j = r j,i, i j. Analogous to Equation (4.7), the spatial average time-frequency matrix can be expressed as D xx (t, f) = A D ss (t, f)a H (4.22)

22 16 where D ss (t, f) = [ Vs1,s 1 (t, f) + Re{V s2,s 1 (t, f)} V s2,s 2 (t, f) + Re{V s1,s 2 (t, f)} ] (4.23) The off-diagonal elements of D ss (t, f) are zero, and the diagonal elements are now made up off both autoterms and crossterms, compare with Equation (4.7). The source separation can now be made with this matrix instead of the matrix defined in Equation (4.7). If the selected t-f point is an autoterm and, no noise is present, Equations (4.22) and (4.7) become equal. This means that the separation algorithm based on the diagonalization of a combined set of spatial time-frequency distributions and the same method but with spatial averaging of the time- frequency distributions, should work almost equally good Combined joint diagonalization and joint anti-diagonalization (JAD) Combined joint diagonalization and joint anti-diagonalization [5] is the most recent attempt to improve the performance of the separation algorithm based on the diagonalization of a combined set of spatial time-frequency distributions. As mentioned previously, the off-diagonal elements of D ss (t, f) are the crossterms and the diagonal elements the autoterms. The D ss (t, f) matrix will usually be a diagonal matrix for each time-frequency points that corresponds to a signals autoterm, and for crossterms anti-diagonal, i.e. the elements on the diagonal are close to zero. This method uses an extension of the method described in the second step of the blind source separation algorithm, chapter 4.2, to obtain the unitary matrix U. At least one crossterm and one autoterm must be selected. It can be described as the minimization of the JD/JAD criterion ( I K ) b H i M k b i 2 L b H i N l b i 2 i=1 k=1 l=1 (4.24)

23 17 over the set of unitary matrices B = [b 1,..., b I ]. U is obtained as a unitary diagonalization matrix of D zz (t, f), for k = 1, 2..., K t-f points corresponding to signal autoterms, and as a unitary anti-diagonalization matrix of D zz (t, f), for l = 1, 2,..., L t-f points corresponding to signal crossterms. This implies that it must be known if a t-f points belong to an autoterm or a crossterm. The suggested method, in [5], to distinguish between autoterms and crossterms is as follows trace{d xx (t, f)} norm{d xx (t, f)} = µ µ >.1 the t-f point is an autoterm µ <.1 the t-f point is a crossterm (4.25) Since trace{d xx (t, f)} = trace{ud ss (t, f)u H } = trace{d ss (t, f)} for t-f points corresponding to crossterms (4.26) where trace{d xx (t, f)} denotes the sum of the diagonal elements of D xx (t, f) and norm{d xx (t, f)} the largest singular value of D xx (t, f). However this may not always work. According to [2, page 122]: The Wigner- Ville distribution goes negative somewhere for every signal, with the exception of the signal given by s(t) = ( α π )1/4 e αt2 /2+jβt 2 /2+jω t. α, β, and ω are real valued numbers. The trace of D xx (t, f) could be negative if a t-f point were selected corresponding to an autoterm, and as the norm is always positive, this would lead to a negative µ and the method above would mistake it for a crossterm. If two source signals are sent and a t-f point that corresponds to an autoterm, of both source signals is chosen, the D xx (t, f) matrix could then have one negative and one positive value on the diagonal, and the trace could then be close to zero. This would also be recognized as a t-f point corresponding to a crossterm. As an alternative to the non-parametric source separation techniques, we discuss in Chapter 5 a parametric method which avoid the selection of autoterms and crossterms.

