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| author | dwinter |
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| date | Tue, 04 Jan 2011 10:02:07 +0100 |
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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> package org.apache.commons.math.transform;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import org.apache.commons.math.FunctionEvaluationException;<a name="line.19"></a> <FONT color="green">020</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.20"></a> <FONT color="green">021</FONT> import org.apache.commons.math.analysis.UnivariateRealFunction;<a name="line.21"></a> <FONT color="green">022</FONT> import org.apache.commons.math.complex.Complex;<a name="line.22"></a> <FONT color="green">023</FONT> <a name="line.23"></a> <FONT color="green">024</FONT> /**<a name="line.24"></a> <FONT color="green">025</FONT> * Implements the <a href="http://documents.wolfram.com/v5/Add-onsLinks/<a name="line.25"></a> <FONT color="green">026</FONT> * StandardPackages/LinearAlgebra/FourierTrig.html">Fast Cosine Transform</a><a name="line.26"></a> <FONT color="green">027</FONT> * for transformation of one-dimensional data sets. For reference, see<a name="line.27"></a> <FONT color="green">028</FONT> * <b>Fast Fourier Transforms</b>, ISBN 0849371635, chapter 3.<a name="line.28"></a> <FONT color="green">029</FONT> * <p><a name="line.29"></a> <FONT color="green">030</FONT> * FCT is its own inverse, up to a multiplier depending on conventions.<a name="line.30"></a> <FONT color="green">031</FONT> * The equations are listed in the comments of the corresponding methods.</p><a name="line.31"></a> <FONT color="green">032</FONT> * <p><a name="line.32"></a> <FONT color="green">033</FONT> * Different from FFT and FST, FCT requires the length of data set to be<a name="line.33"></a> <FONT color="green">034</FONT> * power of 2 plus one. Users should especially pay attention to the<a name="line.34"></a> <FONT color="green">035</FONT> * function transformation on how this affects the sampling.</p><a name="line.35"></a> <FONT color="green">036</FONT> * <p>As of version 2.0 this no longer implements Serializable</p><a name="line.36"></a> <FONT color="green">037</FONT> *<a name="line.37"></a> <FONT color="green">038</FONT> * @version $Revision:670469 $ $Date:2008-06-23 10:01:38 +0200 (lun., 23 juin 2008) $<a name="line.38"></a> <FONT color="green">039</FONT> * @since 1.2<a name="line.39"></a> <FONT color="green">040</FONT> */<a name="line.40"></a> <FONT color="green">041</FONT> public class FastCosineTransformer implements RealTransformer {<a name="line.41"></a> <FONT color="green">042</FONT> <a name="line.42"></a> <FONT color="green">043</FONT> /**<a name="line.43"></a> <FONT color="green">044</FONT> * Construct a default transformer.<a name="line.44"></a> <FONT color="green">045</FONT> */<a name="line.45"></a> <FONT color="green">046</FONT> public FastCosineTransformer() {<a name="line.46"></a> <FONT color="green">047</FONT> super();<a name="line.47"></a> <FONT color="green">048</FONT> }<a name="line.48"></a> <FONT color="green">049</FONT> <a name="line.49"></a> <FONT color="green">050</FONT> /**<a name="line.50"></a> <FONT color="green">051</FONT> * Transform the given real data set.<a name="line.51"></a> <FONT color="green">052</FONT> * <p><a name="line.52"></a> <FONT color="green">053</FONT> * The formula is F<sub>n</sub> = (1/2) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +<a name="line.53"></a> <FONT color="green">054</FONT> * &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)<a name="line.54"></a> <FONT color="green">055</FONT> * </p><a name="line.55"></a> <FONT color="green">056</FONT> *<a name="line.56"></a> <FONT color="green">057</FONT> * @param f the real data array to be transformed<a name="line.57"></a> <FONT color="green">058</FONT> * @return the real transformed array<a name="line.58"></a> <FONT color="green">059</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.59"></a> <FONT color="green">060</FONT> */<a name="line.60"></a> <FONT color="green">061</FONT> public double[] transform(double f[]) throws IllegalArgumentException {<a name="line.61"></a> <FONT color="green">062</FONT> return fct(f);<a name="line.62"></a> <FONT color="green">063</FONT> }<a name="line.63"></a> <FONT color="green">064</FONT> <a name="line.64"></a> <FONT color="green">065</FONT> /**<a name="line.65"></a> <FONT color="green">066</FONT> * Transform the given real function, sampled on the given interval.