# Blind signal separation for convolved non-stationary
signals

Mitsuru Kawamoto, Barros Alan Kardec, Ali Mansour, K. Matsuoka, Ohnishi

Noboru

► To cite this version:

Mitsuru Kawamoto, Barros Alan Kardec, Ali Mansour, K. Matsuoka, Ohnishi Noboru. Blind signal separation for convolved non-stationary signals. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 1999, J82-A (11), pp.1320-1328. hal-00802438

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<figure><img src="image_1.png" /></figure>
# Blind Signal Separation for Convolved Nonstationary Signals

Mitsuru Kawamoto, A.K. Barros, and A. Mansour

Bio-Mimetic Control Research Center, RIKEN, Nagoya, Japan 463-0003

Kiyotoshi Matsuoka

Department of Control Engineering, Kyushu Institute of Technology, Kitakyushu, Japan 804-8550

Noboru Ohnishi

Bio-Mimetic Control Research Center, RIKEN, Nagoya, Japan 463-0003
Graduate School of Engineering, Nagoya University, Nagoya

Blind signal separation is a signal processing technique that can extract the original signals from their mixtures by using only the observed signals when the mixed signals can be observed by multiple sensors. In this paper, we propose a technique for blind signal separation of separating original signals from the observed signals which mix the nonstationary signals, such as speech or music. The proposed method implements blind signal separation by minimizing a nonnegative function that achieves the minimum (zero) when the second-order cross-correlation value of the observed signals becomes zero. The effectiveness of the proposed method is verified by computer simulations and tests that used a mixed speech signal observed in an ordinary room.

Key words: Blind signal separation; nonstationary signal; convolutive mixture; instantaneous mixture; second-order moment

## 1. Introduction

Let speech (source signal), for example, flow from several speakers and have several microphones observe these mixed signals (Fig. 1). The question is, can the source signals be separated and extracted from only these observed

<figure><img src="image_2.png" /><figcaption>Fig. 1. Experimental setup for signal separation.</figcaption></figure>
signals? If the transfer functions between the speakers and the microphones are known, this separation problem can be easily solved. The reason is the source signals are obtained by applying the inverse functions of the transmission functions to the observed signals.

The problem we consider in this paper assumes that the transfer function between the speakers and the microphones is unknown. We also assume that the special statistical properties of the source signals are unknown. However, since a problem with no clues cannot be solved, the separation problem is solved under the assumption that the source signals are mutually statistically independent. Thus, given the condition that there is almost no information on the right side from the microphones in Fig. 1, the problem is to separate the source signals from the observed signals. This kind of signal separation problem is called blind signal separation.

The blind signal separation problem was first proposed by Herault and colleagues [1, 2]. Since then, many techniques for solving this problem have been proposed. Most of these techniques make the transfer functions between the speakers and the microphones an instantaneous mixture, that is,

$$x(t) = As(t) \quad (1)$$

where $x(t) = [x_1(t), \dots, x_N(t)]^T$, $A = [a_{ij}]$, $s(t) = [s_1(t), \dots, s_N(t)]^T$. The source signals $s(t)$ are to be separated from the observed signals $x(t)$ [1–7]. The model in Eq. (1) is easily handled mathematically. However, if blind signal separation in a real environment is considered, the separation may not go well for a technique obtained from the model given by Eq. (1) for the following reason. If the state in Fig. 1 is considered to be the model in Eq. (1), the model has the signals generated simultaneously from two speakers reach their respective microphones simultaneously. However, the situation in Fig. 1 hardly ever occurs.

Alternately, the transfer functions between the speakers and the microphones are convolutive mixtures, that is,

$$x(t) = \bar{A}(z)s(t) \quad (2)$$

and this blind signal separation problem which separates $s(t)$ from $X(t)$ has also been considered [8–17].

$\bar{A}(z) = [\bar{a}_{ij}(z)]$ is the transfer function that takes time delays into account. The observed signal $x_i(t)$ becomes

$$x_i(t) = \sum_{j=1}^{N} \bar{a}_{ij}(z)s_j(t) \quad (i = 1, \dots, N) \quad (3)$$

The $\bar{a}_{ij}(z)$ in Eq. (3) is

$$\bar{a}_{ij}(z) = \sum_{k=-\infty}^{\infty} a_{ij}(k)z^{-k}$$

where $z^{-k}$ is the delay operator for only a delay of $k$, that is, $s_j(t)z^{-k} = s_j(t-k)$. The model given by Eq. (2) is difficult to handle because problems such as stability must be considered. However, we believe a technique capable of performing blind signal separation in a real environment can be obtained. In this paper, we use the model given by Eq. (2).

