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Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming

  • * Corresponding author: Bingo Wing-Kuen Ling

    * Corresponding author: Bingo Wing-Kuen Ling 
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  • It is worth noting that the conventional maximally decimated M-channel mirrored paraunitary linear phase finite impulse response condition is defined in the frequency domain. As the frequency domain is a continuous set, it is expressed as a matrix functional (a continuous function of the frequency) equation. On the other hand, this paper expresses the condition as a finite number of discrete (a set of functions of the sampled frequencies) equations. Besides, this paper proposes to sample the magnitude responses of the filters with the total number of the sampled frequencies being more than the filter lengths. Hence, the frequency selectivities of the filters can be controlled more effectively. This filter bank design problem is formulated as an optimization problem in such a way that the total mirrored paraunitary linear phase error is minimized subject to the specifications on the magnitude responses of the filters at these sampling frequencies. However, this optimization problem is highly nonconvex. To address this difficulty, a norm relaxed sequential quadratic programming approach is applied for finding its local optimal solution. By iterating the above procedures using different initial conditions, a near global optimal solution is obtained. Computer numerical simulation results show that our proposed design outperforms the existing designs.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  $ 20{\log _{10}}\left( {{{\left| {{H_m}\left( \omega \right)} \right|} \over {\sqrt M }}} \right) $ for $ m = 0, \dotsc, 3 $ of the analysis filters in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]

    Figure 2.  (a) $ 10{\log _{10}}\left( {\left| {\left| {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}\left( \omega \right){{\tilde H}_m}\left( \omega \right)} } \right| - 1} \right|} \right) $ and (b) $ 10{\log _{10}}\left( {\left| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}\left( {\omega - {{2\pi k} \over M}} \right){{\tilde H}_m}\left( \omega \right)} } } \right|} \right) $ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]

    Table 1.  $ \mathop {\max }\limits_{\omega \in B_m^p \cup B_m^s} 20{\log _{10}}\left( {\big| {\left| {{H_m}\left( \omega \right)} \right| - \left| {{D_m}\left( \omega \right)} \right|} \big|} \right) $ for $ m = 0, \dotsc, 3 $ of the analysis filters in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]

    The maximum ripple magnitude of the first analysis filter in decibel The maximum ripple magnitude of the second analysis filter in decibel The maximum ripple magnitude of the third analysis filter in decibel The maximum ripple magnitude of the fourth analysis filter in decibel
    Method discussed in [14] -0.3251dB -11.3525dB -11.3525dB -0.3251dB
    Method discussed in [30] -0.7366dB -13.0137dB -12.9932dB -0.7372dB
    Our proposed method -6.1628dB -6.1930dB -6.1930dB -6.1628dB
     | Show Table
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    Table 2.  $ {\log_{10}}\left( {err_{para}\left( l \right)} \right) $ for $ l = 0, \dotsc, L-1 $ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]

    Method discussed in [22] Method discussed in [16] Our proposed method
    $ {\log_{10}}\left( {err_{para}\left( 0 \right)} \right) $ -66.7193dB -141.8743dB 1.1415dB
    $ {\log_{10}}\left( {err_{para}\left( l \right)} \right) $ -141.4721dB -140.8206dB -0.8275dB
     | Show Table
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    Table 3.  $ \mathop {\max }\limits_\omega 10{\log _{10}}\left( {\left| {\left| {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}\left( \omega \right){{\tilde H}_m}\left( \omega \right)} } \right| - 1} \right|} \right) $ and $ \mathop {\max }\limits_\omega 10{\log _{10}}\left( {\left| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}\left( {\omega - {{2\pi k} \over M}} \right){{\tilde H}_m}\left( \omega \right)} } } \right|} \right) $ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]

    Method discussed in [14] Method discussed in [30] Our proposed method
    $ \mathop {\max }\limits_\omega 10{\log _{10}}( | | {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}(\omega){{\tilde H}_m}(\omega)} } | - 1 | ) $ -72.7399dB -142.8249dB -2.9505dB
    $ \mathop {\max }\limits_\omega 10{\log _{10}}( {| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}( {\omega - {{2\pi k} \over M}} ){{\tilde H}_m}( \omega )} } } |}) $ -146.7204dB -145.1184dB -6.0642dB
     | Show Table
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