Discrete Wavelet Filterbank

In a discrete wavelet filterbank, each basis signal is interpreted as the impulse response of a bandpass filter in a constant-Q filter bank:

$$h_k(t) = \frac{1}{\sqrt{a^k}}\, h\left(\frac{t}{a^k}\right), \quad a>1$$

$$\longleftrightarrow\quad H_k(\omega ) = \sqrt{a^k}\, H(a^k\omega )$$

Thus, the $k$ th channel-filter $H_k(\omega )$ is obtained by frequency-scaling (and normalizing for unit energy) the zeroth channel filter $H_0(\omega )$ . The frequency scale-factor is of course equal to the inverse of the time-scale factor.

Recall that in the STFT, channel filter $H_k(\omega )$ is a shift of the zeroth channel-filter $H_0(\omega )$ (which corresponds to ``cosine modulation'' in the time domain).

As the channel-number $k$ increases, the channel impulse response $h_k$ lengthens by the factor $a^k$ ., while the pass-band of its frequency-response $H_k$ narrows by the inverse factor $a^{-k}$ .

Figure 11.32 shows a block diagram of the discrete wavelet filter bank for $a=2$ (the ``dyadic'' or ``octave filter-bank'' case), and Fig.11.33 shows its time-frequency tiling as compared to that of the STFT. The synthesis filters $f_k$ may be used to make a biorthogonal filter bank. If the $h_k$ are orthonormal, then $f_k=h_k$ .

Figure 11.32: Dyadic Biorthogonal Wavelet Filterbank

Figure 11.33: Time-frequency tiling for the Short-Time Fourier Transform (left) and dyadic wavelet filter bank (right).