Underdetermined Blind Identification for Uniform Linear Array by a New Time–Frequency Method

Abstract

This paper proposes a novel underdetermined blind identification method with several new single-source points (SSPs) detection criteria for uniform linear array, where the mixing matrix is complex-valued. These new criteria are based on quadratic time–frequency distribution and employed to detect the SSPs so that the complex-valued mixing matrix can be estimated more precisely. To further enhance the estimation accuracy, a modified peak detection method is presented by exploiting the known source number. Finally, the complex-valued mixing matrix can be obtained by performing a clustering algorithm on samples at selected SSPs. One of the outstanding superiorities for the proposed algorithm is that the new criteria are strict enough for the points that are not the SSPs, which ensures the estimation accuracy of the mixing matrix. The other is that the performance of estimation precision is high even in the noisy case. Numerical simulation results verify the superiority of the proposed algorithm over the existing algorithms.

Appendices

Appendix A

Proof of Theorem 1

We assume that (t, f) is an SSP of the c-th source. According to the Eqs. (2) and (7), we set $\alpha _c =-\pi \sin (\theta _c )$ and can obtain:

$$\begin{aligned} \begin{array}{l} \mathbf{W}_\mathbf{X} (t,f)=\rho _{s_c s_c } (t,f)\mathbf{a}_c \mathbf{a}_c^\mathrm{H} \; \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;=\rho _{s_c s_c } (t,f)\left( {{\begin{array}{c} 1 \\ {e^{j\alpha _c }} \\ \vdots \\ {e^{j(M-1)\alpha _c }} \\ \end{array} }} \right) \Bigg (1 \quad {e^{-j\alpha _c }}\quad \cdots \quad {e^{-j(M-1)\alpha _c }} \Bigg ) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;=\rho _{s_c s_c } (t,f)\left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1&{} {e^{-j\alpha _c }}&{} \cdots &{} {e^{-j(M-1)\alpha _c }} \\ {e^{j\alpha _c }}&{} 1&{} \cdots &{} {e^{-j(M-2)\alpha _c }} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {e^{j(M-1)\alpha _c }}&{} {e^{j(M-2)\alpha _c }}&{} \cdots &{} 1 \\ \end{array} }} \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;=\left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\rho _{x_1 x_1 } (t,f)}&{} {\rho _{x_1 x_2 } (t,f)}&{} \cdots &{} {\rho _{x_1 x_M } (t,f)} \\ {\rho _{x_2 x_1 } (t,f)}&{} {\rho _{x_2 x_2 } (t,f)}&{} \cdots &{} {\rho _{x_2 x_M } (t,f)} \\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ {\rho _{x_M x_1 } (t,f)}&{} {\rho _{x_M x_2 } (t,f)}&{} \cdots &{} {\rho _{x_M x_M } (t,f)} \\ \end{array} }} \right) \end{array} \end{aligned}$$

(15)

Observe the diagonal elements and we can get:

$$\begin{aligned} \rho _{x_1 x_1 } (t,f)=\rho _{x_2 x_2 } (t,f)=\cdots =\rho _{x_M x_M } (t,f) \end{aligned}$$

(16)

That is a condition that SSPs must satisfy. Now we will prove that the condition is strict for other points. We assume that $(t^{\prime },f^{\prime })$ is not the SSP and put the Eqs. (2), (5), (9) into (6) at $(t^{\prime },f^{\prime })$. Then, we will get:

