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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> <a name="line.22"></a> <FONT color="green">023</FONT> /**<a name="line.23"></a> <FONT color="green">024</FONT> * Implements the <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">Fast Hadamard Transform</a> (FHT).<a name="line.24"></a> <FONT color="green">025</FONT> * Transformation of an input vector x to the output vector y.<a name="line.25"></a> <FONT color="green">026</FONT> * <p>In addition to transformation of real vectors, the Hadamard transform can<a name="line.26"></a> <FONT color="green">027</FONT> * transform integer vectors into integer vectors. However, this integer transform<a name="line.27"></a> <FONT color="green">028</FONT> * cannot be inverted directly. Due to a scaling factor it may lead to rational results.<a name="line.28"></a> <FONT color="green">029</FONT> * As an example, the inverse transform of integer vector (0, 1, 0, 1) is rational<a name="line.29"></a> <FONT color="green">030</FONT> * vector (1/2, -1/2, 0, 0).</p><a name="line.30"></a> <FONT color="green">031</FONT> * @version $Revision: 811685 $ $Date: 2009-09-05 13:36:48 -0400 (Sat, 05 Sep 2009) $<a name="line.31"></a> <FONT color="green">032</FONT> * @since 2.0<a name="line.32"></a> <FONT color="green">033</FONT> */<a name="line.33"></a> <FONT color="green">034</FONT> public class FastHadamardTransformer implements RealTransformer {<a name="line.34"></a> <FONT color="green">035</FONT> <a name="line.35"></a> <FONT color="green">036</FONT> /** {@inheritDoc} */<a name="line.36"></a> <FONT color="green">037</FONT> public double[] transform(double f[])<a name="line.37"></a> <FONT color="green">038</FONT> throws IllegalArgumentException {<a name="line.38"></a> <FONT color="green">039</FONT> return fht(f);<a name="line.39"></a> <FONT color="green">040</FONT> }<a name="line.40"></a> <FONT color="green">041</FONT> <a name="line.41"></a> <FONT color="green">042</FONT> /** {@inheritDoc} */<a name="line.42"></a> <FONT color="green">043</FONT> public double[] transform(UnivariateRealFunction f,<a name="line.43"></a> <FONT color="green">044</FONT> double min, double max, int n)<a name="line.44"></a> <FONT color="green">045</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.45"></a> <FONT color="green">046</FONT> return fht(FastFourierTransformer.sample(f, min, max, n));<a name="line.46"></a> <FONT color="green">047</FONT> }<a name="line.47"></a> <FONT color="green">048</FONT> <a name="line.48"></a> <FONT color="green">049</FONT> /** {@inheritDoc} */<a name="line.49"></a> <FONT color="green">050</FONT> public double[] inversetransform(double f[])<a name="line.50"></a> <FONT color="green">051</FONT> throws IllegalArgumentException {<a name="line.51"></a> <FONT color="green">052</FONT> return FastFourierTransformer.scaleArray(fht(f), 1.0 / f.length);<a name="line.52"></a> <FONT color="green">053</FONT> }<a name="line.53"></a> <FONT color="green">054</FONT> <a name="line.54"></a> <FONT color="green">055</FONT> /** {@inheritDoc} */<a name="line.55"></a> <FONT color="green">056</FONT> public double[] inversetransform(UnivariateRealFunction f,<a name="line.56"></a> <FONT color="green">057</FONT> double min, double max, int n)<a name="line.57"></a> <FONT color="green">058</FONT> throws FunctionEvaluationException, IllegalArgumentException {<a name="line.58"></a> <FONT color="green">059</FONT> final double[] unscaled =<a name="line.59"></a> <FONT color="green">060</FONT> fht(FastFourierTransformer.sample(f, min, max, n));<a name="line.60"></a> <FONT color="green">061</FONT> return FastFourierTransformer.scaleArray(unscaled, 1.0 / n);<a name="line.61"></a> <FONT color="green">062</FONT> }<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> * Transform the given real data set.