<a name="line.66"></a> <FONT color="green">067</FONT> * <p><a name="line.67"></a> <FONT color="green">068</FONT> * The formula is F<sub>n</sub> = (1/2) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +<a name="line.68"></a> <FONT color="green">069</FONT> * &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)<a name="line.69"></a> <FONT color="green">070</FONT> * </p><a name="line.70"></a> <FONT color="green">071</FONT> *<a name="line.71"></a> <FONT color="green">072</FONT> * @param f the function to be sampled and transformed<a name="line.72"></a> <FONT color="green">073</FONT> * @param min the lower bound for the interval<a name="line.73"></a> <FONT color="green">074</FONT> * @param max the upper bound for the interval<a name="line.74"></a> <FONT color="green">075</FONT> * @param n the number of sample points<a name="line.75"></a> <FONT color="green">076</FONT> * @return the real transformed array<a name="line.76"></a> <FONT color="green">077</FONT> * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.77"></a> <FONT color="green">078</FONT> * at some point<a name="line.78"></a> <FONT color="green">079</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.79"></a> <FONT color="green">080</FONT> */<a name="line.80"></a> <FONT color="green">081</FONT> public double[] transform(UnivariateRealFunction f,<a name="line.81"></a> <FONT color="green">082</FONT> double min, double max, int n)<a name="line.82"></a> <FONT color="green">083</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.83"></a> <FONT color="green">084</FONT> double data[] = FastFourierTransformer.sample(f, min, max, n);<a name="line.84"></a> <FONT color="green">085</FONT> return fct(data);<a name="line.85"></a> <FONT color="green">086</FONT> }<a name="line.86"></a> <FONT color="green">087</FONT> <a name="line.87"></a> <FONT color="green">088</FONT> /**<a name="line.88"></a> <FONT color="green">089</FONT> * Transform the given real data set.<a name="line.89"></a> <FONT color="green">090</FONT> * <p><a name="line.90"></a> <FONT color="green">091</FONT> * The formula is F<sub>n</sub> = &radic;(1/2N) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +<a name="line.91"></a> <FONT color="green">092</FONT> * &radic;(2/N) &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)<a name="line.92"></a> <FONT color="green">093</FONT> * </p><a name="line.93"></a> <FONT color="green">094</FONT> *<a name="line.94"></a> <FONT color="green">095</FONT> * @param f the real data array to be transformed<a name="line.95"></a> <FONT color="green">096</FONT> * @return the real transformed array<a name="line.96"></a> <FONT color="green">097</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.97"></a> <FONT color="green">098</FONT> */<a name="line.98"></a> <FONT color="green">099</FONT> public double[] transform2(double f[]) throws IllegalArgumentException {<a name="line.99"></a> <FONT color="green">100</FONT> <a name="line.100"></a> <FONT color="green">101</FONT> double scaling_coefficient = Math.sqrt(2.0 / (f.length-1));<a name="line.101"></a> <FONT color="green">102</FONT> return FastFourierTransformer.scaleArray(fct(f), scaling_coefficient);<a name="line.102"></a> <FONT color="green">103</FONT> }<a name="line.103"></a> <FONT color="green">104</FONT> <a name="line.104"></a> <FONT color="green">105</FONT> /**<a name="line.105"></a> <FONT color="green">106</FONT> * Transform the given real function, sampled on the given interval.