$$\begin{aligned}&{} \mathbf{AW}_\mathbf{s} (t^{\prime },f^{\prime })\mathbf{A}^{\hbox {H}} =\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\rho _{x_1 x_1 } (t^{\prime },f^{\prime })}&{} {\rho _{x_1 x_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{x_1 x_M } (t^{\prime },f^{\prime })} \\ {\rho _{x_2 x_1 } (t^{\prime },f^{\prime })}&{} {\rho _{x_2 x_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{x_2 x_M } (t^{\prime },f^{\prime })} \\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ {\rho _{x_M x_1 } (t^{\prime },f^{\prime })}&{} {\rho _{x_M x_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{x_M x_M } (t^{\prime },f^{\prime })} \\ \end{array} }} \right] =\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1&{} 1&{} \cdots &{} 1 \\ {e^{j\alpha _1 }}&{} {e^{j\alpha _2 }}&{} \cdots &{} {e^{j\alpha _N }} \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ {e^{j(M-1)\alpha _1 }}&{} {e^{j\alpha _2 }}&{} \cdots &{} {e^{j\alpha _N }} \\ \end{array} }} \right] \nonumber \\&\quad \quad \times \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\rho _{s_1 s_1 } (t^{\prime },f^{\prime })}&{} {\rho _{s_1 s_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{s_1 s_N } (t^{\prime },f^{\prime })} \\ {\rho _{s_2 s_1 } (t^{\prime },f^{\prime })}&{} {\rho _{s_2 s_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{s_2 s_N } (t^{\prime },f^{\prime })} \\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ {\rho _{s_N s_1 } (t^{\prime },f^{\prime })}&{} {\rho _{s_N s_2 } (t^{\prime },f^{\prime })}&{} \cdots &{} {\rho _{s_N s_N } (t^{\prime },f^{\prime })} \\ \end{array} }} \right] \times \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1&{} {e^{-j\alpha _1 }}&{} \cdots &{} {e^{-j(M-1)\alpha _1 }} \\ 1&{} {e^{-j\alpha _2 }}&{} \cdots &{} {e^{-j\alpha _2 }} \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 1&{} {e^{-j\alpha _N }}&{} \cdots &{} {e^{-j\alpha _N }} \\ \end{array} }} \right] \nonumber \\&\quad =\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{-j\alpha _q }\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} \cdots &{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{-(M-1)j\alpha _q }\rho _{s_p s_q } (t^{\prime },f^{\prime })} } } \\ {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j\alpha _p }\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j(\alpha _p -\alpha _q )}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} \cdots &{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j[\alpha _p -(M-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } } \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j(M-1)\alpha _p }\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j[(M-1)\alpha _p -\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } }&{} \cdots &{} {\sum \limits _{q=1}^N {\sum \limits _{p=1}^N {e^{j[(M-1)\alpha _p -(M-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(17)

If $(t^{\prime },f^{\prime })$ satisfies the Eq. (16), we will obtain:

$$\begin{aligned} \sum _{q=1}^N {\sum _{p=1}^N {\rho _{s_p s_q } (t^{\prime },f^{\prime })} }= & {} \sum _{q=1}^N {\sum _{p=1}^N {e^{j(\alpha _p -\alpha _q )}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } =\cdots \nonumber \\= & {} \sum _{q=1}^N {\sum _{p=1}^N {e^{j[(M-1)\alpha _p -(M-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } \end{aligned}$$

(18)

If the equation above is permanently equal for arbitrary value of $\rho _{s_p s_q } (t^{\prime },f^{\prime }),p,q\in \{1,2,\ldots ,N\},p\ne q$, we will get $\alpha _p =\alpha _q , \quad \forall p, \quad q\in \{1,2,\ldots ,N\},p\ne q$. And $\alpha _i =-\pi \sin (\theta _i ),i\in \{1,\ldots ,N\}$, so $\theta _p =\theta _q , \quad \forall p,q\in \{1,2,\ldots ,N\},p\ne q$, it contradicts with the hypothesis $\theta _i \ne \theta _j (\forall i\ne j)$ that we made before.

A little disturbing point satisfies the Eq. (18), such as the points satisfy the equation below:

$$\begin{aligned} \rho _{s_q s_p } (t^{\prime },f^{\prime })=e^{j(\alpha _p -\alpha _{_q } )}\rho _{s_p s_q } (t^{\prime },f^{\prime })=\cdots =e^{j(M-1)(\alpha _p -\alpha _{_q } )}\rho _{s_p s_q } (t^{\prime },f^{\prime }) \\ \rho _{s_p s_q } (t^{\prime },f^{\prime })=e^{j(\alpha _q -\alpha _{_p } )}\rho _{s_q s_p } (t^{\prime },f^{\prime })=\cdots =e^{j(M-1)(\alpha _q -\alpha _p )}\rho _{s_q s_p } (t^{\prime },f^{\prime }) \end{aligned}$$