<a name="line.65"></a> <FONT color="green">066</FONT> * <p>The integer transform cannot be inverted directly, due to a scaling<a name="line.66"></a> <FONT color="green">067</FONT> * factor it may lead to double results.</p><a name="line.67"></a> <FONT color="green">068</FONT> * @param f the integer data array to be transformed (signal)<a name="line.68"></a> <FONT color="green">069</FONT> * @return the integer transformed array (spectrum)<a name="line.69"></a> <FONT color="green">070</FONT> * @throws IllegalArgumentException if any parameters are invalid<a name="line.70"></a> <FONT color="green">071</FONT> */<a name="line.71"></a> <FONT color="green">072</FONT> public int[] transform(int f[])<a name="line.72"></a> <FONT color="green">073</FONT> throws IllegalArgumentException {<a name="line.73"></a> <FONT color="green">074</FONT> return fht(f);<a name="line.74"></a> <FONT color="green">075</FONT> }<a name="line.75"></a> <FONT color="green">076</FONT> <a name="line.76"></a> <FONT color="green">077</FONT> /**<a name="line.77"></a> <FONT color="green">078</FONT> * The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.<a name="line.78"></a> <FONT color="green">079</FONT> * <br><a name="line.79"></a> <FONT color="green">080</FONT> * Requires <b>Nlog2N = n2</b><sup>n</sup> additions.<a name="line.80"></a> <FONT color="green">081</FONT> * <br><a name="line.81"></a> <FONT color="green">082</FONT> * <br><a name="line.82"></a> <FONT color="green">083</FONT> * <b><u>Short Table of manual calculation for N=8:</u></b><a name="line.83"></a> <FONT color="green">084</FONT> * <ol><a name="line.84"></a> <FONT color="green">085</FONT> * <li><b>x</b> is the input vector we want to transform</li><a name="line.85"></a> <FONT color="green">086</FONT> * <li><b>y</b> is the output vector which is our desired result</li><a name="line.86"></a> <FONT color="green">087</FONT> * <li>a and b are just helper rows</li><a name="line.87"></a> <FONT color="green">088</FONT> * </ol><a name="line.88"></a> <FONT color="green">089</FONT> * <pre><a name="line.89"></a> <FONT color="green">090</FONT> * <code><a name="line.90"></a> <FONT color="green">091</FONT> * +----+----------+---------+----------+<a name="line.91"></a> <FONT color="green">092</FONT> * | <b>x</b> | <b>a</b> | <b>b</b> | <b>y</b> |<a name="line.92"></a> <FONT color="green">093</FONT> * +----+----------+---------+----------+<a name="line.93"></a> <FONT color="green">094</FONT> * | x<sub>0</sub> | a<sub>0</sub>=x<sub>0</sub>+x<sub>1</sub> | b<sub>0</sub>=a<sub>0</sub>+a<sub>1</sub> | y<sub>0</sub>=b<sub>0</sub>+b<sub>1</sub> |<a name="line.94"></a> <FONT color="green">095</FONT> * +----+----------+---------+----------+<a name="line.95"></a> <FONT color="green">096</FONT> * | x<sub>1</sub> | a<sub>1</sub>=x<sub>2</sub>+x<sub>3</sub> | b<sub>0</sub>=a<sub>2</sub>+a<sub>3</sub> | y<sub>0</sub>=b<sub>2</sub>+b<sub>3</sub> |<a name="line.96"></a> <FONT color="green">097</FONT> * +----+----------+---------+----------+<a name="line.97"></a> <FONT color="green">098</FONT> * | x<sub>2</sub> | a<sub>2</sub>=x<sub>4</sub>+x<sub>5</sub> | b<sub>0</sub>=a<sub>4</sub>+a<sub>5</sub> | y<sub>0</sub>=b<sub>4</sub>+b<sub>5</sub> |<a name="line.98"></a> <FONT color="green">099</FONT> * +----+----------+---------+----------+<a name="line.99"></a> <FONT color="green">100</FONT> * | x<sub>3</sub> | a<sub>3</sub>=x<sub>6</sub>+x<sub>7</sub> | b<sub>0</sub>=a<sub>6</sub>+a<sub>7</sub> | y<sub>0</sub>=b<sub>6</sub>+b<sub>7</sub> |<a name="line.100"></a> <FONT color="green">101</FONT> * +----+----------+---------+----------+<a name="line.101"></a> <FONT color="green">102</FONT> * | x<sub>4</sub> | a<sub>0</sub>=x<sub>0</sub>-x<sub>1</sub> | b<sub>0</sub>=a<sub>0</sub>-a<sub>1</sub> | y<sub>0</sub>=b<sub>0</sub>-b<sub>1</sub> |<a name="line.102"></a> <FONT color="green">103</FONT> * +----+----------+---------+----------+<a name="line.103"></a> <FONT