24 Chapter 5 Parametric source separation The avoid the problem of selecting time-frequency points, a parametric approach to recover the source signals will now be suggested. The Discrete Polynomial-Phase Transform presented in [6] is being used. The main issue is to show that efficient source separation can be done using the Discrete Polynomial-Phase Transform, and to compare the performance of the discrete polynomial-phase transform based method with the non-parametric methods described earlier. Another parametric source separation technique is presented in [9]. The polynomial operator, of order P, is defined as DP P [s(n), τ] = P 1 p= [s p (n pτ)] (P 1 p ) n R (5.1) where { s(n) p even [s p (n)] = s (n) p odd (5.2) s (n) denotes the conjugate of s(n) and τ is called the delay parameter. The discrete polynomial-phase transform operator, of order P, is defined as DPT P [s(n), w, τ] = T 1 n=(p 1)τ 18 DP P [s(n), τ]e jwn (5.3)

25 19 where denotes the sampling interval. If DPT P is applied to a polynomial-phased signal of the same order, P, it will produce a single spectral line corresponding to the highest order polynomial coefficient. After estimating this coefficient, and filtering out the coefficient from the received signal, the DPT P 1 is applied and the second highest coefficient can be estimated, and so on, until all coefficients are estimated. In this thesis the consequence of using this method for source separation is investigated. The coefficients of one source signal are being estimated in the presence of the coefficients of other source signals. To remove ambiguity in the estimation process we require that [8]: a q,i τ q 1 q π q! q q = 1, 2,..., Q i (5.4) 5.1 An example using the discrete polynomialphase operator The discrete polynomial-phase operator of order 2 and 1, DPT q=2,1, applied on the signal s(t) = e jπ(.5+.3t+.3t2), using τ = T (as recommended in [7]), and T = 256 q samples at 2dB signal-to-noise ratio (SNR), can be seen in Figure 5.1. The peak frequency for DPT 2, using τ = 128, is at ω = This gives us estimation of the highest polynomial coefficient as a 2 = ω = fact{q} τ q =.29998π, where fact{q} denotes the factorial of q. The second highest polynomial phase coefficient, using τ = 256, can be obtain in a similar way, as a 1 = =.35π. The interested reader may read further regarding the polynomial phase transform operator in [6], [7], and [8].

26 2 14 q=2 3 q= Frequency Frequency Figure 5.1: DPT q=2,1, τ = 128, 256, applied to the signal received by an antenna element at 2dB SNR.

27 The source separation algorithm using the discrete polynomial-phase transform The algorithm is based on the data model described in Equation (5.5), where x(t) is the noisy linear mixture of source signals, s(t), received by the linear array sensors. x(t) = As(t) + n(t) t R (5.5) where A is the mixing matrix and the additive noise n(t) is modulated as a stationary, temporally white, zero-mean, complex random process, independent of the source signals. To be able to estimate the mixing matrix at least two antenna elements are needed, regardless of the number of sources. The algorithm can however recover the source signals with only one antenna element, regardless of the number of sources. The phase coefficients of the source signals are described as a q,i,m, where I = number of sources, i = 1, 2,..., I Q = the polynomial order of the source signals with the highest polynomial order, q = 1, 2,..., Q M = number of antenna elements, m = 1, 2,..., M The source separation algorithm based on the discrete polynomial phase transform, that will be used, is an extension of a multicomponent signal analysis method, presented in [6] and [7]. The polynomial order, Q = max{q i }, of the source signals with the highest polynomial order is estimated (if it is unknown) with the model order selection algorithm, chapter The source separation algorithm can be summarized as follows. The signal received by the first antenna element is used. The highest coefficient, a Q,1,1, for the first source signal is estimated, then the second highest coefficient, a Q 1,1,1, for the same source signal, and so on, until all coefficients a q,1,1, q = Q, Q 1,...2, 1, the constant amplitude, b 1,1, and the lowest polynomial-phase coefficient, c 1,1, of the received signal are estimated. c i,m includes a,i and the shift in the phase which corresponds