where $\forall p,q\in \{1,2,\ldots ,N\},p\ne q$. In order to satisfy the equation above, there will be a strict condition that is shown below:

$$\begin{aligned}&\rho _{s_q s_p } (t^{\prime },f^{\prime }),\rho _{s_p s_q } (t^{\prime },f^{\prime })\ne 0 \nonumber \\&\frac{\rho _{s_q s_p } (t^{\prime },f^{\prime })}{\rho _{s_p s_q } (t^{\prime },f^{\prime })}=e^{j(\alpha _p -\alpha _{_q } )},\forall p,q\in \{1,2,\ldots ,N\},p\ne q \end{aligned}$$

(19)

The condition is too strict for disturbing points. So far, we do not consider the case with$\rho _{s_p s_q } (t^{\prime },f^{\prime })=0$, because at disturbing point there will exist many $p,q\in \{1,2,\ldots ,N\}$ where$\rho _{s_p s_q } (t^{\prime },f^{\prime })\ne 0$, and these $p,q\in \{1,\ldots ,N\}$ satisfying (19) is strict all the same.

In conclusion, the Eq. (16) can be served as a criterion to detect the SSPs. But the Eq. (16) must not be satisfied sternly, so we introduce a positive threshold $\varepsilon _1$ which is close to 0. The modified criterion can be described by:

$$\begin{aligned} \sum _{{\begin{array}{l} {i,j=1} \\ {i\ne j} \\ \end{array} }} ^M {\left| {\rho _{x_i x_i } (t,f)-\rho _{x_j x_j } (t,f)} \right| } \le \varepsilon _1 \end{aligned}$$

This completes the proof of the Theorem 1. $\square$

Appendix B

Proof of Theorem 2

We assume that (t, f) is an SSP of the c-th source. According to the Eq. (15), we can obtain M equations:

$$\begin{aligned} \left[ {{\begin{array}{c} {\rho _{x_1 x_1 } (t,f)} \\ {\rho _{x_2 x_1 } (t,f)} \\ \vdots \\ {\rho _{x_M x_1 } (t,f)} \\ \end{array} }} \right]= & {} \rho _{s_c s_c } (t,f)\left[ {{\begin{array}{c} 1 \\ {e^{j\alpha _c }} \\ \vdots \\ {e^{j(M-1)\alpha _c }} \\ \end{array} }} \right] \\ \left[ {{\begin{array}{cc} {\rho _{x_1 x_2 } (t,f)} \\ {\rho _{x_2 x_2 } (t,f)} \\ \vdots \\ {\rho _{x_M x_2 } (t,f)} \\ \end{array} }} \right]= & {} \rho _{s_c s_c } (t,f)\left[ {{\begin{array}{c} {e^{-j\alpha _c }} \\ 1 \\ \vdots \\ {e^{j(M-2)\alpha _c }} \\ \end{array} }} \right] \\ \vdots \\ \left[ {{\begin{array}{cc} {\rho _{x_1 x_M } (t,f)} \\ {\rho _{x_2 x_M } (t,f)} \\ \vdots \\ {\rho _{x_M x_M } (t,f)} \\ \end{array} }} \right]= & {} \rho _{s_c s_c } (t,f)\left[ {{\begin{array}{c} {e^{-j(M-1)\alpha _c }} \\ {e^{-j(M-2)\alpha _c }} \\ \vdots \\ 1 \\ \end{array} }} \right] \end{aligned}$$

The M equation above can be written to be a unitary equation:

$$\begin{aligned} \left[ {{\begin{array}{cc} {\rho _{x_1 x_k } (t,f)} \\ {\rho _{x_2 x_k } (t,f)} \\ \vdots \\ {\rho _{x_M x_k } (t,f)} \\ \end{array} }} \right] =\rho _{s_c s_c } (t,f)\left[ {{\begin{array}{cc} {e^{j(1-k)\alpha _c }} \\ {e^{j(2-k)\alpha _c }} \\ \vdots \\ {e^{j(M-k)\alpha _c }} \\ \end{array} }} \right] \end{aligned}$$

