Filter Design with Matlab

Introduction - Time-Frequency Analysis and Design - Filter Bank Design - Conclusion

Nov. 9th Talk: Matlab and the Ptolemy/Matlab Interface

Nov. 30th Talk: Tool Collaboration with Matlab in Design

Introduction

• Review last week's talk
• Outline this week's talk

Time-Frequency Analysis and Design Using Tcl/Tk/Matlab

• Allows users to analyze and synthesize time-frequency signals using a graphical user interface
• The user fills rectangular bins in the time-frequency domain using the mouse to interact with Tk
• Menu commands for creating and displaying signals trigger Matlab commands sent by Tcl.
• Tcl/Matlab interface
• The Tcl/Matlab interface uses Unix pipes, which is not robust, but works in this context because the Tcl/Tk script creates the Matlab commands, and not the user

Design of Two-Channel FIR Filter Banks

• Paraunitary filter banks
  • Unitary matrix:

     H
    U  U = c I
    

  • If U is square:

     H      -1
    U  = c U
    

  • Let c=1 (normal):

     H    -1       -1    H
    U  = U    or  U   = U
    

  • Note: real unitary matrix is orthogonal; if also c=1, then orthonormal
  • Fourier Transform Unitary Matrix:

     -1  j w     T  j w *
    X  (e   ) = X (e   )
    

  • Substitute z = exp(j w) to get Paraunitary:

     -1       T  -1
    X  (z) = X (z  )
    

    Subscript * means replace all coefficients by their complex conjugates

• Two channel filter bank
  • We want the z-transform response with input X(z) to be a delayed version of X(z):

                              -(N-1)
    T(z) X(z) + A(z) X(-z) = z       X(z)
    

  • Eliminate aliasing when A(z) = 0 (quadrature mirror filters), so

                                        -(N-1)
    T(z) = G0(z) H0(z) + G1(z) H1(z) = z
    

  • T(z) introduces magnitude and phase errors, which can be nearly eliminated (for near perfect reconstruction filter banks) or completely eliminate (for perfect reconstruction filter banks)
  • The insight is that if G0(z)H0(z) is a valid halfband lp filter and G1(z)H1(z) is a valid halfband hp filter then T(z) is just a delay. The smith and barnwell solution is to work backwards and define a valid halfband lp filter and then find the spectral factors and assign them to G0(z) and H0(z).
• Perfect reconstruction filter bank design by using direct spectral factorization
  • design halfband FIR filter (windows, Parks-McClellan)

  • raise magnitude response to be non-negative so that it is a power spectrum

  • root the all-zero transfer function polynomial

  • quantize roots

  • split roots between minimum phase (analysis filter) and maximum phase (synthesis filter)

  • Problems: numerical errors in higher-order polynomials
• Matlab demos
  • First, set the Matlab path

    setenv MATLABPATH /users/williamc/matlab/2channelfb/cqmf
    

  • Design of filters in a two-channel perfect reconstruction: 66th order, passband cutoff response of 2*0.2 normalized between 0 and 1, and a passband deviation of 0.0005:

    conj_qmf(66,[ 0.2 0.3], 0.00005, 1)
    

Conclusion