28 22 to the time delay when the signal travels the extra time to reach antenna element number m, compared to traveling to antenna element number 1. The first estimated source signal is filtered out of the received signal. Now the coefficients of the second source signal are estimated and filtered out and so forth until all the desired source signals coefficients are estimated. The algorithm then repeats the process, but this time for the second antenna element, and so forth. The source separation algorithm suggested in this thesis will now be described more in detailed. 1. Initialization for using the signals received at the mth antenna element. Let z i (t) = x m (t), i = 1, and m = 1 for t =, 1,..., T 1 2. Initialization for estimating the parameters of the ith source signal. Let q = Q and z q i (t) = z i (t) 3. Let τ q = T q (as recommended in [7]) and compute â q,i,m = 1 q!(τ q ) q 1 arg max{ DPT q[z (q) i (t), w, τ q ] } (5.6) where arg max{f(w)} is the value of w at the maximum of f(w). Let z (q 1) i (t) = z q i (t)e jâ q,i,m(t ) q Substitute q = q 1. If q 1, repeat Estimating the lowest phase coefficient of the ith received signal. { T 1 } ĉ i,m = phase zi (t) t= (5.7) 5. Estimating the constant amplitude of the ith source signal. ˆbi,m = 1 T 1 T zi (t) (5.8) t=

29 23 6. If i = I go to 8, else filter out the ith component. z i+1 (t) = ( z i (t) 1 T T 1 t= z i (t) z i (t) ˆb i,m e j(ĉ i,m+ Q q=1 âq,i,m(t )) 7. Substitute i = i + 1 and go to 2. ) e j Q q=1 âq,i,m(t ) q = 8. If m = M exit algorithm, else substitute m = m + 1 and go to 1. An average over all antenna elements, m, can now be made for the polynomialphase coefficients â q,i,m and ˆb i,m. Passing z () i through a lowpass filter before entering stage 4 did not improve the performance of the algorithm. However the estimation can be improved if an initial estimate of all source signals are calculated and then all source signals, but one, are filtered out from the received signals and the remaining source signal can then be estimated without so much influence from the other source signals. This would, of course, take more time, which may not always be available. The sorting of the estimated source signals is critical. Antenna 1 estimates the sources, for example, in the following order: s 3 (t), s 2 (t), s 1 (t). The fifth antenna estimates the sources in another order, for example: s 2 (t), s 3 (t), s 1 (t). This leads to an incorrect estimation if the algorithm is unable to sort the estimated signals, i.e. detect that the estimation order has been changed. If the sources are well separated in the time-frequency domain the sorting can be done. If the source signals are not well separated this method fails. Furthermore if the highest polynomial coefficient of one source signal is estimated correctly, all the other parameters of that signal usually will be estimated correctly Model order selection Since the source signals have constant amplitude the following method can be used to determine the polynomial order, Q, that should be used in the proposed parametric source separation technique. 1. Initialization for using the signals received at the first antenna element.

30 24 Let z i (t) = x 1 (t), and i = 1 for t =, 1,..., T 1 2. Initialization. Set P to be higher then the polynomial order of the source signal with the highest polynomial order. 3. Initialization for estimating the parameters of the ith source signal. Let q = P and z q i (t) = z i (t) 4. Let τ q = T q (as recommended in [7]) and compute â q,i,p = 1 q!(τ q ) q 1 arg max{ DPT q[z (q) i (t), w, τ q ] } (5.9) where arg max{f(w)} is the value of w at the maximum of f(w). Let z (q 1) i (t) = z q i (t)e jâ q,i,p (t ) q Substitute q = q 1. If q 1, repeat Estimating the lowest phase coefficient of the ith received signal. ĉ i,p = phase { T 1 t= z i (t) } (5.1) 6. Estimating the constant amplitude of the ith source signal. ˆbi,P = 1 T 1 T t= 7. If i = I go to 9, else filter out the ith component. z i+1 (t) = ( z i (t) 1 T 8. Substitute i = i + 1 and go to 3. T 1 t= z i (t) z i (t) (5.11) ) e j Q q=1 âq,i,p (t ) q (5.12) 9. If P = 1 go to 1, else substitute P = P 1, i = 1, z i (t) = x 1 (t) and go to Set Q = P corresponding to max P { I i=1 ˆbi,P }