(20)

where $k=1,2,\ldots ,M$. Then we separate these complexes into real part and imaginary part where $\rho _{s_c s_c } (t,f)$ and $\rho _{x_k x_k } (t,f)$ are the real value seen from the Eq. (3).

$$\begin{aligned}&\mathrm{Re}\left( {\left[ {{\begin{array}{c} {\rho _{x_1 x_k } (t,f)} \\ {\rho _{x_2 x_k } (t,f)} \\ \vdots \\ {\rho _{x_M x_k } (t,f)} \\ \end{array} }} \right] } \right) + \mathrm{Im}\left( {\left[ {{\begin{array}{c} {\rho _{x_1 x_k } (t,f)} \\ {\rho _{x_2 x_k } (t,f)} \\ \vdots \\ {\rho _{x_M x_k } (t,f)} \\ \end{array} }} \right] } \right) \\&\quad =\left( {\mathrm{Re}\left( {\left[ {{\begin{array}{c} {e^{j(1-k)\alpha _c }} \\ {e^{j(2-k)\alpha _c }} \\ \vdots \\ {e^{j(M-k)\alpha _c }} \\ \end{array} }} \right] } \right) +\mathrm{Im}\left( {\left[ {{\begin{array}{c} {e^{j(1-k)\alpha _c }} \\ {e^{j(2-k)\alpha _c }} \\ \vdots \\ {e^{j(M-k)\alpha _c }} \\ \end{array} }} \right] } \right) } \right) \times \rho _{s_c s_c } (t,f) \end{aligned}$$

And we can get:

$$\begin{aligned}&\mathrm{Re}\left( {\left[ {{\begin{array}{c} {\rho _{x_1 x_k } (t,f)} \\ {\rho _{x_2 x_k } (t,f)} \\ \vdots \\ {\rho _{x_M x_k } (t,f)} \\ \end{array} }} \right] } \right) =\left( {\mathrm{Re}\left( {\left[ {{\begin{array}{c} {e^{j(1-k)\alpha _c }} \\ {e^{j(2-k)\alpha _c }} \\ \vdots \\ {e^{j(M-k)\alpha _c }} \\ \end{array} }} \right] } \right) } \right) \times \rho _{s_c s_c } (t,f) \\&\mathrm{Im}\left( {\left[ {{\begin{array}{c} {\rho _{x_1 x_k } (t,f)} \\ {\rho _{x_2 x_k } (t,f)} \\ \vdots \\ {\rho _{x_M x_k } (t,f)} \\ \end{array} }} \right] } \right) =\left( {\mathrm{Im}\left( {\left[ {{\begin{array}{c} {e^{j(1-k)\alpha _c }} \\ {e^{j(2-k)\alpha _c }} \\ \vdots \\ {e^{j(M-k)\alpha _c }} \\ \end{array} }} \right] } \right) } \right) \times \rho _{s_c s_c } (t,f) \end{aligned}$$

So we can obtain:

$$\begin{aligned} \mathrm{Re}(\rho _{x_m x_k } (t,f))= & {} \mathrm{Re}(e^{j(m-k)\alpha _c })\times \rho _{s_c s_c } (t,f) \nonumber \\ \mathrm{Im}(\rho _{x_m x_k } (t,f))= & {} \mathrm{Im}(e^{j(m-k)\alpha _c })\times \rho _{s_c s_c } (t,f) \nonumber \\ \rho _{x_k x_k } (t,f)= & {} \rho _{s_c s_c } (t,f) \end{aligned}$$

(21)

where $m=1,2,\ldots ,M$. We add the square of the first two equations of (21) together and notice that $\left| {e^{j(m-k)\alpha _c }} \right| =1$. So, we can obtain:

$$\begin{aligned} \rho _{s_c s_c } (t,f)= & {} \sqrt{{\mathrm{Re}}^{2}\left( \rho _{x_m x_k } (t,f)\right) +{\mathrm{Im}}^{2}\left( \rho _{x_m x_k } (t,f)\right) } \nonumber \\= & {} \left| {\rho _{x_m x_k } (t,f)} \right| \nonumber \\= & {} \rho _{x_k x_k } (t,f) \end{aligned}$$

