Design of Oversampled Interleaved DFT Modulated Filter Bank Using 2Block Gauss-Seidel Method

Abstract

In this paper, we present several new properties of the recently introduced interleaved DFT modulated filter bank and an efficient algorithm for designing the filter bank. The periodicity and symmetry properties of the overall transfer function and aliasing transfer functions are stated. Then the design of the filter bank is formulated into a constrained optimization problem that jointly minimizes the overall distortion and aliasing distortion subject to fixed bounds on the stopband energy, transition-band energy, and passband flatness of the prototype filters. The constrained optimization problem is solved by the 2block Gauss-Seidel method, which alternatively optimizes the analysis PF pair and the synthesis PF pair. Since the overall distortion and aliasing distortion are jointly minimized, the proposed algorithm can lead to filter banks with small reconstruction error, even when the filter banks behave with a low redundancy ratio and short PFs. The convergence of the proposed algorithm is proved. Numerical examples and comparisons with the existing method are included to demonstrate the performance of the proposed algorithm.

Appendix: Efficient Calculation of Matrices Φ a,y ,a=0,…,[K/2] and Vectors r y for y=p,q

With (6), (12a), and (12b), matrix Φ _a,p can be written as

$$\begin{aligned} \boldsymbol{\Phi}_{a,\mathbf{p}} =& \int_{0}^{4\pi /M} \bigl\{ \mathbf{R}_{a}\mathbf{pp}^{T}\mathbf{R}_{a}^{\dagger} \bigr\} \,d\omega \\ =& \left [ \begin{array}{c@{\quad}c} \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{0,a}(\omega )\mathbf{p}_{0}\mathbf{p}_{0}^{T}\boldsymbol{\Theta}_{0,a}^{\dagger} (\omega )\,d\omega & \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{0,a}(\omega )\mathbf{p}_{0}\mathbf{p}_{1}^{T}\boldsymbol{\Theta}_{1,a}^{\dagger} (\omega )\,d\omega \\ \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{1,a}(\omega )\mathbf{p}_{1}\mathbf{p}_{0}^{T}\boldsymbol{\Theta}_{0,a}^{\dagger} (\omega )\,d\omega & \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{1,a}(\omega )\mathbf{p}_{1}\mathbf{p}_{1}^{T}\boldsymbol{\Theta}_{1,a}^{\dagger} (\omega )\,d\omega \end{array} \right ] \end{aligned}$$

(A.1)

Denote $\boldsymbol{\Phi}_{a,\mathbf{p}}^{k_{0},k_{1}} = \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{k_{0},a}(\omega )\mathbf{p}_{0}\mathbf{p}_{0}^{T}\boldsymbol{\Theta}_{k_{1},a}^{\dagger} (\omega )\,d\omega,k_{0},k_{1} = 0,1$, and its (s,t)th element is evaluated by

$$\begin{aligned} &\boldsymbol{\Phi}_{a,\mathbf{p}}^{k_{0},k_{1}}[s,t] = \sum _{k,l = 0}^{ML - 1} p_{k_{0}}(k)p_{k_{1}}(l) \int_{0}^{4\pi /M} \boldsymbol{\Theta}_{k_{0},a}( \omega )[s,k]\boldsymbol{\Theta}_{k_{1},a}^{ *} (\omega )[l,t] \,d\omega \\ &\quad s,t = 0, \ldots,ML - 1 \end{aligned}$$

(A.2)

In (A.2), the integral is independent of the PFs and is only required to be calculated once. In the integral, element $\boldsymbol{\Theta}_{k_{0},a}(\omega )[s,k]$ can be calculated by invoking the property of the DFT:

$$\begin{aligned} \boldsymbol{\Theta}_{k_{0},a}(\omega )[s,k] =& \frac{1}{K}\sum _{m = 0}^{M/2 - 1} e^{ - j(\omega - \frac{(4m + 2k_{0})\pi}{M})} e^{ - js(\omega - \frac{(4m + 2k_{0})\pi}{M})}e^{ - jk(\omega - \frac{(4m + 2k_{0})\pi}{M} - \frac{2a\pi}{K})} \\ =& \frac{1}{K}e^{ - j\omega (1 + s + k)}W_{M}^{ - k_{0}(1 + s + k)}W_{K}^{ - ak} \sum_{m = 0}^{M/2 - 1} W_{M}^{ - 2m(1 + s + k)} \\ =& \left \{ \begin{array}{l} \frac{M}{2K}e^{ - j\omega (1 + s + k)}W_{M}^{ - k_{0}(1 + s + k)}W_{K}^{ - ak},(1 + s + k)\bmod (M/2) = 0 \\ 0,\quad\mathrm{otherwise} \end{array} \right . \end{aligned}$$