31 Estimating the mixing matrix and the lowest phase coefficient of the source signals The mixing matrix and the lowest phase coefficient of the source signals can be recovered up to a fixed permutation and some complex factors. The signals are received by an equispaced linear antenna array with the inter-element spacing d =.5λ, where λ is the RF wavelength. At least two antenna elements are needed to estimate the mixing matrix, regardless of the number of sources. Figure 5.2: Incoming plane wave to an antenna array with 5 elements. The first antenna array element will receive each source signal τ i, i = 1,..., I seconds before the second element. This time delay will correspond to a shift in the phase of each source signal when observed by each antenna. This phase shift can be estimated from φ i = 2πd λ sin θ i (5.13) where θ i is the direction of arrival of the source signals. Under the assumption that the distance, d, is much smaller than the distance from the source to the antenna element, the first antenna array element can be

32 26 taken as the reference element and the mixing matrix can be assumed to be a matrix of steering vectors, i.e. A = e jω 1d e jω 2d... e jω Id e jω 12d e jω 22d... e jω I2d e jω 1(M 1)d e jω 2(M 1)d... e jω I(M 1)d where M is the number of antenna elements, λ the RF wavelength, (5.14) and ω i = 2π sin θ λ i The mixing matrix can simply be described as e jk 1 e jk 2... e jk I A = e j2k 1 e j2k 2... e j2k I e j(m 1)k 1 e j(m 1)k 2... e j(m 1)k I where k i = π sin θ i, if d = λ 2. (5.15) Therefore it is known that π k i π. Furthermore, it is also known that c i,m includes a,i and the shift in the phase, which corresponds to the time delay when the signal travels the extra time to reach antenna element m, compared to traveling to antenna element number 1, i.e. ĉ i,m has been wrapped, i.e and c i,m = a,i + (m 1)k i (5.16) π < ĉ i,m π (5.17) < a,i + (m 1)k i < m = 1, 2..., M (5.18) The following system of equations has to be solved e jĉ i,1 e jĉ i,2 e jĉ i,3. e jĉ i,m = = =. = e ja,i e j(a,i+k i ) e j(a,i+2k i ). e j(a,i+(m 1)k i ) (5.19)

33 27 Before solving the system of equations mentioned, using a least-square (LS) method, we have to unwrap the estimates ĉ i,m. Then â,i and ˆk i can be estimated with 1 ĉ i,1 1 1 ĉ i,2 1 2 [ ] â,i ĉ i,3.. ˆk i 1 M 1 D =. ĉ i,m B C (5.2) D = (B H B) 1 B H C (5.21) If it is known that k i is negative, i.e. < θ i < 18 and c i,1 > c i,2 > c i,3 and so on, the unwrapping could be done easily with the following method If ĉ i,e < ĉ i,e+1 subtract 2π from ĉ i,e+1. (5.22) If it is known that k i is positive, i.e. 18 < θ i < 36 and c i,1 < c i,2 < c i,3 and so on, the unwrapping could be done easily with the following method If ĉ i,e > ĉ i,e+1 add 2π to ĉ i,e+1. (5.23) If k i is equal to zero, i.e. θ i = or θ i = 18, all c i,m would be equal and no unwrapping would need to be done. To determine the sign of k i use the following algorithm if < ĉ i,m and < ĉ i,m+1 ĉ i,m < ĉ i,m+1 k i is positive ĉ i,m > ĉ i,m+1 k i is negative if ĉ i,m < and ĉ i,m+1 < ĉ i,m < ĉ i,m+1 k i is negative ĉ i,m > ĉ i,m+1 k i is positive