(22)

Equation (22) indicates that $\left| {\rho _{x_m x_k } (t,f)} \right|$, namely the modulus or the absolute value of mixture signals WVD, are the same for the $\forall m,k\in \{1,\ldots ,M\}$ including the case that $m=k$ (that is $\rho _{x_k x_k } (t,f))$.

That is a condition that SSPs must satisfy. Now we will prove that the condition is strict for other points. We assume that $(t^{\prime },f^{\prime })$ is the disturbing point. From (17), if $(t^{\prime },f^{\prime })$ satisfies the condition in (22), we can obtain:

$$\begin{aligned}&\left| {\sum _{q=1}^N {\sum _{p=1}^N {e^{j[(m-1)\alpha _p -(k-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } } \right| \nonumber \\&=\sum _{q=1}^N {\sum _{p=1}^N {e^{j[(k-1)\alpha _p -(k-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} } \end{aligned}$$

(23)

where $m,k\in \{1,2,\ldots ,M\},m\ne k$. If we set $\sum _{q=1}^N {e^{j[(k-1)\alpha _p -(k-1)\alpha _q ]}\rho _{s_p s_q } (t^{\prime },f^{\prime })} =\mathbf{\rho }_p$, (23) will turn to be:

$$\begin{aligned} \left| {\sum _{p=1}^N {e^{j(m-k)\alpha _p }{} \mathbf{\rho }_p } } \right| =\sum _{p=1}^N {\mathbf{\rho }_p } \end{aligned}$$

(24)

$\mathbf{\rho }_p$ is arbitrary for different sources. So, it can be inferred that $\alpha _1 =\alpha _2 =\cdots =\alpha _N$. $\alpha _i =-\pi \sin (\theta _i ),i\in \{1,2,\ldots ,N\}$, so, $\theta _p =\theta _q , \quad \forall p,q\in \{1,2,\ldots ,N\},p\ne q$. That is contradictory with the hypothesis that $\theta _i \ne \theta _j (\forall i\ne j)$. Hence, the condition in (22) is also strict for disturbing points except for a quit few special cases.

Since $\rho _{x_m x_k } (t,f)$ and $\rho _{x_k x_m } (t,f)$ are conjugate symmetry, they have the same modulus. So, we can simplify the condition in (22) to be

$$\begin{aligned} \rho _{x_1 x_1 } (t,f)=\left| {\rho _{x_m x_k } (t,f)} \right| ,m,k\in \{1,2,\ldots ,N\},m>k, \end{aligned}$$

(25)

The selected points may not strictly satisfy (25), so we have to modify (25) to be:

$$\begin{aligned} \sum _{\mathop {m,k=1}\limits _{m\ne k} }^M {\left| {\left| {\rho _{x_m x_k } (t,f)} \right| -\rho _{x_k x_k } (t,f)} \right| } \le \varepsilon _2 \end{aligned}$$

where $\varepsilon _2$ is a positive threshold that is close to 0.

From (21), we can get the estimation of DOAs. The phase angle of $\rho _{x_m x_k } (t,f)$ is same with that of $e^{j(m-k)\alpha _c }$, namely, $\mathrm{angle}(\rho _{x_m x_k } (t,f))=(m-k)\alpha _c$ where $\alpha _c =-\pi \sin (\theta _c ),c\in \{1,2,\ldots ,N\}$. So, we get:

$$\begin{aligned} \arcsin \left( \frac{1}{-\pi (m-k)}\mathrm{angle}\left( \rho _{x_m x_k } (t,f)\right) \right) =\theta _c \end{aligned}$$

where $m,k\in \{1,2,\ldots ,M\},m\ne k$. The equation above is the estimation of DOA.

This completes the proof of the Theorem 2. $\square$