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		<title>Convolution in the Frequency Domain</title>
		<link>http://plusha.com/2006/01/03/convolution-in-the-frequency-domain/</link>
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		<pubDate>Tue, 03 Jan 2006 01:09:24 +0000</pubDate>
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				<category><![CDATA[Computer]]></category>
		<category><![CDATA[numerical]]></category>

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		<description><![CDATA[주파수 영역에서의 (circular) convolution 길이 n인 real wavelet1과 길이 m인 real wavelet2를 각각 complex로 만들어준 후 길이 m+n-1이 되도록 Zero padding을 해준다. 각각의 wavelet들을 FFT한 후에 k=m+n-1 output(1:k)=cwave1(1:k)*cwave2(1:k) !곱하기 그리고 나서 inverse Fourier transform 하면 Time domain에서의 convolution 결과와 같게 나온다. 01: L=m+n-1 02: cw1(1:n)=cmplx(wave1(1:n),0.) 03: cw1(n+1:L)=cmplx(0.,0.) 04: 05: cw2(1:m)=cmplx(wave2(1:m),0.) 06: cw2(m+1:L)=cmplx(0.,0.) 07: 08: call [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=plusha.com&#038;blog=174108&#038;post=7&#038;subd=plusha&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
			<content:encoded><![CDATA[<div>주파수 영역에서의 (circular) convolution</div>
<p>길이 n인 real wavelet1과<br />
길이 m인 real wavelet2를<br />
각각 complex로 만들어준 후 길이 m+n-1이 되도록<br />
Zero padding을 해준다.</p>
<p>각각의 wavelet들을 FFT한 후에</p>
<p>k=m+n-1<br />
output(1:k)=cwave1(1:k)*cwave2(1:k) !곱하기</p>
<p>그리고 나서 inverse Fourier transform 하면<br />
Time domain에서의 convolution 결과와 같게 나온다.</p>
<pre><tt><code><font color="#000000">01:</font>   L<font color="#990000">=</font>m<font color="#990000">+</font>n<font color="#990000">-</font><font color="#993399">1</font>
<font color="#000000">02:</font>   <b><font color="#000000">cw1</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>n<font color="#990000">)=</font><b><font color="#000000">cmplx</font></b><font color="#990000">(</font><b><font color="#000000">wave1</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>n<font color="#990000">),</font><font color="#993399">0</font><font color="#990000">.)</font>
<font color="#000000">03:</font>   <b><font color="#000000">cw1</font></b><font color="#990000">(</font>n<font color="#990000">+</font><font color="#993399">1</font><font color="#990000">:</font>L<font color="#990000">)=</font><b><font color="#000000">cmplx</font></b><font color="#990000">(</font><font color="#993399">0</font><font color="#990000">.,</font><font color="#993399">0</font><font color="#990000">.)</font>
<font color="#000000">04:</font>   
<font color="#000000">05:</font>   <b><font color="#000000">cw2</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>m<font color="#990000">)=</font><b><font color="#000000">cmplx</font></b><font color="#990000">(</font><b><font color="#000000">wave2</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>m<font color="#990000">),</font><font color="#993399">0</font><font color="#990000">.)</font>
<font color="#000000">06:</font>   <b><font color="#000000">cw2</font></b><font color="#990000">(</font>m<font color="#990000">+</font><font color="#993399">1</font><font color="#990000">:</font>L<font color="#990000">)=</font><b><font color="#000000">cmplx</font></b><font color="#990000">(</font><font color="#993399">0</font><font color="#990000">.,</font><font color="#993399">0</font><font color="#990000">.)</font>
<font color="#000000">07:</font>   
<font color="#000000">08:</font>   <b><font color="#0000FF">call</font></b> <b><font color="#000000">fft</font></b><font color="#990000">(</font>cw1<font color="#990000">,</font>L<font color="#990000">)</font>
<font color="#000000">09:</font>   <b><font color="#0000FF">call</font></b> <b><font color="#000000">fft</font></b><font color="#990000">(</font>cw2<font color="#990000">,</font>L<font color="#990000">)</font>
<font color="#000000">10:</font>   
<font color="#000000">11:</font>   <b><font color="#000000">conv</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>L<font color="#990000">)=</font><b><font color="#000000">cw1</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>L<font color="#990000">)</font><font color="#990000">*</font><b><font color="#000000">cw2</font></b><font color="#990000">(</font><font color="#993399">1</font><font color="#990000">:</font>L<font color="#990000">)</font>
<font color="#000000">12:</font>   
<font color="#000000">13:</font>   <b><font color="#0000FF">call</font></b> <b><font color="#000000">ift</font></b><font color="#990000">(</font>conv<font color="#990000">,</font>L<font color="#990000">)</font></code>
</tt></pre>
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