color="green">104</FONT> * | x<sub>5</sub> | a<sub>1</sub>=x<sub>2</sub>-x<sub>3</sub> | b<sub>0</sub>=a<sub>2</sub>-a<sub>3</sub> | y<sub>0</sub>=b<sub>2</sub>-b<sub>3</sub> |<a name="line.104"></a> <FONT color="green">105</FONT> * +----+----------+---------+----------+<a name="line.105"></a> <FONT color="green">106</FONT> * | x<sub>6</sub> | a<sub>2</sub>=x<sub>4</sub>-x<sub>5</sub> | b<sub>0</sub>=a<sub>4</sub>-a<sub>5</sub> | y<sub>0</sub>=b<sub>4</sub>-b<sub>5</sub> |<a name="line.106"></a> <FONT color="green">107</FONT> * +----+----------+---------+----------+<a name="line.107"></a> <FONT color="green">108</FONT> * | x<sub>7</sub> | a<sub>3</sub>=x<sub>6</sub>-x<sub>7</sub> | b<sub>0</sub>=a<sub>6</sub>-a<sub>7</sub> | y<sub>0</sub>=b<sub>6</sub>-b<sub>7</sub> |<a name="line.108"></a> <FONT color="green">109</FONT> * +----+----------+---------+----------+<a name="line.109"></a> <FONT color="green">110</FONT> * </code><a name="line.110"></a> <FONT color="green">111</FONT> * </pre><a name="line.111"></a> <FONT color="green">112</FONT> *<a name="line.112"></a> <FONT color="green">113</FONT> * <b><u>How it works</u></b><a name="line.113"></a> <FONT color="green">114</FONT> * <ol><a name="line.114"></a> <FONT color="green">115</FONT> * <li>Construct a matrix with N rows and n+1 columns<br> <b>hadm[n+1][N]</b><a name="line.115"></a> <FONT color="green">116</FONT> * <br><i>(If I use [x][y] it always means [row-offset][column-offset] of a Matrix with n rows and m columns. Its entries go from M[0][0] to M[n][m])</i></li><a name="line.116"></a> <FONT color="green">117</FONT> * <li>Place the input vector <b>x[N]</b> in the first column of the matrix <b>hadm</b></li><a name="line.117"></a> <FONT color="green">118</FONT> * <li>The entries of the submatrix D<sub>top</sub> are calculated as follows.<a name="line.118"></a> <FONT color="green">119</FONT> * <br>D<sub>top</sub> goes from entry [0][1] to [N/2-1][n+1].<a name="line.119"></a> <FONT color="green">120</FONT> * <br>The columns of D<sub>top</sub> are the pairwise mutually exclusive sums of the previous column<a name="line.120"></a> <FONT color="green">121</FONT> * </li><a name="line.121"></a> <FONT color="green">122</FONT> * <li>The entries of the submatrix D<sub>bottom</sub> are calculated as follows.<a name="line.122"></a> <FONT color="green">123</FONT> * <br>D<sub>bottom</sub> goes from entry [N/2][1] to [N][n+1].<a name="line.123"></a> <FONT color="green">124</FONT> * <br>The columns of D<sub>bottom</sub> are the pairwise differences of the previous column<a name="line.124"></a> <FONT color="green">125</FONT> * </li><a name="line.125"></a> <FONT color="green">126</FONT> * <li>How D<sub>top</sub> and D<sub>bottom</sub> you can understand best with the example for N=8 above.<a name="line.126"></a> <FONT color="green">127</FONT> * <li>The output vector y is now in the last column of <b>hadm</b></li><a name="line.127"></a> <FONT color="green">128</FONT> * <li><i>Algorithm from: http://www.archive.chipcenter.com/dsp/DSP000517F1.html</i></li><a name="line.128"></a> <FONT color="green">129</FONT> * </ol><a name="line.129"></a> <FONT color="green">130</FONT> * <br><a name="line.130"></a> <FONT color="green">131</FONT> * <b><u>Visually</u></b><a name="line.131"></a> <FONT color="green">132</FONT> * <pre><a name="line.132"></a> <FONT color="green">133</FONT> * +--------+---+---+---+-----+---+<a name="line.133"></a> <FONT color="green">134</FONT> * | 0 | 1 | 2 | 3 | ... |n+1|<a name="line.134"></a> <FONT color="green">135</FONT> * +------+--------+---+---+---+-----+---+<a name="line.135"></a> <FONT color="green">136</FONT> * |0 | x<sub>0</sub> | /\ |<a name="line.136"></a> <FONT color="green">137</FONT> * |1 | x<sub>1</sub> | || |<a name="line.137"></a> <FONT color="green">138</FONT> * |2 | x<sub>2</sub> | <= D<sub>top</sub> => |<a name="line.138"></a> <FONT color="green">139</FONT> * |... | ... | || |<a