Almost all conventional methods for blind signal separation [for both models given by Eqs. (1) and (2)] use the non-Gaussian characteristics of signals. Specifically, there are techniques using the high-order correlations of the observed signals [1–4] and either super-Gaussian or sub-Gaussian source signals [5, 8–11]. In a technique that evaluates high-order correlations, since many signals which have nearly Gaussian distributions exist in the natural environment, accurately estimating the high-order correlations of the observed signals becomes difficult. And for the techniques using the super-Gaussian or sub-Gaussian properties, it is complicated to discover an algorithm for these statistical properties from the observed signals.

In this paper, we propose a method of implementing blind signal separation that does not use the non-Gaussian characteristics (high-order correlation, super-Gaussian, sub-Gaussian properties) of the signal but uses the nonstationarity of the signals by decorrelating the cross-correlation (second-order correlation) of the observed signals. Matsuoka and colleagues proposed a method for the situation given by Eq. (1) that uses the nonstationarity of the signal [7]. The method proposed in this paper is an extension of the method given in Ref. 7 that uses the transfer functions between the speakers and microphones even for the situation given by Eq. (2). Thus, this technique can be applied in the experiments to separate the original nonstationary signals such as speech or music from their mixtures observed in a normal room.

Chan and colleagues [12], Ehlers and Schuster [13], and Gerven and colleagues [14, 15] have proposed techniques to implement blind signal separation by decorrelating the cross-correlation of the observed signals for the model given by Eq. (2). Our proposed method, which differs from the previously reported ones, is implemented by only evaluating the cross-correlation values of only the observed signals at the same time. In addition, our method can achieve blind signal separation even when $\bar{A}(z)$ is a nonminimum phase system. The effectiveness of this technique is verified by computer simulation and in a separation test using two speakers and two microphones in a normal room.

The signal processing techniques in blind signal separation are required for the problem of separating a specific signal from multiple signals, for example, measuring the signals in living bodies, manufacturing mechanisms for voice control by a specific person, speakerphone conversations, and fault diagnosis of machines by sound. A person can listen to the conversation of a specific person
while several people are talking at the same time, in other words, the hearing function called the “cocktail party effect.” Modeling this kind of function would be valuable.

## 2. Source Signals

The source signals $s(t)$ are mutually statistically independent and have zero mean. The autocorrelation matrix $R(t, \tau)$ of the source signals $s(t)$ is

$$ 
\begin{aligned}
\mathbf{R}(t, \tau) \\
&= E[\mathbf{s}(t)\mathbf{s}(t-\tau)] \\
&= \mathrm{diag}\{E[\mathbf{s}_1(t)\mathbf{s}_1(t-\tau)], \dots, \\
&\qquad E[\mathbf{s}_N(t)\mathbf{s}_N(t-\tau)]\} \\
&\equiv \mathrm{diag}\{\mathbf{r}_1(t, \tau), \dots, \mathbf{r}_N(t, \tau)\}
\end{aligned}
\quad (4) $$

where $\mathrm{diag}\{\dots\}$ denotes a diagonal matrix having the diagonal elements $\{\dots\}$. $E[x]$ denotes the ensemble average of $x$.

Furthermore, the source signals are nonstationary signals having the following properties.

### Assumption 1

Elements $r_i(t, \tau)$, $i = 1, \dots, N$, of the autocorrelation matrix $R(t, \tau)$ change with time independently of each other for all $\tau$.

The observed signals $\mathbf{x}(t)$ are given by the process in Eq. (2). And $\bar{\mathbf{A}}(z)$ satisfies the following.

### Assumption 2

The poles and zeros of $\bar{\mathbf{A}}(z)$ do not exist on the unit circle $|z| = 1$.

This assumption means that we consider the method which can achieve blind signal separation even when $\bar{\mathbf{A}}(z)$ is a nonminimum phase system.

