A Complex-Valued Mixing Matrix Estimation Algorithm for Underdetermined Blind Source Separation

Abstract

This paper considers complex-valued mixing matrix estimation in underdetermined blind source separation. An effective estimation algorithm based on both single-source-point (SSP) detection and modified dynamic data field clustering is proposed. First, array-processing-based time–frequency SSP detection is applied to improve signal sparsity, therein utilizing the real and imaginary components of the observed signals in the time–frequency domain. The algorithm can be applied to the estimation of complex-valued mixing matrix based on L-shaped arrays and uniform circular arrays. Then, to overcome the limitation that the clustering performance of traditional algorithms is affected by noise, data cleansing detection is introduced to reselect the SSPs with high potential energy as representative objects to achieve preliminary data classification. Finally, a dynamic data field clustering algorithm is adopted to move and merge the representative objects until all column vectors of the mixing matrix are estimated. Simulation results show that the proposed method can effectively estimate complex-valued mixing matrices with high accuracy, especially in real-world noncooperative cases without prior knowledge.

Appendices

Appendix A: The Defined Variables

Consider that there is only one effective source $s_n(t)$ with a relatively large value at any TF point $(t_1,f_1)$; then, Eq. (5) can be described as

$$\begin{aligned} \left\{ \begin{array}{ll} X_m^x(t_1,f_1) = a_{mn}^xS_n(t_1,f_1)\\ X_m^y(t_1,f_1) = a_{mn}^yS_n(t_1,f_1).\\ \end{array} \right. \end{aligned}$$

(26)

Then, we obtain

$$\begin{aligned} \begin{aligned} Re\left[ X_m^x(t_1,f_1)\right]&= Re(a^x_{mn})Re\left[ S_n(t_1,f_1)\right] -Im(a^x_{mn})Im\left[ S_n(t_1,f_1)\right] \end{aligned} \end{aligned}$$

(27)

$$\begin{aligned} \begin{aligned} Im\left[ X_m^x(t_1,f_1)\right]&= Im(a^x_{mn})Re\left[ S_n(t_1,f_1)\right] +Re(a^x_{mn})Im\left[ S_n(t_1,f_1)\right] \end{aligned} \end{aligned}$$

(28)

$$\begin{aligned} \begin{aligned} Re\left[ X_m^y(t_1,f_1)\right]&= Re(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] -Im(a^y_{mn})Im\left[ S_n(t_1,f_1)\right] \end{aligned} \end{aligned}$$

(29)

$$\begin{aligned} \begin{aligned} Im\left[ X_m^y(t_1,f_1)\right]&= Im(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] +Re(a^y_{mn})Im\left[ S_n(t_1,f_1)\right] . \end{aligned} \end{aligned}$$

(30)

From Eqs. (27), (28), (29) and (30), the following equations are obtained:

$$\begin{aligned} \begin{aligned}&Re\left[ X_m^x(t_1,f_1)\right] Re\left[ X_m^y(t_1,f_1)\right] \\&\quad = Re(a^x_{mn}) Re(a^y_{mn})Re^2\left[ S_n(t_1,f_1)\right] +Im(a^y_{mn})Im(a^x_{mn})Im^2\left[ S_n(t_1,f_1)\right] \\&\qquad -Re(a^x_{mn})Im(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\&\qquad -Re(a^y_{mn})Im(a^x_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\ \end{aligned}\qquad \end{aligned}$$

(31)

$$\begin{aligned} \begin{aligned}&Im\left[ X_m^x(t_1,f_1)\right] Im\left[ X_m^y(t_1,f_1)\right] \\&\quad = Im(a^x_{mn}) Im(a^y_{mn})Re^2\left[ S_n(t_1,f_1)\right] +Re(a^y_{mn})Re(a^x_{mn})Im^2\left[ S_n(t_1,f_1)\right] \\&\qquad +Im(a^x_{mn})Re(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\&\qquad +Re(a^x_{mn})Im(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\ \end{aligned}\qquad \end{aligned}$$

(32)

$$\begin{aligned} \begin{aligned}&Re\left[ X_m^x(t_1,f_1)\right] Im\left[ X_m^y(t_1,f_1)\right] \\&\quad = Re(a^x_{mn}) Im(a^y_{mn})Re^2\left[ S_n(t_1,f_1)\right] -Im(a^x_{mn})Re(a^y_{mn})Im^2\left[ S_n(t_1,f_1)\right] \\&\qquad +Re(a^x_{mn})Re(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\&\qquad -Im(a^y_{mn})Im(a^x_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\ \end{aligned}\qquad \end{aligned}$$

