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<a name="DWT-Definitions"></a>
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Next: <a href="DWT-Initialization.html#DWT-Initialization" accesskey="n" rel="next">DWT Initialization</a>, Up: <a href="Wavelet-Transforms.html#Wavelet-Transforms" accesskey="u" rel="up">Wavelet Transforms</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Definitions-1"></a>
<h3 class="section">32.1 Definitions</h3>
<a name="index-DWT_002c-mathematical-definition"></a>

<p>The continuous wavelet transform and its inverse are defined by
the relations,
</p>
<div class="example">
<pre class="example">w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt
</pre></div>

<p>and,
</p>
<div class="example">
<pre class="example">f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds
</pre></div>

<p>where the basis functions <em>\psi_{s,\tau}</em> are obtained by scaling
and translation from a single function, referred to as the <em>mother
wavelet</em>.
</p>
<p>The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (<em>s</em>,
<em>\tau</em>).  The frequency and time axes are sampled <em>dyadically</em>
on scales of <em>2^j</em> through a level parameter <em>j</em>.
The resulting family of functions <em>{\psi_{j,n}}</em>
constitutes an orthonormal
basis for square-integrable signals.  
</p>
<p>The discrete wavelet transform is an <em>O(N)</em> algorithm, and is also
referred to as the <em>fast wavelet transform</em>.
</p>



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