## 3. Separation Process

The network in Fig. 2 extracts the source signals $s(t)$ from the observed signals $\mathbf{x}(t)$. The network in Fig. 2 is called the signal separation network. The signal separation network is a feedforward network. The input signals to the network are the observed signals $\mathbf{x}(t)$. The output signals $y_i(t)$ are

$$ 
\begin{aligned}
y_i(t) = x_i(t - L) &+ \sum_{j=1}^{N} \sum_{k=0}^{M} b_{ij}(k) x_j(t - k) \\
& (i = 1, \dots, N; 0 \le L < M)
\end{aligned}
\quad (5) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Signal separation network.</figcaption></figure>

$$ = \sum_{j=1}^{N} \bar{b}_{ij}(z) x_j(t) \quad (6) $$

where $b_{ij}(z) = \sum_{k=0}^{M} b_{ij}(k)z^{-k}$, $i$ and $j = 1, \dots, N$ for $i \neq j$, and $b_{ii}(z) = z^{-L}$, $i = 1, \dots, N$ and $0 \le L < M$.

In a vector representation, Eq. (6) can be written as

$$ \mathbf{y}(t) = \bar{\mathbf{B}}(z)\mathbf{x}(t) \quad (7) $$

where $\mathbf{y}(t) = [y_1(t), \dots, y_N(t)]^T$, $\mathbf{B}(z) = [b_{ij}(z)] = \sum_{k=0}^{M} \mathbf{B}(k)z^{-k}$, and $\mathbf{B}(k) = [b_{ij}(k)]$.

If Eq. (2) is substituted into Eq. (7), the latter becomes

$$ \mathbf{y}(t) = \bar{\mathbf{B}}(z)\bar{\mathbf{A}}(z)\mathbf{s}(t) \quad (8) $$

If $C \equiv B(z)A(z)$ is $D(z)P$, the output signal of the signal separation network becomes

$$ \mathbf{y}(t) = \mathbf{D}(z)\mathbf{P}\mathbf{s}(t) (\equiv \bar{\mathbf{s}}(t) = [\bar{\mathbf{s}}_1(t), \dots, \bar{\mathbf{s}}_N(t)]^T) $$

where $P$ is a permutation matrix and $D(z)$ is the diagonal matrix expressed by

$$ \mathbf{D}(z) = \mathrm{diag} \left\{ \sum_{k=-\infty}^{\infty} d_1(k) z^{-k}, \dots, \sum_{k=-\infty}^{\infty} d_N(k) z^{-k} \right\} $$

The output signals from the signal separation network become the source signals passed through some filter. $\bar{s}_i(t)$, for $i = 1, \dots, N$, can be regarded as the source signals. The reason is $\bar{s}_i(t)$, for $i = 1, \dots, N$, are mutually statistically
independent. The objective is to find matrix $B(z)_0$ that satisfies the relationship $C(z) = D(z)P$.

## 4. Blind Signal Separation

When the mixing process between the source signals and the microphones was the instantaneous mixture given by Eq. (1), and the source signals were nonstationary signals, Matsuoka and colleagues showed that blind signal separation is possible by minimizing the following cost function [7].

$$Q = \frac{1}{2} \left\{ \sum_{i=1}^{N} \log E[y_i(t)^2] - \log \det E[y(t)y(t)^T] \right\} \quad (9)$$

where $\det X$ is the determinant of matrix $X$. Refer to Ref. 7 for the output signals $y(t)$ used in Eq. (9).

In this paper, this idea is applied to the convolutive mixture expressed by Eq. (2). To this end, Eq. (9) is transformed to

$$Q(t, \bar{B}(z)) = \frac{1}{2} \left\{ \sum_{i=1}^{N} \log E[y_i(t-L)^2] - \log \det E[y(t-L)y(t-L)^T] \right\} \quad (10)$$

$L$ in Eq. (10) is the same time delay as $L$ in Eq. (5). Our proposed method regards the $t-L$ time as $t=0$. Thus, $[x(t), \dots, x(t-L+1)]$ can be treated as future values, and $[x(t-L-1), \dots, x(t-M)]$ as past values. For that reason, our method can handle a nonminimum phase system. Equation (10) is a nonnegative function that takes the minimum value (zero) only when $y_i(t-L)$ and $y_j(t-L)$ are decorrelated, that is, $E[y_i(t-L)y_j(t-L)] = 0$, for $i$ and $j=1, \dots, N$ and $i \neq j$. Equation (10), which differs from the evaluation function proposed in Refs. 12 and 13, evaluates only one set of cross-correlations, $E[y_i(t-L)y_j(t-L)]$ ($i, j=1, \dots, N; i \neq j$), and data outside that set, for example, $E[y_i(t)y_j(t-\tau)]$ ($i, j=1, \dots, N; i \neq j; \forall \tau$) are not taken into account. Matrix $B(z)_0$ that satisfies the relationship $C(z) = D(z)P$ is found by minimizing Eq. (10). Equation (10) is minimized by the following steepest descent method:

