An Eigenfilter-Based Approach to the Design of Time-Frequency Localization Optimized Two-Channel Linear Phase Biorthogonal Filter Banks

Abstract

We present a novel eigenfilter-based approach to the design of time-frequency optimized, linear-phase, biorthogonal FIR filter banks. We first design a linear-phase, low-pass analysis filter, followed by a complementary linear-phase, low-pass synthesis filter. The optimality criterion used is uncertainty-based time-frequency localization, where the objective function is a convex combination of time variance and frequency variance of the respective filters. The objective function to be minimized is formulated in a convex-quadratic form and the perfect reconstruction (PR) and vanishing moment (VM) conditions are imposed in the eigen design of filters as a set of linear equality constraints. The PR and VM conditions are expressed in the time domain matrix formulation, so that these can directly be incorporated into the eigenfilter design. Using the Rayleigh principle, the optimal filter is obtained as an eigenvector corresponding to the minimum eigenvalue of the real symmetric positive-definite matrix associated with the optimization criterion. Thus, our formulation reduces the design problem of time-frequency optimal filter banks to an eigenfilter-based problem. Furthermore, the filter banks designed in this manner are found to be regular and are valid candidates for wavelet filter banks, allowing for the construction of linear phase wavelets. We present a few examples to show that the smooth wavelets can be constructed using the proposed method.

Appendices

Appendix

Derivation of Matrix Formulation for Time and Frequency Variance for Real Symmetric Discrete-Time Sequences

1.1 Frequency Variance Measure

From the Eq. (31) the frequency variance of the zero phase, low-pass, real FIR filter $h(n)$ of length $2N+1$, $N\in \mathbb {N}$ is given by

$$\begin{aligned} \sigma _{\omega }^{2}=\mathbf {a^{T}}\mathbf {Q}\mathbf {a}, \end{aligned}$$

(50)

where the matrix $\mathbf {Q}$ and the vector $\mathbf {a}$ are defined in the Eqs. (31) and (26), respectively. We define the matrix $\mathbf {E}$ as

$$\begin{aligned} \mathbf {E}=\omega ^{2}\left[ \begin{array}{c} 1\\ 2\cos (\omega )\\ 2\cos (2\omega )\\ \vdots \\ 2\cos (N\omega )\end{array}\right] \left[ \begin{array}{ccccc} 1&2\cos (\omega )&2\cos (2\omega )&\dots&2\cos (N\omega )\end{array}\right] \end{aligned}$$

(51)

The matrix $\mathbf {E}$ can also be expressed as follows:

$$\begin{aligned} \mathbf {E}(\omega )=\omega ^{2}\mathbf {c}(\omega )\mathbf {c}^{T}(\omega ) \end{aligned}$$

(52)

The vector $\mathbf {c}(\omega )$ is defined in (27). The $(k,l)^{th}$ element of the matrix $\mathbf {E}$ is given as

$$\begin{aligned}{}[\mathbf {E}]_{k,l}={\left\{ \begin{array}{ll} \omega ^{2} &{} k=l=0\\ \omega ^{2}\{\cos (k+l)\omega +\cos (k-l)\omega \} &{} k=0\; \text{ or }\; l=0\; \text{ except }\; k=l=0 \\ 2\omega ^{2}\{\cos (k+l)\omega +\cos (k-l)\omega \} &{} otherwise \end{array}\right. }\quad \quad \nonumber \\ \end{aligned}$$

(53)

where $0 \le k,l \le N$. The matrix $\mathbf {Q}$ corresponding to the frequency variance is related to the matrix $\mathbf {E}$ as

$$\begin{aligned} \mathbf {Q}(\omega )=\frac{1}{\pi }\int _{0}^{\pi }\mathbf {E}(\omega )d\omega \end{aligned}$$

(54)

The $(k,l)^{th}\;\text{ element }$ of the matrix $\mathbf {Q}$ can be given as

$$\begin{aligned}{}[\mathbf {Q}]_{k,l}={\left\{ \begin{array}{ll} \frac{1}{\pi }\int _{0}^{\pi }\omega ^{2}d\omega =\frac{\pi ^{2}}{3} &{} k=l=0\\ \frac{1}{\pi }\int _{0}^{\pi }4\omega ^{2}\cos ^2(k\omega )d\omega =\frac{2\pi ^{2}}{3}+\frac{1}{k^2} &{} k=l\; except\; k=l=0 \end{array}\right. } \end{aligned}$$

(55)

$$\begin{aligned}{}[\mathbf {Q}]_{k,l}={\left\{ \begin{array}{ll} \int _{0}^{\pi }\omega ^{2}[\cos (k+l)\omega +\cos (k-l)\omega ]\frac{d\omega }{\pi } &{} k=0\; or \; l\!=\!0\; except\; k\!=\!l\!=\!0\!\!\!\\ \int _{0}^{\pi }2\omega ^{2}[\cos (k+l)\omega +\cos (k-l)\omega ]\frac{d\omega }{\pi } &{} 1\le k,l \le N,\; except\; k=l \end{array}\right. }\nonumber \\ \end{aligned}$$

(56)

