直接法设计均匀与非均匀滤波器组
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 应用科技  2018, Vol. 45 Issue (3): 14-18  DOI: 10.11991/yykj.201612008 0

### 引用本文

ZHANG Chunjie, TIAN Chunyu. Design of uniform and non-uniform filter banks by direct method[J]. Applied Science and Technology, 2018, 45(3), 14-18. DOI: 10.11991/yykj.201612008.

### 文章历史

Design of uniform and non-uniform filter banks by direct method
ZHANG Chunjie, TIAN Chunyu
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: As for the reconstruction design of the filter banks, the existing algorithms can only be used to design uniform or non-uniform filter banks, it is hard to design two different types of filter banks at the same time, therefore, the flexibility is not enough. A new method of designing all filter banks directly was proposed in the paper. In the method, the reconstruction conditions of filter banks are derived from the frequency domain and expressed in the form of matrix. According to the distribution of the given frequency bands, all channel filters in the analysis modules are designed simultaneously; then the optimum solution of the reconstruction matrix equation can be obtained by the iterative optimization theory, the filtering coefficients of all passages in the comprehensive modules can be further obtained and the necessary filter banks can be designed. Simulation results show that, the design method can be used to design the uniform filter banks and the non-uniform filter banks simultaneously, realize high-efficiency division of frequency band, the flexibility is high, in addition, the method can ensure the linear phase characteristics of each channel filter.
Key words: design of filter banks    software radio    approximate reconstruction    iterative optimization    linear-phase    least square solution    aliasing distortion    FIR filter

1 滤波器组重构条件推导

 \begin{aligned}Y\left( z \right) = & \sum\limits_{m = 0}^{M - 1} {\frac{1}{{{n_m}}}{F_k}\left( z \right)\sum\limits_{l = 0}^{{n_m} - 1} {X\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right){H_m}\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right)} } = \\& \sum\limits_{m = 0}^{M - 1} {\frac{1}{{{n_m}}}{F_m}\left( z \right)X\left( z \right){H_m}\left( z \right)} + \\& \sum\limits_{m = 0}^{M - 1} {\frac{1}{{{n_m}}}{F_m}\left( z \right)\sum\limits_{l = 1}^{{n_m} - 1} {X\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right){H_m}\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right)} } = \\& X\left( z \right){T_0}\left( z \right) + \sum\limits_{l = 1}^{{n_m} - 1} {X\left( {z{e^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right){T_l}\left( z \right)} \end{aligned}

 ${T_0}\left( z \right) = \sum\limits_{m = 0}^{M - 1} {\frac{1}{{{n_m}}}{F_k}\left( z \right)} \sum\limits_{l = 0}^{{n_m} - 1} {X\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right){H_m}\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right)}$ (1)
 ${T_l}\left( z \right) = \sum\limits_{m = 0}^{M - 1} {\frac{1}{{{n_m}}}{F_m}\left( z \right){H_m}\left( {z{{\rm{e}}^{ - {\rm{j}}\frac{{2{\rm{\pi }}l}}{{{n_m}}}}}} \right)} \;\;\; l = 1,2, \cdots ,M$ (2)

${T_0}\left( z \right)$ 称为整个滤波系统的失真函数，体现了整个滤波系统的幅频相位特性， ${T_l}\left( z \right)$ 称为整个系统的混叠失真函数，对应的混叠信号是 $X\left( {z{{\rm{e}}^{{{ - {\rm{j}}2\pi l} / {{n_m}}}}}} \right)$

${T_0}\left( z \right)$ 是一个纯延迟，即 ${T_0}\left( z \right) = {z^{ - \varDelta }}$ ${T_l}\left( z \right){\rm{ = }}0$ 时，整个系统可实现准确重构，Δ表示时延长度。结合式(1)、(2)，将重构条件表示为矩阵方程的形式，即

 ${{Ax}} = {{b}}$ (3)

 ${{{A}}_{L\left( {2N - 1} \right) \times MN}} = \left[\!\!\!\!\! {\begin{array}{*{20}{c}}{{\beta _{1,0}}\Im \left( {{{{h}}_1}} \right)} \!\! & \!\! {{\beta _{2,0}}\Im \left( {{{{h}}_2}} \right)} \!\! & \!\! \cdots & {{\beta _{M,0}}\Im \left( {{{{h}}_M}} \right)}\\{{\beta _{1,1}}\Im \left( {{{{h}}_1} \otimes {{{\varLambda }}_1}} \right)} \!\! & \!\! {{\beta _{2,1}}\Im \left( {{{{h}}_2} \otimes {{{\varLambda }}_1}} \right)} \!\! & \!\! \cdots & {{\beta _{M,1}}\Im \left( {{{{h}}_M} \otimes {{{\varLambda }}_1}} \right)}\\{{\beta _{1,2}}\Im \left( {{{{h}}_1} \otimes {{{\varLambda }}_2}} \right)} \!\! & \!\! {{\beta _{2,2}}\Im \left( {{{{h}}_2} \otimes {{{\varLambda }}_2}} \right)} \!\! & \!\! \cdots & \!\! {{\beta _{M,2}}\Im \left( {{{{h}}_M} \otimes {{{\varLambda }}_2}} \right)}\\ \vdots \!\! & \!\! \cdots \!\! & \!\! \cdots \!\! & \!\! \vdots \\{{\beta _{1,L - 1}}\Im \left( {{{{h}}_1} \otimes {{{\varLambda }}_{L - 1}}} \right)} \!\! & \!\! {{\beta _{2,L - 1}}\Im \left( {{{{h}}_2} \otimes {{{\varLambda }}_{L - 1}}} \right)} \!\! & \!\! \cdots \!\! & \!\! {{\beta _{M,L - 1}}\Im \left( {{{{h}}_M} \otimes {{{\varLambda }}_{L - 1}}} \right)}\end{array}} \!\!\!\!\! \right]$