(33)

$$\begin{aligned} \begin{aligned}&Im\left[ X_m^x(t_1,f_1)\right] Re\left[ X_m^y(t_1,f_1)\right] \\&\quad = Im(a^x_{mn})Re(a^y_{mn})Re^2\left[ S_n(t_1,f_1)\right] -Im(a^y_{mn})Re(a^x_{mn})Im^2\left[ S_n(t_1,f_1)\right] \\&\qquad -Im(a^x_{mn})Im(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] \\&\qquad +Re(a^x_{mn})Re(a^y_{mn})Re\left[ S_n(t_1,f_1)\right] Im\left[ S_n(t_1,f_1)\right] .\qquad \\ \end{aligned} \end{aligned}$$

(34)

Then, we define the variables

$$\begin{aligned} \begin{aligned} Z_n&=Re\left[ X_m^x(t_1,f_1)\right] Re\left[ X_m^y(t_1,f_1)\right] +Im\left[ X_m^x(t_1,f_1)\right] Im\left[ X_m^y(t_1,f_1)\right] \\&=\left| \begin{array}{ll} Re\left[ X^x_m(t_1,f_1)\right] &{} -Im\left[ X^x_m(t_1,f_1)\right] \\ Im\left[ X^y_m(t_1,f_1)\right] &{} Re\left[ X^y_m(t_1,f_1)\right] \\ \end{array} \right| \end{aligned} \end{aligned}$$

(35)

$$\begin{aligned} \begin{aligned} Y_n&= Re\left[ X_m^x(t_1,f_1)\right] Im\left[ X_m^y(t_1,f_1)\right] - Im\left[ X_m^x(t_1,f_1)\right] Re\left[ X_m^y(t_1,f_1)\right] \\&=\left| \begin{array}{ll} Re\left[ X^x_m(t_1,f_1)\right] &{} Im\left[ X^x_m(t_1,f_1)\right] \\ Re\left[ X^y_m(t_1,f_1)\right] &{} Im\left[ X^y_m(t_1,f_1)\right] \\ \end{array} \right| . \end{aligned} \end{aligned}$$

(36)

Based on Eqs. (3) and (4) and the above formulas, we obtain the following equation after simplification:

$$\begin{aligned} \begin{aligned} \frac{Z_n}{Y_n}&=\frac{Re(a^x_{mn})Re(a^y_{mn}) + Im(a^x_{mn})Im(a^y_{mn})}{Re(a^x_{mn})Im(a^y_{mn}) - Im(a^x_{mn})Re(a^y_{mn})}\\&=\frac{1}{\tan [u(\sin \phi _n \sin \theta _n -\cos \phi _n \sin \theta _n)]} \\&=g(\phi _n,\theta _n), \end{aligned} \end{aligned}$$

(37)

where $u=2 \pi d(m-1)/\lambda$. $g(\phi _n,\theta _n)$ is a function of $\phi _n$ and $\theta _n$; this means that the signals from one source in the TF domain have the same slopes using $Y_n$ and $Z_n$ as the coordinate axis x and y, respectively. Thus, the newly defined variables can cluster signals from one source in the TF domain.

The derivation is completed.

Appendix B: Extending to the Uniform Circular Array Model

Figure 9 describes the model of the uniform circular array with radius R. According to [5, 22], this special uniform circular array is called the “uniform circular array with a center sensor.” Comparing with the conventional uniform circular array, this array has a narrower main lobe and a lower side lobe such that interference outside the main lobe can be suppressed more effectively. Assume that there are M sensors and N far-field sources. The reference sensor is positioned at the center point of the circular array. The nth source two-dimensional DOA is $\left( \theta _n,\phi _n\right)$, where $\phi _n$ and $\theta _n$ denote the azimuth and elevation.

Fig. 9

Model of the uniform circular array

The time delay for the nth source to the mth sensor is $\tau _{mn}= \cos ( \frac{2\pi }{M-1}(m-2) - \phi _n)sin( \theta _n)\frac{ R}{v}$; then, we can obtain the mixing matrix A

$$\begin{aligned} {\mathbf {A}}= \begin{bmatrix} 1&\cdots&1 \\ e^{j 2\pi f\tau _{21}}&\cdots&e^{j 2\pi f\tau _{2N}}\\ \vdots&\ddots&\vdots \\ e^{j 2\pi f\tau _{M1}}&\cdots&e^{j 2\pi f\tau _{MN}} \\ \end{bmatrix} = \begin{bmatrix} {\begin{matrix} 1 &{} \cdots &{} 1 \\ e^{j\xi _1 \cos \phi _1 } &{} \cdots &{} e^{j\xi _N \cos \phi _N } \\ \vdots &{} \ddots &{}\vdots \\ e^{j\xi _1 \cos \left( \frac{2\pi }{M-1} (M-2) - \phi _1\right) } &{} \cdots &{} e^{j\xi _N \cos \left( \frac{2\pi }{M-1} (M-2) - \phi _N\right) } \\ \end{matrix}} \end{bmatrix} \end{aligned}$$