In order to evaluate the integrals in the Eq. (56), we substitute $k+l=m$ and $k-l=s$. Thus (56) boils down to

$$\begin{aligned}{}[\mathbf {Q}]_{k,l}={\left\{ \begin{array}{ll} \int _{0}^{\pi }\omega ^{2}[\cos (m\omega )+\cos (s\omega )]\frac{d\omega }{\pi } &{} k=0\; or \; l=0\; except\; k=l=0\\ \int _{0}^{\pi }2\omega ^{2}[\cos (m\omega )+\cos (s\omega )]\frac{d\omega }{\pi } &{} 1\le k,l \le N,\; except\; k=l \end{array}\right. }\nonumber \\ \end{aligned}$$

(57)

The indefinite integral $I=\int \omega ^{2}\cos (m\omega )d\omega$ is evaluated as

$$\begin{aligned} I&= \int \omega ^{2}\cos (m\omega )d\omega \nonumber \\&= \frac{\omega ^{2}\sin (m\omega )}{m}+\frac{2\omega \cos (m\omega )}{m^{2}}-\frac{2\sin (m\omega )}{m^{2}}, m\ne 0 \end{aligned}$$

(58)

On substituting limits in the integral of the expression (58) we get

$$\begin{aligned} \int _{0}^{\pi }\omega ^{2}\cos (m\omega )d\omega =\frac{2\pi (-1)^{m}}{m^{2}},\;m \ne 0 \end{aligned}$$

(59)

Using (59), (57), and (55), we obtain

$$\begin{aligned}{}[\mathbf {Q}]_{k,l}={\left\{ \begin{array}{ll} \frac{\pi ^{2}}{3} &{} k=l=0\\ \frac{2 \pi ^{2}}{3}+\frac{1}{k^2} &{} k=l\; except\; k=l=0\\ \frac{4(-1)^{k+l}}{(k+l)^2} &{} k=0\; or\; l=0,\; except\; k=l=0\\ 8(-1)^{k+l}\frac{(k^{2}+l^{2})}{(k^{2}-l^{2})^{2}} &{} otherwise \end{array}\right. } \end{aligned}$$

(60)

where $0\le k,l\le N$.

1.2 Time Variance Measure

From the Eq. (34) the time variance of the zero phase, low-pass, real FIR filter $h(n)$ of length $2N+1$, $N\in \mathbb {N}$ is expressed as

$$\begin{aligned} \sigma _{n}^{2}=\mathbf {a^{T}Pa}, \end{aligned}$$

(61)

where the matrix $\mathbf {P}$ and the vector $\mathbf {a}$ are defined in the Eqs. (34) and (26), respectively. We define the matrix $\mathbf {F}$ as

$$\begin{aligned} \mathbf {F}=\left[ \begin{array}{c} 0\\ -2\sin (\omega )\\ -4\sin (2\omega )\\ \vdots \\ -2N\sin (N\omega )\end{array}\right] \left[ \begin{array}{ccccc} 0&-2\sin (\omega )&-4\sin (2\omega )&\dots&-2N\sin (N\omega )\end{array}\right] \end{aligned}$$

(62)

$$\begin{aligned} \mathbf {F}=\mathbf {f}(\omega )\mathbf {f^{T}}(\omega ) \end{aligned}$$

The vector $\mathbf {f}(\omega )$ is defined in (28). The $(k,l)^{th}$ element of the matrix $\mathbf {F}$ is given as

$$\begin{aligned}{}[\mathbf {F}]_{k,l}=4kl\sin (k\omega )\sin (l\omega ) \end{aligned}$$

The matrix $\mathbf {P}$ corresponding to time variance is related to the matrix $\mathbf {F}$ as

$$\begin{aligned} \mathbf {P}(\omega )=\frac{1}{\pi }\int _{0}^{\pi }\mathbf {F}(\omega )d\omega \end{aligned}$$

The $(k,l)^{th}$ element of matrix $\mathbf {P}$ is

$$\begin{aligned}{}[\mathbf {P}]_{k,l}&= \frac{1}{\pi }\int _{0}^{\pi }4kl\sin (k\omega )\sin (l\omega )d\omega \nonumber \\&= \frac{1}{\pi }\int _{0}^{\pi }kl[2\cos (k-l)\omega -2\cos (k+l)\omega ]d\omega \end{aligned}$$

(63)

In order to evaluate the integral in the Eq. (63), we substitute $k+l=m$ and $k-l=s$. Thus (63) boils down to

$$\begin{aligned}{}[\mathbf {P}]_{k,l}=\frac{1}{\pi }\int _{0}^{\pi }kl\{2\cos (s\omega )-2\cos (m\omega )\}d\omega \end{aligned}$$

(64)

The value of the integral $\frac{1}{\pi }\int _{0}^{\pi }\cos (m\omega )d\omega$ is evaluated as

$$\begin{aligned} \frac{1}{\pi }\int _{0}^{\pi }\cos (m\omega )d\omega =0,m\ne 0 \end{aligned}$$

(65)

Using (65) and (64), we get

$$\begin{aligned}{}[\mathbf {P}]_{k,l}={\left\{ \begin{array}{ll} 0 &{} k \ne l \\ 2k^{2} &{} k=l \end{array}\right. } \end{aligned}$$

(66)

Thus, the matrix $\mathbf {P}$ can be expressed as

$$\begin{aligned} \mathbf {[P]}_{k,l}=2k^{2}\delta (k-l),0\le k,l\le N \end{aligned}$$

It is to be noted that the matrix $\mathbf {P}$ can be obtained directly using (5) in time domain; however, for the sake of completeness we derived the matrix $\mathbf {P}$ using frequency-domain approach and Parseval’s identity.