$$\Delta B(k) \doteq -\alpha \frac{\partial Q(t, \bar{B}(z))}{\partial B(k)} = -\alpha \left[ \frac{\partial Q(t, \bar{B}(z))}{\partial b_{ij}(k)} \right] \quad (11)$$

where $\alpha$ is a small positive constant. The symbol $\doteq$ in Eq. (11) means that only the nondiagonal elements on the right-hand side of Eq. (11) are identical to the ones on the

left-hand side. If the right-hand side of equation (11) is calculated, the following equation is obtained:

$$\Delta B(k) \doteq \alpha z^{-k} \left\{ I - (\mathrm{diag} E[y(t-L)y(t-L)^T])^{-1} \times E[y(t-L)y(t-L)^T] \right\} \bar{B}(z)^{-T} \quad (k = 0, \dots, M) \quad (12)$$

The derivation of Eq. (12) is shown in the Appendix. diag $X$ is the diagonal matrix having the diagonal elements of matrix $X$ as its elements.

In practice, $E[y(t-L)y(t-L)^T]$ is replaced by its instantaneous value $y(t-L)y(t-L)^T$. The moving average $\phi_i(t)$ is used to estimate $\mathrm{diag} E[y(t-L)y(t-L)^T]$:

$$\phi_i(t) = \beta \phi_i(t-1) + (1-\beta)y_i(t-L)^2 \quad (i = 1, \dots, N; 0 < \beta < 1) \quad (13)$$

Therefore, Eq. (12) becomes

$$\Delta B(k) \doteq \alpha z^{-k} \left\{ I - \Phi(t)^{-1} \times y(t-L)y(t-L)^T \right\} \bar{B}(z)^{-T} \quad (k = 0, \dots, M) \quad (14)$$

where $\Phi(t) = \mathrm{diag}\{\phi_1(t), \dots, \phi_N(t)\}$. Blind separation of nonstationary signals is implemented by updating $B(k)$, for $k=0, \dots, M$, in Eqs. (13) and (14).

## 5. Experimental Results

We shall show two test results of blind separation implemented by using the proposed method. The number of source signals is set to $N=2$ in both cases. Equation (14) is represented by

$$\begin{align}
\Delta b_{12}(k) &= -\frac{\alpha y_1(t-L)y_2(t-k)}{(1-z^{2L}b_{12}(z)b_{21}(z))\phi_1(t)} \nonumber \\
\Delta b_{21}(k) &= -\frac{\alpha y_2(t-L)y_1(t-k)}{(1-z^{2L}b_{12}(z)b_{21}(z))\phi_2(t)} \tag{15} \\
(k = 0, \dots, M) \nonumber
\end{align}$$

We do not use the algorithm given by Eq. (15), but rather the equations that remove the common term $1/(1-z^{2L}b_{12}(z)b_{21}(z))$:

$$\begin{align}
\Delta b_{12}(k) &= -\alpha \frac{y_1(t-L)y_2(t-k)}{\phi_1(t)} \nonumber \\
\Delta b_{21}(k) &= -\alpha \frac{y_2(t-L)y_1(t-k)}{\phi_2(t)} \tag{16}
\end{align}$$
Whether the common term $1/(1-z^{2L}b_{12}(z)b_{21}(z))$ can be deleted in the algorithm in Eq. (15) has not been clarified mathematically. However, the tests presented below demonstrate the ability of the simplified algorithm (16) to perform blind signal separation.

### Example 1

The source signals $s_1(t)$ and $s_2(t)$ used in Example 1 were different male voices. The convolutive mixture matrix $\bar{A}(z)$ was

$$ \bar{A}(z) = \begin{bmatrix} z^{-1} + 0.3z^{-2} & 0.4 + 0.4z^{-1} + 0.3z^{-2} \\ 0.4 + 0.3z^{-1} + 0.2z^{-2} & 1 + 0.4z^{-1} \end{bmatrix} $$

The poles of $\bar{A}(z)^{-1}$ are 4.962, 0.355, -0.41 + 0.21i, and -0.41 + 0.21i. $\bar{A}(z)^{-1}$ has one pole outside the unit circle $|z| = 1$, from which we deduce that $A(z)$ is a nonminimum phase system. Parameters $M$ and $L$ in Eq. (5) were 15 and 8, respectively. $\alpha$ in Eq. (16) and $\beta$ in Eq. (13) were $\alpha = 0.0015$ and $\beta = 0.8$. The initial values of $b_{12}(k)$ and $b_{21}(k)$, for $k = 0, \dots, 15$, and $\phi_i(t)$ were 0, 0, and 1, respectively.

