Subband- and Wavelet-Based Coding - Digital Signal Processing: Fundamentals and Applications - page 630

Filter design constraint equations for the lowpass filter H 0 ðzÞ :

RðzÞ¼H 0 ðzÞH 0 ðz 1 Þ

RðzÞþRðzÞ¼ 1
rð 2 nÞ¼ 0 : 5 dðnÞ

Equations for the other filters:

H 1 ðzÞ¼z ðN 1 Þ H 0 ðz 1

Þ

G 0 ðzÞ¼H 1 ðzÞ
G 1 ðzÞ¼H 0 ðzÞ

Design of the analysis and synthesis filters to satisfy the above conditions is very challenging. Smith
and Barnwell (1984) were the first to show that perfect reconstruction in a two-band filter bank is
possible when the linear phase of the FIR filter requirement is relaxed. The Smith-Barnwell filters are
called the conjugate quadrature filters (PR-CQF). Eight- and 16-tap PR-CQF coefficients are listed in
Table 13.1 . As shown in Table 13.1 , the filter coefficients are not symmetric; hence, the obtained
analysis filter does not have a linear phase. The detailed design of Smith-Barnwell filters can be found
in their research paper (Smith and Barnwell, 1984) and the design of other types of analysis and
synthesis filters can be found in Akansu and Haddad (1992).
Now let us verify the filter constraint in the following example.

Table 13.1 SmitheBarnwell PR-CQF Filters
8 Taps 16 Taps

0.0348975582178515 0.02193598203004352

0.01098301946252854 0.001578616497663704

0.06286453934951963 0.06025449102875281

0.223907720892568 0.0118906596205391

0.556856993531445 0.137537915636625

0.357976304997285 0.05745450056390939

0.02390027056113145 0.321670296165893

0.07594096379188282 0.528720271545339

0.295779674500919
0.0002043110845170894
0.0290669978946796

0.03533486088708146

0.006821045322743358
0.02606678468264118
0.001033363491944126

0.01435930957477529