(A.3)

Analogously, element $\boldsymbol{\Theta}_{k_{1},a}^{ *} (\omega )[l,t]$ is equivalent to

$$ \boldsymbol{\Theta}_{k_{1},a}^{ *} (\omega )[l,t] = \left \{ \begin{array}{l} \frac{M}{2K}e^{j\omega (1 + l + t)}W_{M}^{k_{1}(1 + l + t)}W_{K}^{al},\ (1 + l + t)\bmod (M/2) = 0 \\ 0,\quad\mbox{otherwise } \end{array} \right . $$

(A.4)

Consequently, the integral in (A.2) is equal to

$$ I_{A2} = \left \{ \begin{array}{l} \frac{M\pi}{K^{2}}( - 1)^{\frac{2(1 + s + k)}{M}(k_{1} - k_{0})}W_{K}^{a(l - k)},(1 + s + k)\bmod (\frac{M}{2}) = 0\ \mathrm{and}\ l + t = s + k \\ 0,\quad\mbox{otherwise } \end{array} \right . $$

(A.5)

From (A.2) and (A.5), only a few multiplications are required in the computation of $\boldsymbol{\Phi}_{a,\mathbf{p}}^{k_{0},k_{1}}[s,t]$, resulting in an efficient calculation of matrix Φ _a,p . The calculation of matrices Φ _a,q can be performed similarly.

Also, in terms of the symmetry of matrices Θ _0,0(ω) and Θ _1,0(ω), vector r _p is

$$\begin{aligned} \mathbf{r}_{\mathbf{p}} =& \operatorname{Re} \left \{ \int _{0}^{4\pi /M} e^{jD\omega} \left [ \begin{array}{c@{\quad}c} \boldsymbol{\Theta}_{0,0}(\omega ) & \mathbf{0} \\ \mathbf{0} & \boldsymbol{\Theta}_{1,0}(\omega ) \end{array} \right ]^{T}\mathbf{p}\,d\omega \right \} \\ =& \operatorname{Re} \left \{ \left [ \begin{array}{c} \int_{0}^{4\pi /M} e^{jD\omega} \boldsymbol{\Theta}_{0,0}(\omega )\mathbf{p}_{0}\,d\omega \\ \int_{0}^{4\pi /M} e^{jD\omega} \boldsymbol{\Theta}_{1,0}(\omega )\mathbf{p}_{1}\,d\omega \end{array} \right ] \right \} \end{aligned}$$

(A.6)

Denote $\mathbf{r}_{k,\mathbf{p}} = \int_{0}^{4\pi /M} e^{jD\omega} \boldsymbol{\Theta}_{k,0}(\omega )\mathbf{p}_{k}\,d\omega,k = 0,1$, whose lth element is calculated by

$$ \mathbf{r}_{k,\mathbf{p}}[l] = \sum_{n = 0}^{ML - 1} p_{k}(n) \int_{0}^{4\pi /M} e^{jD\omega} \boldsymbol{\Theta}_{k,0}(\omega )[l,n]\,d\omega $$

(A.7)

From [17], the delay D is a multiple of M, i.e., D=Mn _d . Thus, the integral in (A.7) is evaluated by

$$ I_{A7} = \left \{ \begin{array}{l} \frac{2\pi}{K},(1 + l + n)\bmod (\frac{M}{2}) = 0\ \mbox{and}\ D = 1 + l + n \\ 0,\quad\mbox{otherwise } \end{array} \right . $$

(A.8)

Therefore, r _k,p can be calculated quickly, resulting in an efficient evaluation of r _p . Vector r _q can be analogously computed.