34 28 ĉ i,m < < ĉ i,m+1 ĉ i,m+1 ĉ i,m < π k i is negative ĉ i,m+1 ĉ i,m > π k i is positive ĉ i,m+1 < < ĉ i,m ĉ i,m ĉ i,m+1 < π k i is positive ĉ i,m ĉ i,m+1 > π k i is negative The reader may verify the algorithm using the unit circle shown in Figure 5.3. The problem of determine the sign of k i can be described as the problem of determining if α or β is the smallest angle. α is the angle between c i,m and c i,m+1 traversing counter clockwise and β is the angle between c i,m and c i,m+1 traversing clockwise. Figure 5.3: The unit circle The algorithm mentioned above can be used several times if more than two antenna elements are used to compare the different c i,m, m = 1, 2,..., M. If k i is close to zero or π, the algorithm would not always give the same answer, due to the presence of noise. If one of the estimated coefficients, ĉ i,m+1 and ĉ i,m, are close to π and the other one is close to π we assume that k i is π. If ĉ i,m ĉ i,m+1 we assume k i is zero otherwise we assume k i = π.

35 29 Once ˆk i is estimated, the direction of arrival, θ i, of the source signals can be estimated as ˆθ i = arcsin ( πˆk i ) (5.24) This will give two possible direction of arrival for each source signal, i.e. we would not be able to know if the source signals are arriving from [ 9-9 ], or from the other side of the antenna array [9-27 ].

36 Chapter 6 Simulations and results 6.1 Performance Index The interference-to-signal ratio (ISR) used to compare the different techniques is defined as [4] I p,q = E ( Âdiagonal{Â# A}A) p,q 2 (6.1) ISR = 1 log{ p q I p,q } (6.2) where diagonal{x} denotes the elements on the diagonal of X and log{x} indicates the base 1 logarithm of X. The ISR is given in db and a low ISR indicates good performance. 6.2 Comparison between the different source separation techniques The different source separation techniques that will be compared are DPPT : The method using the discrete polynomial-phase transform. JD : The separation algorithm based on the joint diagonalization of a combined set of spatial time-frequency distributions. 3

37 31 SA: JD with spatial averaging of the time- frequency distributions. JAD : JD combined with joint anti-diagonalization. In all the following simulations 2 source signals were sent from the directions of arrival 5 and 1. For the non-parametric methods the time-frequency points are selected at the times 32 and 92, and their respective frequencies were chosen at the maximum of the absolute value of the Wigner-Ville distribution of the signals received at the centre array sensor for each time unless otherwise stated. If the polynomial order of the source signal with the highest polynomial order is not known the order is assumed to be less then Two linear FM signals In the simulations two linear FM signals, s 1 (t) and s 2 (t) (Figure 6.1), were used as source signals. s 1 (t) = e jπ(.5+.3t+.3t2 ) s 2 (t) = e jπ(.2+.1t+.2t2 ) (6.3).5 Sources frequency time in samples Figure 6.1: The individual Wigner-Ville distribution of the sources. An average over 1 runs per SNR was simulated, 128 samples were used. A total

38 32 of 5 antenna array sensors were used. For SA, a subarray of 4 elements were added, which made a total of 9 antenna sensors. The simulations were done with SNR=2, 19,..., db. The performance of the DPPT was investigated when the polynomial order of the source signal with the highest polynomial order was known and when it was not known and had to be estimated. Figure 6.2 shows the performance of the different techniques when there is no distinguishing between autoterms and crossterms. The model order selection algorithm worked very good. Between a SNR of 2 and 1 db, SA and DPPT perform very well, however in Figure 6.3 it can be seen that the DPPT has greater stability in it s performance, since the variance of the ISR (ISR in db) for SA is much higher. 5 ISR in db DPPT, order known DPPT, order unknown JD Spatial averaging SNR in db Figure 6.2: SNR in db (integer) vs. ISR in db Autoterms If it was possible to select only autoterms, corresponding to s 2, using the same settings (chapter 6.2.1) as before JD and SA outperforms DPPT, see Figure 6.4 and 6.5. Crossterms If only crossterms are chosen, both JD and SA mostly fails Figure 6.6.

