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

where and range over the integers, and 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:

(12.120) |

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

(12.121) |

Then

(12.122) |

which does not depend on .

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