<a name="line.106"></a> <FONT color="green">107</FONT> * <p><a name="line.107"></a> <FONT color="green">108</FONT> * The formula is F<sub>n</sub> = &radic;(1/2N) [f<sub>0</sub> + (-1)<sup>n</sup> f<sub>N</sub>] +<a name="line.108"></a> <FONT color="green">109</FONT> * &radic;(2/N) &sum;<sub>k=1</sub><sup>N-1</sup> f<sub>k</sub> cos(&pi; nk/N)<a name="line.109"></a> <FONT color="green">110</FONT> *<a name="line.110"></a> <FONT color="green">111</FONT> * </p><a name="line.111"></a> <FONT color="green">112</FONT> *<a name="line.112"></a> <FONT color="green">113</FONT> * @param f the function to be sampled and transformed<a name="line.113"></a> <FONT color="green">114</FONT> * @param min the lower bound for the interval<a name="line.114"></a> <FONT color="green">115</FONT> * @param max the upper bound for the interval<a name="line.115"></a> <FONT color="green">116</FONT> * @param n the number of sample points<a name="line.116"></a> <FONT color="green">117</FONT> * @return the real transformed array<a name="line.117"></a> <FONT color="green">118</FONT> * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.118"></a> <FONT color="green">119</FONT> * at some point<a name="line.119"></a> <FONT color="green">120</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.120"></a> <FONT color="green">121</FONT> */<a name="line.121"></a> <FONT color="green">122</FONT> public double[] transform2(UnivariateRealFunction f,<a name="line.122"></a> <FONT color="green">123</FONT> double min, double max, int n)<a name="line.123"></a> <FONT color="green">124</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.124"></a> <FONT color="green">125</FONT> <a name="line.125"></a> <FONT color="green">126</FONT> double data[] = FastFourierTransformer.sample(f, min, max, n);<a name="line.126"></a> <FONT color="green">127</FONT> double scaling_coefficient = Math.sqrt(2.0 / (n-1));<a name="line.127"></a> <FONT color="green">128</FONT> return FastFourierTransformer.scaleArray(fct(data), scaling_coefficient);<a name="line.128"></a> <FONT color="green">129</FONT> }<a name="line.129"></a> <FONT color="green">130</FONT> <a name="line.130"></a> <FONT color="green">131</FONT> /**<a name="line.131"></a> <FONT color="green">132</FONT> * Inversely transform the given real data set.<a name="line.132"></a> <FONT color="green">133</FONT> * <p><a name="line.133"></a> <FONT color="green">134</FONT> * The formula is f<sub>k</sub> = (1/N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +<a name="line.134"></a> <FONT color="green">135</FONT> * (2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)<a name="line.135"></a> <FONT color="green">136</FONT> * </p><a name="line.136"></a> <FONT color="green">137</FONT> *<a name="line.137"></a> <FONT color="green">138</FONT> * @param f the real data array to be inversely transformed<a name="line.138"></a> <FONT color="green">139</FONT> * @return the real inversely transformed array<a name="line.139"></a> <FONT color="green">140</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.140"></a> <FONT color="green">141</FONT> */<a name="line.141"></a> <FONT color="green">142</FONT> public double[] inversetransform(double f[]) throws IllegalArgumentException {<a name="line.142"></a> <FONT color="green">143</FONT> <a name="line.143"></a> <FONT color="green">144</FONT> double scaling_coefficient = 2.0 / (f.length - 1);<a name="line.144"></a> <FONT color="green">145</FONT> return FastFourierTransformer.scaleArray(fct(f), scaling_coefficient);<a name="line.145"></a> <FONT color="green">146</FONT> }<a name="line.146"></a> <FONT color="green">147</FONT> <a name="line.147"></a> <FONT color="green">148</FONT> /**<a name="line.148"></a> <FONT color="green">149</FONT> * Inversely transform the given real function, sampled on the given interval.<a name="line.149"></a> <FONT color="green">150</FONT> * <p><a name="line.150"></a> <FONT color="green">151</FONT> * The formula is f<sub>k</sub> = (1/N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +<a name="line.151"></a> <FONT color="green">152</FONT> * (2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)<a name="line.152"></a> <FONT color="green">153</FONT> * </p><a name="line.153"></a> <FONT color="green">154</FONT> *<a name="line.154"></a> <FONT color="green">155</FONT> * @param f the function to be sampled and inversely transformed<a name="line.155"></a> <FONT color="green">156</FONT> * @param min the lower bound for the interval<a name="line.156"></a> <FONT color="green">157</FONT> * @param max the upper bound for the interval<a name="line.157"></a> <FONT color="green">158</FONT> * @param n the number of sample points<a name="line.158"></a> <FONT color="green">159</FONT> * @return the real inversely transformed array<a name="line.159"></a> <FONT color="green">160</FONT> * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.160"></a> <FONT color="green">161</FONT> * at some point<a