A Promising Technique for Blind Identification: The Generic Statistics

Abstract

The generic statistics, related to lower-order derivatives of the log characteristic function (CAF) evaluating at the processing points located away from the origin, are frequently used in multivariate statistical signal processing. In comparison with the conventional statistics, e.g., cumulants, the generic statistics have the following advantages: (a) they can offer the structural simplicity and controllable statistical stability of lower-order statistics, and retain higher-order statistical information; (b) if the derivatives of the log CAF were evaluated at all (infinitely many) possible processing-points, a complete description of the joint CAF would be obtained. Furthermore, we show in this paper that even if a random process is symmetrically distributed, the odd-order generic statistics are not equal to zero, while in such a case the odd-order cumulants are equal to zero. For these reasons, a family of blind identification (BI) methods, in which the mixing matrix is obtained by decomposing the tensor constructed by the higher order derivatives of the log CAF of the observations, is proposed to achieve BI of underdetermined mixtures. Simulation results show that the BI methods based on generic statistics have superior performance to the existing cumulant-based method, such as FOOBI method, especially, when the SNR of the observations is high and/or the data block is short.

Appendices

Appendix A

1.1 Proof of Eq. (7)

In this Appendix, we show the computational details of equation in (7). First, the differentiation of (7) with respect to $u_{q_{1}}$ gives

$$\begin{aligned} \frac{\partial \psi _{x} (\mathbf{u})}{\partial u_{q_{1}} }= & {} \sum \nolimits _{p=1}^{P} {\frac{\partial \left( {\varphi _{p} \left( \sum \nolimits _{q} {A_{qp} u_{q}}\right) } \right) }{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q} } } \right) }\frac{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q}} } \right) }{\partial u_{q_{1}}}} \nonumber \\= & {} \sum \nolimits _{p=1}^{P} {A_{q_{1 p}}} \frac{\partial \left( {\varphi _{p} \left( \sum \nolimits _{q} {A_{qp} u_{q}}\right) } \right) }{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q}}} \right) } \end{aligned}$$

(17)

Similarly, the differentiation of (7) with respect to $(u_{q_{1}} ,u_{q_{2}} ,\ldots ,u_{q_{K}})$ gives

$$\begin{aligned} \frac{\partial ^{(K)}\psi _{x} (\mathbf{u})}{\partial u_{q_1 } \partial u_{q_{2}} \cdots \partial u_{q_{K}}}= & {} \sum \nolimits _{p=1}^P {A_{q_{1 p}} \cdots A_{q_{K-1}p} \frac{\partial ^{(K)}\left( {\varphi _{p} \left( \sum \nolimits _{q} {A_{qp} u_{q} } \right) } \right) }{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q} } } \right) ^{K}}\frac{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q}}} \right) }{\partial u_{q_{K}}}} \nonumber \\= & {} \sum \nolimits _{p=1}^{P} {A_{q_{1 p}} \cdots A_{q_{K p}} \frac{\partial ^{(K)}\left( {\varphi _{p} \left( \sum \nolimits _{q} {A_{qp} u_{q} } \right) } \right) }{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q} } } \right) ^{K}}} \end{aligned}$$

(18)

Defining $C_{K,p} ={\partial ^{(K)}\left( {\varphi _{p} (\sum \nolimits _{q} {A_{qp} u_{q} } )} \right) }\big /{\partial \left( {\sum \nolimits _{q} {A_{qp} u_{q}}} \right) ^{K}}$, we can rewrite (18) in a more compact form as in (7).