Figure 3 shows the elements $\bar{c}_{11}(z)$, $\bar{c}_{12}(z)$, $\bar{c}_{21}(z)$, $\bar{c}_{22}(z)$ in matrix $C(z) (= B(z)_0 A(z))$. The elements $\bar{c}_{ij}(z)$ are given by

$$ \bar{c}_{ij}(z) = \sum_{k=0}^{17} c_{ij}(k) z^{-k} $$

$B(z)_0$ was found after 30,000 iterations using our learning algorithm. Figure 3 reveals that the nondiagonal elements $\bar{c}_{12}(z)$ and $\bar{c}_{21}(z)$ are nearly equal to zero. Thus, one can see that the proposed algorithm could separate the source signals from their convolutive mixtures.

### Example 2

In Example 2, we applied the proposed method to the experiment which extracts source signals from their mixtures observed in an ordinary room. The source signals $s_1(t)$ and $s_2(t)$ are the voices of the same man, but the conversations are different. Figure 4 shows the placement of the speakers and microphones used in the test. The source signals $s_1(t)$ and $s_2(t)$ are output from Speaker 1 and Speaker 2, respectively. The observed signals $x_1(t)$ and $x_2(t)$ are the signals observed by Mic. 1 and Mic. 2. Parameters $M$ and

<figure><img src="image_4.png" /><figcaption>Fig. 3. Elements $\bar{c}_{11}(z)$, $\bar{c}_{12}(z)$, $\bar{c}_{21}(z)$, and $\bar{c}_{22}(z)$ of matrix $\bar{C}(z)$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Configuration of the two speakers and two microphones.</figcaption></figure>

$L$ in Eq. (5) were set to 800 and 100, respectively. $\alpha$ in Eq. (16) was set to $\alpha = 0.00001$, and $\beta$ in Eq. (13) to $\beta = 0.9$. The initial values of $b_{12}(k)$ and $b_{21}(k)$, for $k = 0, \dots, 800$, and $\phi_i(t)$ were 0, 0, and 1, respectively. Figure 5 shows signals $s_i(t)$, $x_i(t)$, and $y_i(t)$, for $i = 1, 2$. It can be seen that the output signals $y_i(t)$ are close to the source signals $s_i(t)$. Therefore, one can see that our method could separate the source signals from their mixtures observed in a normal room.

## 6. Conclusions

In this paper, we have proposed a blind signal separation technique that could separate nonstationary signals from their mixtures when nonstationary signals, for example, speech or music, were generated from multiple signal sources and these mixed signals could be observed by multiple sensors. The proposed technique is an extension

<figure><img src="image_6.png" /><figcaption>Fig. 5. Signals $s_i(t)$, $x_i(t)$, $y_i(t)$ ($i = 1, 2$).</figcaption></figure>
of the technique proposed by Matsuoka and colleagues [7]
for Eq. (1) to enable its application even to Eq. (2).

Our proposed algorithm achieves blind signal sepa-
ration by decorrelating the cross-correlations of the ob-
served signals. Techniques to decorrelate the
cross-correlations have been proposed in Refs. 12 to 15, but
our technique differs from those conventional ones. It fea-
tures

* the ability to handle a nonminimum phase system, and

* evaluating only the correlations $E[y_i(t-L)y_j(t-L)]$ at the same time.

When the sounds flowing from the speakers in an
ordinary room are observed by microphones, the transfer
function between the speakers and the microphones be-
comes a nonminimum phase system [19]. Thus, for blind
signal separation in a real environment, a nonminimum
phase problem must always be solved. If the transfer func-
tion A(z) between the speakers and the microphones is a
nonminimum phase system, the future values of the ob-
served signals must be considered when estimating the
system [20]. Therefore, we handled future and past values
by regarding the time t - L as t = 0. We showed the ability
to perform blind signal separation by the proposed algo-
rithm even when matrix A(z) is a nonminimum phase
system (see Example 1). We also demonstrated the solution
by applying our technique to the problem of blind signal
separation in a real environment (see Example 2). The
conventional techniques in Refs. 12 to 15 do not handle the
nonminimum phase problem.