39 DPPT, order known DPPT, order unknown JD Spatial averaging Variance of ISR SNR in db Figure 6.3: Variance of ISR vs. SNR in db One autoterm and one crossterm To test the JAD we use one crossterm and one autoterm. The JD uses only the autoterm, see Figure 6.7. The ISR for the JAD is circa 2 db lower than for JD.

40 34 Only autoterms 5 1 ISR in db DPPT, order known DPPT, order unknown JD Spatial averaging SNR in db Figure 6.4: SNR in db (integer) vs. ISR in db (only autoterms were selected) 7 Only autoterms 6 DPPT, order known DPPT, order unknown JD Spatial averaging Variance of ISR SNR in db Figure 6.5: Variance of ISR vs. SNR in db (only autoterms were selected)

41 35 5 Only crossterms ISR in db DPPT, order known DPPT, order unknown JD Spatial averaging SNR in db Figure 6.6: SNR in db (integer) vs. ISR in db (only crossterms were selected) 5 1 DPPT, order known DPPT, order unknown JD with one autoterm Spatial averaging with one crossterm and one autoterm JAD with one crossterm and one autoterm ISR in db SNR in db Figure 6.7: SNR in db (integer) vs. ISR in db (one crossterm and one autoterm)

42 Computational Complexity The question of which method is the fastest is of big interest in real time applications. If the polynomial order of the source signal with the highest polynomial JD Spatial averaging JAD Number of samples Figure 6.8: Relative time to compute each method compared to DPPT (order known), for 2 t-f points and 5 antenna elements (SA uses 9 antenna elements), using 128, 256, and 512 samples. order is known the DPPT is always faster than the non-parametric methods, Figure 6.8 and Table 6.1, for 2 t-f points and 5 antenna elements. As the sampling size increases this becomes more and more clear. The calculation of the timefrequency distributions is very time consuming, compared to the use of the discrete polynomial-phase operator. Sample size DPPT JD SA JAD Table 6.1: Relative time to compute each method compared to DPPT (order known), for 2 t-f points with three different sample sizes. As the number of antenna elements increases the difference in time to compute

43 37 Number of antenna elements DPPT JD SA JAD Table 6.2: Relative time to compute each method compared to DPPT (order known), for 2 t-f points and 128 samples, using 3-1 antenna elements. each method, compared to the DPPT, decreases, Figure 6.9 and Table JD Spatial averaging JAD Number of antenna elements Figure 6.9: Relative time to compute each method compared to DPPT (order known), for 2 t-f points and 128 samples, using 3-1 antenna elements. If the polynomial order of the source signal with the highest polynomial order is not known and the model order selection algorithm must be used (using P = 7) the DPPT is mostly still faster then the non-parametric methods, see Figure 6.1 and Table 6.3. To be able to recover the source signals the DPPT requires only one antenna element and at least two antenna elements to estimate the mixing matrix, regardless of the number of source signals. The non-parametric method on

44 JD Spatial averaging JAD Number of samples Figure 6.1: Relative time to compute each method compared to DPPT (order unknown), for 2 t-f points and 5 antenna elements, using 128, 256, and 512 samples. Sample size DPPT JD SA JAD Table 6.3: Relative time to compute each method compared to DPPT (order unknown), for 2 t-f points with three different sample sizes. the other hand need more antenna elements then source signals. The simulations above used 5 antenna elements (SA 9).

