Discrete Wavelet Transform

The discrete wavelet transform is a discrete-time, discrete-frequency counterpart of the continuous wavelet transform of the previous section:

$$X(k,n) = s^{-k/2} \int_{-\infty}^\infty x(t) h\left(nT-a^{-k}t\right) dt$$

$$= \int_{-\infty}^\infty x(t) h\left(na^k T-t\right) dt$$

where $n$ and $k$ range over the integers, and $h$ is the mother wavelet, interpreted here as a (continuous) filter impulse response.

The inverse transform is, as always, the signal expansion in terms of the orthonormal basis set:

$$x(t) = \sum_k \sum_n X(k,n) \underbrace{\varphi _{kn}(t)}_{\hbox{basis}}$$ (12.120)

We can show that discrete wavelet transforms are constant-Q by defining the center frequency of the $k$ th basis signal as the geometric mean of its bandlimits $\omega_1$ and $\omega _2$ , i.e.,

$$\omega _c(k) \isdef \sqrt{\omega _1(k)\,\omega _2(k)} \eqsp \sqrt{a^k\omega _1(0)\,a^k\omega _2(0)} \eqsp a^k\omega _c(0).$$ (12.121)

Then

$$Q(k) \isdefs \frac{\omega _c(k)}{\omega _2(k) - \omega _1(k)} \eqsp \frac{a^k\omega _c(0)}{a^k\omega _2(0) - a^k\omega _1(0)} \eqsp Q(0)$$ (12.122)

which does not depend on $k$ .