The evaluation function we proposed only evaluates
the correlations $E[y_i(t-L)y_j(t-L)]$ at the same time. The
conventional techniques, for example in Ref. 13, consider
the correlations at different times in addition to the correla-
tions at the same time. As a result, since many correlations
had to be evaluated, the estimation error of the correlations
has a significant effect on the estimation of the transfer
function A(z) and worsens the estimate of A(z). In Ref. 12,
since the source signals are assumed to be stationary sig-
nals, the correlations of only the unknown parameters
$a_{ij}(k)$ of matrix A(z) must be evaluated. However, in a
practical problem, the number of correlations for evaluation
cannot be determined because matrix A(z) is unknown. Our
technique does not have this problem because only the
correlations $E[y_i(t-L)y_j(t-L)]$ at the same time are evalu-
ated.

In this paper, we experimentally demonstrated the
ability to perform blind signal separation with a simplified
algorithm (16). In the future, we will examine theoretically
whether algorithm (12) or (16) is stable, and solve the
problem which handles $B(z)^{-T}$ in Eq. (14) for $N \ge 3$ (three
or more source signals).

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## APPENDIX

The calculation on the right-hand side of Eq. (11) is shown below. Equation (11) can be transformed to

$$ 
\begin{aligned}
& \frac{\partial Q(t, \bar{B}(z))}{\partial B(k)} \\
&= \frac{1}{2} \frac{\partial}{\partial B(k)} \left( \sum_{i=1}^{N} \log E[y_i(t - L)^2] \right. \\
& \qquad \left. - \log(\det \mathbf{R}_y(t)) \right) \\
&= \frac{1}{2} \frac{\partial}{\partial B(k)} \left( \sum_{i=1}^{N} \log(\mathbf{e}^{(i)} \mathbf{R}_y(t) \mathbf{e}^{(i)T}) \right)
\end{aligned}
$$

$$ 
\begin{aligned}
& - \log(\det \mathbf{R}_y(t)) \\
= & \frac{1}{2} \frac{\partial}{\partial \mathbf{B}(k)} \left( \sum_{i=1}^{N} \log(\mathbf{e}^{(i)} \sum_{n=0}^{M} \mathbf{B}(n) z^{-n} \mathbf{R}_x(t)) \right. \\
& \left. \times \sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n} \mathbf{e}^{(i)T} \right) \\
& - 2 \log(\det(\sum_{n=0}^{M} \mathbf{B}(n) z^{-n})) - \log(\det \mathbf{R}_x(t))
\end{aligned}
\quad (A.1) $$

where $\mathbf{e}^{(i)}$, $\mathbf{R}_y(t)$, and $\mathbf{R}_x(t)$ are

$$ \mathbf{e}^{(i)} = [0, \dots, 0, 1 \text{ (i-th element)}, 0, \dots, 0] $$

$$ \mathbf{R}_y(t) = E[y(t - L)y(t - L)^T] $$

$$ \mathbf{R}_x(t) = E[x(t - L)x(t - L)^T] $$

If the right-hand side of Eq. (A.1) is solved for

$$ 
\begin{aligned}
& \frac{\partial Q(t, \bar{B}(z))}{\partial B(k)} \\
= & z^{-k} \sum_{i=1}^{N} \frac{\mathbf{e}^{(i)T} \mathbf{e}^{(i)} (\sum_{n=0}^{M} \mathbf{B}(n) z^{-n}) \mathbf{R}_x(t) (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})}{\mathbf{e}^{(i)} (\sum_{n=0}^{M} \mathbf{B}(n) z^{-n}) \mathbf{R}_x(t) (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n}) \mathbf{e}^{(i)T}} \\
& \times (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})^{-1} \\
& - z^{-k} (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})^{-1} \\
= & z^{-k} \sum_{i=1}^{N} \frac{\mathbf{e}^{(i)T} \mathbf{e}^{(i)} \mathbf{R}_y(t)}{\mathbf{e}^{(i)} \mathbf{R}_y(t) \mathbf{e}^{(i)T}} (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})^{-1} \\
& - z^{-k} (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})^{-1} \\
= & -z^{-k} \left\{ I - (\text{diag} \mathbf{R}_y(t))^{-1} \mathbf{R}_y(t) \right\} \\
& \times (\sum_{n=0}^{M} \mathbf{B}(n)^T z^{-n})^{-1}
\end{aligned}
\quad (A.2) $$

we obtain Eq. (12). See Ref. 18 for the differential of the matrix.