 {\alpha _{m,l}} = \left\{ {\begin{aligned}& {1,} \quad \quad {{{\rm e}^{{{{\rm j}2\pi * l} / L}}} \in \left\{ {{{\rm e}^{{\rm j}2\pi \left( 1 \right)/{n_m}}},{{\rm e}^{{\rm j}2\pi \left( 2 \right)/{n_m}}}, \cdots ,{{\rm e}^{{\rm j}2\pi \left( {{n_m} - 1} \right)/{n_m}}}} \right\}}\\& {0,} \quad \quad {{\text{其他}}}\end{aligned}} \right.

βm, l定义为

 ${\beta _{m,l}} = \left\{ {\begin{array}{*{20}{l}} {{1 / {{n_m},}}}&{l = 0} \\ {{{{\alpha _{m,l}}} / {{n_m},}}}&{l \ne 0} \end{array}} \right.$ (4)

 $\Im \left( {{{\left[ {{a_1},{a_2}, \cdots ,{a_N}} \right]}^{\rm{T}}}} \right) = \left[ {\begin{array}{*{20}{c}} {{a_1}}&0&0& \cdots &0 \\ {{a_2}}&{{a_1}}&0& \cdots &0 \\ {{a_3}}&{{a_2}}&{{a_1}}& \cdots &0 \\ \vdots & \cdots & \cdots & \cdots & \vdots \\ {{a_N}}&{{a_{N - 1}}}&{{a_{N - 2}}}& \cdots &{{a_1}} \\ 0&{{a_N}}&{{a_{N - 1}}}& \cdots &{{a_2}} \\ 0&0&{{a_N}}& \cdots &{{a_3}} \\ 0&0&0& \cdots & \vdots \\ 0&0&0& \cdots &{{a_N}} \end{array}} \right]$

${{{h}}_m}\left( {m = 0,2, \cdots ,M - 1} \right)$ 是分析模块中第 $m$ 个通道的滤波器系数，即 ${{{h}}_m} = {\left( {{h_{m,0}},{h_{m,1}}, \cdots ,{h_{m,N - 1}}} \right)^{\rm{T}}}$ 。运算符 $\otimes$ 表示2个矩阵的对应元素相乘。运算符 ${{{\varLambda }}_i}$ 定义为

 ${{{\varLambda }}_i} = {\left[ {{{\rm e}^{{{{\rm j}2\pi i \cdot \left( 0 \right)} / L}}} \;\; {{\rm e}^{{{{\rm j}2\pi i \cdot \left( 1 \right)} / L}}} \;\; {{\rm e}^{{{{\rm j}2\pi i \cdot \left( 2 \right)} / L}}} \;\; \cdots \;\; {{\rm e}^{{{{\rm j}2\pi i \cdot \left( {N - 1} \right)} / L}}}} \right]^{\rm{T}}}$

 $Q = {Q_{{\rm{PR}}}} + {Q_{\rm{MRE}}} = \left\| {{{Ax}} - {{b}}} \right\|_2^2 + \sum\limits_{m = 1}^M {{Q_{{\rm{MRE}},m}}}$ (5)

 ${Q_{{\rm{MRE}},m}} = \sum\limits_{i = 1}^\rho {\left\| {\left| {{H_m}\left( {{{\rm e}^{{\rm j}w}}} \right)} \right| - {r_m}\left( {{w_i}} \right)} \right\|} _2^2$ (6)

 ${r_m}\left( {{w_i}} \right) = \left\{ {\begin{array}{*{20}{l}}{1,} & {{w_i}{\text{是}}{H_m}\left( {{{\rm e}^{\rm j}{^w}}} \right){\text{通带内的频率点}}}\\{0,} & {{\text{其他}}}\end{array}} \right.$

2 滤波器组设计步骤

1) 初始化各通道滤波器系数长度 $N$ 、分析模块各通道原始FIR滤波器频率特性、需要满足的误差值 ${Q_D}$ 、频率采样点个数 $\rho$ 、各通道抽取插值速率 $\left[ {{n_1},{n_2}, \cdots ,{n_{M - 1}}} \right]$ ，并求出其最小公倍数L

2) 采用等波纹逼近法设计系统前端分析模块中各通道原始FIR滤波器系数 ${{ h}_m}\left( n \right)\left( {m = 1,2, \cdots ,M} \right)$ ，并结合其他定义参数、运算符生分析矩阵A

3) 通过求解方程 ${{Ax}} = {{b}}$ 的最小二乘解，得到综合模块中各通道滤波器系数 ${{x}}$

4) 通过式(5)计算误差目标函数Q

5) 如果求解的误差 $Q$ 小于给定的误差 ${Q_D}$ ，终止该算法，否则通过迭代更新分析模块中的滤波器系数，并返回步骤2)，直到误差 $Q \leqslant {Q_D}$

3 性能仿真