name="line.161"></a> <FONT color="green">162</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.162"></a> <FONT color="green">163</FONT> */<a name="line.163"></a> <FONT color="green">164</FONT> public double[] inversetransform(UnivariateRealFunction f,<a name="line.164"></a> <FONT color="green">165</FONT> double min, double max, int n)<a name="line.165"></a> <FONT color="green">166</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.166"></a> <FONT color="green">167</FONT> <a name="line.167"></a> <FONT color="green">168</FONT> double data[] = FastFourierTransformer.sample(f, min, max, n);<a name="line.168"></a> <FONT color="green">169</FONT> double scaling_coefficient = 2.0 / (n - 1);<a name="line.169"></a> <FONT color="green">170</FONT> return FastFourierTransformer.scaleArray(fct(data), scaling_coefficient);<a name="line.170"></a> <FONT color="green">171</FONT> }<a name="line.171"></a> <FONT color="green">172</FONT> <a name="line.172"></a> <FONT color="green">173</FONT> /**<a name="line.173"></a> <FONT color="green">174</FONT> * Inversely transform the given real data set.<a name="line.174"></a> <FONT color="green">175</FONT> * <p><a name="line.175"></a> <FONT color="green">176</FONT> * The formula is f<sub>k</sub> = &radic;(1/2N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +<a name="line.176"></a> <FONT color="green">177</FONT> * &radic;(2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)<a name="line.177"></a> <FONT color="green">178</FONT> * </p><a name="line.178"></a> <FONT color="green">179</FONT> *<a name="line.179"></a> <FONT color="green">180</FONT> * @param f the real data array to be inversely transformed<a name="line.180"></a> <FONT color="green">181</FONT> * @return the real inversely transformed array<a name="line.181"></a> <FONT color="green">182</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.182"></a> <FONT color="green">183</FONT> */<a name="line.183"></a> <FONT color="green">184</FONT> public double[] inversetransform2(double f[]) throws IllegalArgumentException {<a name="line.184"></a> <FONT color="green">185</FONT> return transform2(f);<a name="line.185"></a> <FONT color="green">186</FONT> }<a name="line.186"></a> <FONT color="green">187</FONT> <a name="line.187"></a> <FONT color="green">188</FONT> /**<a name="line.188"></a> <FONT color="green">189</FONT> * Inversely transform the given real function, sampled on the given interval.<a name="line.189"></a> <FONT color="green">190</FONT> * <p><a name="line.190"></a> <FONT color="green">191</FONT> * The formula is f<sub>k</sub> = &radic;(1/2N) [F<sub>0</sub> + (-1)<sup>k</sup> F<sub>N</sub>] +<a name="line.191"></a> <FONT color="green">192</FONT> * &radic;(2/N) &sum;<sub>n=1</sub><sup>N-1</sup> F<sub>n</sub> cos(&pi; nk/N)<a name="line.192"></a> <FONT color="green">193</FONT> * </p><a name="line.193"></a> <FONT color="green">194</FONT> *<a name="line.194"></a> <FONT color="green">195</FONT> * @param f the function to be sampled and inversely transformed<a name="line.195"></a> <FONT color="green">196</FONT> * @param min the lower bound for the interval<a name="line.196"></a> <FONT color="green">197</FONT> * @param max the upper bound for the interval<a name="line.197"></a> <FONT color="green">198</FONT> * @param n the number of sample points<a name="line.198"></a> <FONT color="green">199</FONT> * @return the real inversely transformed array<a name="line.199"></a> <FONT color="green">200</FONT> * @throws FunctionEvaluationException if function cannot be evaluated<a name="line.200"></a> <FONT color="green">201</FONT> * at some point<a name="line.201"></a> <FONT color="green">202</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.202"></a> <FONT color="green">203</FONT> */<a name="line.203"></a> <FONT color="green">204</FONT> public double[] inversetransform2(UnivariateRealFunction f,<a name="line.204"></a> <FONT color="green">205</FONT> double min, double max, int n)<a name="line.205"></a> <FONT color="green">206</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.206"></a> <FONT color="green">207</FONT> <a name="line.207"></a> <FONT color="green">208</FONT> return transform2(f, min, max, n);<a name="line.208"></a> <FONT color="green">209</FONT> }<a name="line.209"></a> <FONT color="green">210</FONT> <a name="line.210"></a> <FONT color="green">211</FONT> /**<a name="line.211"></a> <FONT color="green">212</FONT> * Perform the FCT algorithm (including inverse).