Inverting non-minimum phase FIR transfer functions with application to reverberant speech

Abstract

The problem of removing reverberation in speech is considered when the room impulse response is non-minimum phase. The method does not require multiple sensors or the use of all-pass transfer function networks as in previous techniques. Instead we use spectral factorization of large polynomials. Spectral factorization gives a method of reflecting the non-minimum phase roots without having to calculate their actual values.

Appendices

Appendix 1

To divide the two polynomials of equal degree $\frac{B^{*}}{\Delta ^{*}}.$

Without losing generality we consider the equivalent problem of dividing two polynomials of equal degree into a power series which have unity zeroth coefficients.

$$\begin{aligned} \frac{b^{*}}{a^{*}}=1+p_1 z+p_2 z^{2}+...+p_\ell z^{\ell } \end{aligned}$$

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Where $b^{*} = 1 + b_{1} z + b_{2} z^{2} + ... + b_{n} z^{n}$ and $a^{*} = 1 + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n}$

Multiplying out (a1) and comparing coefficients of z yields the simple recursion

$$\begin{aligned} p_0&= 1\nonumber \\ p_i&= b_i -\sum _{j=1}^{\max (n,i)} {p_{i-j} a_j },\quad i\le n \nonumber \\&= -\sum _{j=1}^{\max (n,i)} {p_{i-j} a_j },\quad i\le n,\;i>n \end{aligned}$$

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For example, for polynomials

$$\begin{aligned} B^{*}&= B_0 +B_1 z+B_2 z^{2}+ \cdots +B_n z^{n} \\ \Delta ^{*}&= A_0 +A_1 z+A_2 z^{2}+ \cdots +A_n z^{n} \end{aligned}$$

We can remove the zeroth coefficients thus and define $a^{*}$ and $b^{*}$ polynomials with $a^{*}(0) = b^{*}(0) = 1$

$$\begin{aligned} B^{*}&= B_0 (1+\frac{B_1 }{B_0 }z+\frac{B_2 }{B_0 }z^{2}+ \cdots +\frac{B_n }{B_0 }z^{n})=B_0 b^{*} \\ \Delta ^{*}&= A_0 (1+\frac{A_1 }{A_0 }z+\frac{A_2 }{A_0 }z^{2}+ \cdots +\frac{A_n }{A_0 }z^{n})=A_0 a^{*} \end{aligned}$$

Then use the recursion (a2) on $a^{*}$ and $b^{*}$. We then multiply all the $p_{i}, i = 0, 1, 2, \ldots \ell$ terms by the constant $\frac{B_0 }{A_0 }.$

Appendix 2: Method of spectral factorization of large polynomials

A Laurent series can be easily factorized using many methods, but here we use a feedback method used in earlier work (Moir 2009) which is shown below. The technique uses negative feedback to force the output of a negative-feedback system to be the same as its input. In this case the input “signal” is one half of the Laurent series and convolution takes effect in the feedback path as illustrated in Fig. 8 below.

Fig. 8

Spectral factorization using feedback

The method is essentially easy in principle. A known (or desired) Laurent series is entered at the set-point and the loop continues until the spectral-factor convolved with itself (the reverse coefficients) become equal to the desired values. This is akin to a control-system with $\Delta (z^{-1})\Delta (z)$ in its feedback path. The gain of the loop forces the spectral factor to the desired value which we already can calculate from the known impulse response $B(z^{-1})B(z)$.

1.1 Algorithm: Feedback spectral factorization of Laurent series

Consider a Laurent series $l_{-n} \hbox {z}^{n}+ \cdots +l_{-2} \hbox {z}^{2}+l_{-1} \hbox {z}^{1}+l_0 +l_1 z^{-1}+l_2 z^{-2}+ \cdots l_n \hbox {z}^{-n}=\varvec{\Delta } (z^{-1})\varvec{\Delta } (z).$

Initialise vectors ${\varvec{W}}_0 ,{\varvec{V}}_0$ of length n+1 to some initial values. (possibly zero). Assume that the Laurent series vector ${\varvec{L}}$ is already known or can be estimated. Here the Laurent series is found from half of the two-sided impulse response data when multiplying $B(z^{-1})B(z)$. We only require one half of the symmetric Laurent series, the terms in negative power of z including the zeroth term. Hence write ${\varvec{L}}=[l_0 ,l_1 ,\ldots l_n]$.

Use a constant gain (step-size) $\left| \eta \right| <1$. Care must be taken not to make the gain too large as then the algorithm will become unstable. Too small a value will result in a slow rate of convergence.

Step 1 :

Form the convolution vector ${\varvec{V}}_k ={\varvec{W}}_k *{\varvec{W}}_{-k}$ where its individual elements are found from the elements of ${\varvec{W}}_k$ as $\hbox {v}_i =\sum _{j=0}^{n-i} {w_i } w_{i+j} ,i=1,2, \ldots n$

Step 2 :

Update the error vector $\varvec{\varphi }_k ={\varvec{L}}-{\varvec{V}}_k$

Step 3 :

Update the weight vector ${\varvec{W}}_{k+1} ={\varvec{W}}_k +\eta \varvec{\varphi }_k$

Step 4 :

Stopping criterion Test for convergence $J = \sum _{j=0}^{n} (l_{j} - v_{j})^{2} < \delta$, for some small $\delta$, where $l_{j}, j = 1, 2...n$ are the known (or desired) Laurent series coefficients of the vector L. Stop if convergent else return to step 1. The elements of the vector ${\varvec{W}}_k$ are now the spectral factor polynomial coefficients. The driving noise variance is estimated from the zeroth element of the vector $\sigma ^{2}={\varvec{W}} (0)^{2}$ which can be assumed to be unity and any gain term absorbed into the spectral factor.