45 If the polynomial order, Q = max{q i }, is not known The parametric method is limited to polynomial-phase signals. Also the polynomial order of the source signal with the highest polynomial order has to be known or estimated correctly. The non-parametric methods work regardless of if the polynomial order is known and for other signal models. If the polynomial order of the received signal is not known and not estimated, the non-parametric methods would work but the DPPT method would not. If only one source signal is sent to an antenna array then the parameters of that signal can be estimated correctly even if the polynomial order of the sent signal is not known or estimated, for high SNR. If, for example, the true polynomial order of the signal is Q, the receiver makes an assumption that the polynomial order of the source signal is lower then P. P > Q is chosen to be within the operation range, defined in [8], of discrete polynomial phase transform. First we compute the DPT q=p 1, and, as long as q is greater than the true order, the DPT will have a spectral line at zero frequency. The assumed polynomial order q is decreased until the DPT has a distinct spectral line at a nonzero frequency. In this case, with only one source signal, the estimation of the polynomial coefficients works fine. To test this the signal s 1 is used under the same conditions as before. Figure 6.12 shows the estimated coefficients, a 2 and a 1, for s 1 at different values of the SNR, when Q = 2 and P = 6. Note the spectral lines at zero frequency for q = 5, 4, 3, when DPT q is applied at the signal received by the first antenna element at SNR 2dB, Figure If both s 1 and s 2 were sent under the same conditions as before, the estimated coefficients are not estimated correctly, Figure It can be seen in the Figure 6.14 that the highest peak is not at zero frequency, for q = 5, 4, 3, due to crossterms. But if the polynomial order of the source signals was known or estimated correctly, then they would be estimated correctly, Figure 6.16 and The crossterms are the two smaller peaks in Figure 6.16 (top left). If the the source signals were not well separated in the time-frequency domain these crossterms would become much

46 4 6 q=5 7 q=4 45 q=3 7 q=2 14 q= Figure 6.11: DPT q, q = 5, 4,..., 1, applied at the signal received by the first antenna element at SNR 2dB. Q = 2, P = 6, one source signal. higher and the estimation of the polynomial coefficients would fail. The conclusion can be made that the polynomial order of the source signals has to be known or estimated correctly, if two or more source signals are sent when using the parametric source separation algorithm suggested in this thesis.

47 x 1 3 Q=2 P= Highest coefficient / Pi Second highest coefficient / Pi SNR in db SNR in db Figure 6.12: The estimated coefficients a 2 and a 1. Q = 2, P = 6, one source signal. 4 x Highest coefficient / Pi Second highest coefficient / Pi SNR in db SNR in db Figure 6.13: The estimated coefficients a 2,i and a 1,i. Q = 2, P = 6, two source signals.

48 x 14 q=5 1 q=4 1 q=3 35 q=2 4 q= x Figure 6.14: DPT q, q = 5, 4,..., 1, applied to the signal received by the first antenna element at SNR 2dB. Q = 2, P = 6, two source signals. 4 x Highest coefficient / Pi Second highest coefficient / Pi SNR in db SNR in db Figure 6.15: The estimated coefficients a 2,i and a 1,i. The polynomial order of the two source signals are known.

49 43 7 q=2 14 q= Figure 6.16: DPT q, q = 5, 4,..., 1, applied to the signal received by the first antenna element at SNR=2dB. The polynomial order of the two source signals are known.

50 6.5 If the polynomial orders, Q i, of the source signals are not equal If the polynomial orders, Q i, are not equal for all source signals but the polynomial order of the signal with the highest polynomial order is known, or estimated correctly, the algorithm using the DPPT can work. This will be illustrated using the signals 44 s 1 (t) = e jπ(.5+.3t+.3 t2 ) s 2 (t) = e jπ(.2+.1 t) q=2 q= Frequency Figure 6.17: DPT q, q = 2, 1, applied to the signal received by the first antenna element at SNR=2dB. The polynomial order of the source signal with the highest polynomial order is known. The ISR was db for SNR=2dB, which is very good. The DPT q, q = 2, 1, applied to the signal received by the first antenna element can be seen in Figure 6.17