<a name="line.212"></a> <FONT color="green">213</FONT> *<a name="line.213"></a> <FONT color="green">214</FONT> * @param f the real data array to be transformed<a name="line.214"></a> <FONT color="green">215</FONT> * @return the real transformed array<a name="line.215"></a> <FONT color="green">216</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.216"></a> <FONT color="green">217</FONT> */<a name="line.217"></a> <FONT color="green">218</FONT> protected double[] fct(double f[])<a name="line.218"></a> <FONT color="green">219</FONT> throws IllegalArgumentException {<a name="line.219"></a> <FONT color="green">220</FONT> <a name="line.220"></a> <FONT color="green">221</FONT> final double transformed[] = new double[f.length];<a name="line.221"></a> <FONT color="green">222</FONT> <a name="line.222"></a> <FONT color="green">223</FONT> final int n = f.length - 1;<a name="line.223"></a> <FONT color="green">224</FONT> if (!FastFourierTransformer.isPowerOf2(n)) {<a name="line.224"></a> <FONT color="green">225</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.225"></a> <FONT color="green">226</FONT> "{0} is not a power of 2 plus one",<a name="line.226"></a> <FONT color="green">227</FONT> f.length);<a name="line.227"></a> <FONT color="green">228</FONT> }<a name="line.228"></a> <FONT color="green">229</FONT> if (n == 1) { // trivial case<a name="line.229"></a> <FONT color="green">230</FONT> transformed[0] = 0.5 * (f[0] + f[1]);<a name="line.230"></a> <FONT color="green">231</FONT> transformed[1] = 0.5 * (f[0] - f[1]);<a name="line.231"></a> <FONT color="green">232</FONT> return transformed;<a name="line.232"></a> <FONT color="green">233</FONT> }<a name="line.233"></a> <FONT color="green">234</FONT> <a name="line.234"></a> <FONT color="green">235</FONT> // construct a new array and perform FFT on it<a name="line.235"></a> <FONT color="green">236</FONT> final double[] x = new double[n];<a name="line.236"></a> <FONT color="green">237</FONT> x[0] = 0.5 * (f[0] + f[n]);<a name="line.237"></a> <FONT color="green">238</FONT> x[n >> 1] = f[n >> 1];<a name="line.238"></a> <FONT color="green">239</FONT> double t1 = 0.5 * (f[0] - f[n]); // temporary variable for transformed[1]<a name="line.239"></a> <FONT color="green">240</FONT> for (int i = 1; i < (n >> 1); i++) {<a name="line.240"></a> <FONT color="green">241</FONT> final double a = 0.5 * (f[i] + f[n-i]);<a name="line.241"></a> <FONT color="green">242</FONT> final double b = Math.sin(i * Math.PI / n) * (f[i] - f[n-i]);<a name="line.242"></a> <FONT color="green">243</FONT> final double c = Math.cos(i * Math.PI / n) * (f[i] - f[n-i]);<a name="line.243"></a> <FONT color="green">244</FONT> x[i] = a - b;<a name="line.244"></a> <FONT color="green">245</FONT> x[n-i] = a + b;<a name="line.245"></a> <FONT color="green">246</FONT> t1 += c;<a name="line.246"></a> <FONT color="green">247</FONT> }<a name="line.247"></a> <FONT color="green">248</FONT> FastFourierTransformer transformer = new FastFourierTransformer();<a name="line.248"></a> <FONT color="green">249</FONT> Complex y[] = transformer.transform(x);<a name="line.249"></a> <FONT color="green">250</FONT> <a name="line.250"></a> <FONT color="green">251</FONT> // reconstruct the FCT result for the original array<a name="line.251"></a> <FONT color="green">252</FONT> transformed[0] = y[0].getReal();<a name="line.252"></a> <FONT color="green">253</FONT> transformed[1] = t1;<a name="line.253"></a> <FONT color="green">254</FONT> for (int i = 1; i < (n >> 1); i++) {<a name="line.254"></a> <FONT color="green">255</FONT> transformed[2 * i] = y[i].getReal();<a name="line.255"></a> <FONT color="green">256</FONT> transformed[2 * i + 1] = transformed[2 * i - 1] - y[i].getImaginary();<a name="line.256"></a> <FONT color="green">257</FONT> }<a name="line.257"></a> <FONT color="green">258</FONT> transformed[n] = y[n >> 1].getReal();<a name="line.258"></a> <FONT color="green">259</FONT> <a name="line.259"></a> <FONT color="green">260</FONT> return transformed;<a name="line.260"></a> <FONT color="green">261</FONT> }<a name="line.261"></a> <FONT color="green">262</FONT> }<a name="line.262"></a> </PRE> </BODY> </HTML>