(38)

where $\xi _N=2\pi R \sin \theta _N/ \lambda$. Set the center of the array as the reference sensor. ${\mathbf {A}}$ has a similar structure as the mixing matrix of ULA, which means that $a_{1n}=1$ and $\Vert a_{mn}\Vert =1 (m\not =1)$, and the above prerequisites are fundamental to Eq. (6) [14]. Therefore, we can obtain the same conclusion for the SSP identification.

However, the SSPs cannot be clustered because the mixing matrix ${\mathbf {A}}$ contains complex values for circular arrays. Therefore, as in the L-shaped algorithm, we define $Y_n$ and $Z_n$. Consider only one effective source $s_n(t)$ with a relatively large value at any TF point $(t_1,f_1)$.

$$\begin{aligned} Y_n= & {} \left| \begin{array}{ll} Re\left[ X_m(t_1,f_1)\right] &{} Im\left[ X_m(t_1,f_1)\right] \\ Re\left[ X_{m+1}(t_1,f_1)\right] &{} Im\left[ X_{m+1}(t_1,f_1)\right] \\ \end{array} \right| \end{aligned}$$

(39)

$$\begin{aligned} Z_n= & {} \left| \begin{array}{ll} Re\left[ X_m(t_1,f_1)\right] &{} -Im\left[ X_m(t_1,f_1)\right] \\ Im\left[ X_{m+1}(t_1,f_1)\right] &{} Re\left[ X_{m+1}(t_1,f_1)\right] \\ \end{array} \right| , \end{aligned}$$

(40)

where $X_m(t_1,f_1)$ and $X_{m+1}(t_1,f_1)$ represent the STFT coefficients of the mth and $(m+1)$th sensor signals, respectively, and $2 \le m \le M$.

Similar to Eqs. (11) and (12), we can obtain

$$\begin{aligned} \left[ \frac{Z_n}{Y_n}\right] _m&= \frac{Re(a_{mn})Re(a_{(m+1)n})+Im(a_{(m+1)n})Im(a_{mn})}{Re(a_{mn})Im(a_{(m+1)n})-Re(a_{(m+1)n})Im(a_{mn})}\nonumber \\&=\frac{1}{tan\left\{ \xi _n\left[ \cos \left( \frac{2\pi (m-1)}{(M-1)}-\phi _n\right) - \cos \left( \frac{2\pi (m-2)}{(M-1)}-\phi _n\right) \right] \right\} }\nonumber \\&=\frac{1}{tan\left\{ 2 \xi _n \sin \left( \frac{\pi }{M-1}\right) \left[ \sin \left( \frac{\pi }{(M-1)}(2m-3)-\phi _n\right) \right] \right\} }&m\ge 2 \end{aligned}$$

(41)

$$\begin{aligned} \left[ \frac{Z_n}{Y_n}\right] _1&= \frac{Re(a_{mn})Re(a_{1n})+Im(a_{1n})Im(a_{mn})}{Re(a_{mn})Im(a_{1n})-Re(a_{1n})Im(a_{mn})}\nonumber \\&= -\frac{1}{tan\left( \xi _n \cos \left[ \frac{2\pi }{M-1}(m-2)-\phi _n\right] \right. }&m = 1, \end{aligned}$$

(42)

where $\xi _n=2\pi R \sin \theta _n/ \lambda$. $Re(a_{1n})=1$ and $Im(a_{1n})=0$.

Equations (41) and (42) are functions of $\phi _n$ and $\theta _n$. This means that the SSPs in the TF domain have the same slopes when using $Y_n$ and $Z_n$ as the coordinate axes x and y, respectively. To validate the above derivation, we perform a simulation. The parameter setting and sources are the same as in Experiment 2. The results are averaged over 50 Monte Carlo trials under the 20 dB SNR condition. The NMSE of TIFROM, DUET and the proposed algorithms based on uniform circular arrays is $-33.1, -34.3 and -35.8$ dB, respectively. Similar to the L-shaped array, the simulation results show the applicability of the proposed algorithm to uniform circular arrays.