Appendix B

1.1 Proof of Theorem 1

Lemma

Suppose z is a random variable with a $K\hbox {th}$-order cumulant. Then, for any $b\in {R}, z+b$ has a $K\hbox {th}$-order cumulant and

$$\begin{aligned} C_{K,z} (z+b)=\left\{ {\begin{array}{ll} C_{K,z} (z)+b,&{}\hbox { if }K=1 \\ C_{K,z} (z),&{}\hbox { if }K\ne 1 \\ \end{array}} \right. \end{aligned}$$

(19)

Proof

According to the definition of the generating function in (2), we obtain

$$\begin{aligned} \phi _{z+b} (u)= & {} {E}[\exp (v(z+b))]=\exp (vb){E}[\exp (vz)] \\\Rightarrow & {} \varphi _{z+b} (v)=vb+\varphi _{z} (v) \\\Rightarrow & {} {{\hbox {d}}^{(K)}\varphi _{z+b} (v)}\big /{{\hbox {d}}v^{K}}={{\hbox {d}}^{(K)}(vb)}\big /{{\hbox {d}}v^{K}}+{{\hbox {d}}^{(K)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{K}} \end{aligned}$$

Since $C_{K,z} ={{\hbox {d}}^{(K)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{K}}\left| {_{v=0} } \right.$, Eq. (19) holds. $\square$

Suppose z is a random process which is symmetrically distributed. According to Lemma, the cumulants of z are not affected by adding a constant to z. Hence, we can further assume that z has zero mean.

Because the probability density function f(z) and the mean of random variable z are symmetric and zero, respectively, then, we obtain

$$\begin{aligned} \phi _{z}^{(2k+1)}(v)\left| {_{v=0} } \right. ={E}[z^{2k+1}]=\int _{z} {z^{2k+1}f(z)dz} =0 \end{aligned}$$

(20)

On the other hand, exploiting the relationship between GF and CGF, we derive

$$\begin{aligned} \varphi _{z} (v)=\log {E}(\exp (vz))=\log (\phi _{z} (v)) \end{aligned}$$

(21)

Differentiating (21) with respect to u gives

$$\begin{aligned} \phi _{z}^{\prime }(v)=\phi _{z} (v)\varphi _{z}^{\prime }(v) \end{aligned}$$

(22)

and evaluating (22) at $v=0$ gives $M_{1,z} =C_{1,z} =0$. Thus, Theorem 1 holds when the order is 1.

Differentiating (21) $2k+1$ times and evaluating at $v=0$ gives

$$\begin{aligned} \phi _{z}^{(2k+1)}(0)=\sum \limits _{j=0}^{2k} {\phi _{z}^{(j)}(0)\varphi _{z}^{(2k+1-j)}(0)} \end{aligned}$$

(23)

If $j=2m+1, m=1,\ldots ,k$, using the property shown in Eq. (20), we obtain $\phi _{z}^{(j)}(0)\varphi _{z}^{(2k+1-j)}(0)=0$; else if $j=2m, m=1,\ldots ,k$, then, $2k+1-j$ is odd. Therefore, $\phi _{z}^{(j)}(0)\varphi _{z}^{(2k+1-j)}(0)=0$ also holds true based on assumption that Theorem 1 holds true when the order is lower than $2k+1$ and odd. It is interesting to mention that it agrees with the mathematical induction idea.

In all, Theorem 1 follows.

Appendix C

1.1 Proof of Theorem 2

Suppose z is a random process which is symmetrically distributed. According to Theorem 1, its odd-order cumulant is equal to zero, i.e.,

$$\begin{aligned} C_{2k-1,z} ={{\hbox {d}}^{(2k-1)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{2k-1}}\left| {_{v=0} } \right. =0 \end{aligned}$$

(24)

Note that its 2k-order cumulants are not null. That is

$$\begin{aligned} C_{2k,z} ={{\hbox {d}}^{(2k)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{2k}}\left| {_{v=0} } \right. \ne 0 \end{aligned}$$

(25)

Therefore, ${{\hbox {d}}^{(2k-1)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{2k-1}}$ is a monotone function for v near 0. Thus, we obtain

$$\begin{aligned} {{\hbox {d}}^{(2k-1)}\varphi _{z} (v)}\big /{{\hbox {d}}v^{2k-1}}\left| {_{v\rightarrow 0} } \right.= & {} {\hbox {d}}^{(2k-1)}\varphi _{z} (0)+{\hbox {d}}^{(2k)}\varphi _{z} (0)\times v\nonumber \\\ne & {} 0 \end{aligned}$$

(26)

Thus, Theorem 2 holds true.