name="line.139"></a> <FONT color="green">140</FONT> * |N/2-1 | x<sub>N/2-1</sub> | \/ |<a name="line.140"></a> <FONT color="green">141</FONT> * +------+--------+---+---+---+-----+---+<a name="line.141"></a> <FONT color="green">142</FONT> * |N/2 | x<sub>N/2</sub> | /\ |<a name="line.142"></a> <FONT color="green">143</FONT> * |N/2+1 | x<sub>N/2+1</sub> | || |<a name="line.143"></a> <FONT color="green">144</FONT> * |N/2+2 | x<sub>N/2+2</sub> | <= D<sub>bottom</sub> => | which is in the last column of the matrix<a name="line.144"></a> <FONT color="green">145</FONT> * |... | ... | || |<a name="line.145"></a> <FONT color="green">146</FONT> * |N | x<sub>N/2</sub> | \/ |<a name="line.146"></a> <FONT color="green">147</FONT> * +------+--------+---+---+---+-----+---+<a name="line.147"></a> <FONT color="green">148</FONT> * </pre><a name="line.148"></a> <FONT color="green">149</FONT> *<a name="line.149"></a> <FONT color="green">150</FONT> * @param x input vector<a name="line.150"></a> <FONT color="green">151</FONT> * @return y output vector<a name="line.151"></a> <FONT color="green">152</FONT> * @exception IllegalArgumentException if input array is not a power of 2<a name="line.152"></a> <FONT color="green">153</FONT> */<a name="line.153"></a> <FONT color="green">154</FONT> protected double[] fht(double x[]) throws IllegalArgumentException {<a name="line.154"></a> <FONT color="green">155</FONT> <a name="line.155"></a> <FONT color="green">156</FONT> // n is the row count of the input vector x<a name="line.156"></a> <FONT color="green">157</FONT> final int n = x.length;<a name="line.157"></a> <FONT color="green">158</FONT> final int halfN = n / 2;<a name="line.158"></a> <FONT color="green">159</FONT> <a name="line.159"></a> <FONT color="green">160</FONT> // n has to be of the form n = 2^p !!<a name="line.160"></a> <FONT color="green">161</FONT> if (!FastFourierTransformer.isPowerOf2(n)) {<a name="line.161"></a> <FONT color="green">162</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.162"></a> <FONT color="green">163</FONT> "{0} is not a power of 2",<a name="line.163"></a> <FONT color="green">164</FONT> n);<a name="line.164"></a> <FONT color="green">165</FONT> }<a name="line.165"></a> <FONT color="green">166</FONT> <a name="line.166"></a> <FONT color="green">167</FONT> // Instead of creating a matrix with p+1 columns and n rows<a name="line.167"></a> <FONT color="green">168</FONT> // we will use two single dimension arrays which we will use in an alternating way.<a name="line.168"></a> <FONT color="green">169</FONT> double[] yPrevious = new double[n];<a name="line.169"></a> <FONT color="green">170</FONT> double[] yCurrent = x.clone();<a name="line.170"></a> <FONT color="green">171</FONT> <a name="line.171"></a> <FONT color="green">172</FONT> // iterate from left to right (column)<a name="line.172"></a> <FONT color="green">173</FONT> for (int j = 1; j < n; j <<= 1) {<a name="line.173"></a> <FONT color="green">174</FONT> <a name="line.174"></a> <FONT color="green">175</FONT> // switch columns<a name="line.175"></a> <FONT color="green">176</FONT> final double[] yTmp = yCurrent;<a name="line.176"></a> <FONT color="green">177</FONT> yCurrent = yPrevious;<a name="line.177"></a> <FONT color="green">178</FONT> yPrevious = yTmp;<a name="line.178"></a> <FONT color="green">179</FONT> <a name="line.179"></a> <FONT color="green">180</FONT> // iterate from top to bottom (row)<a name="line.180"></a> <FONT color="green">181</FONT> for (int i = 0; i < halfN; ++i) {<a name="line.181"></a> <FONT color="green">182</FONT> // D<sub>top</sub><a name="line.182"></a> <FONT color="green">183</FONT> // The top part works with addition<a name="line.183"></a> <FONT color="green">184</FONT> final int twoI = 2 * i;<a name="line.184"></a> <FONT color="green">185</FONT> yCurrent[i] = yPrevious[twoI] + yPrevious[twoI + 1];<a name="line.185"></a> <FONT color="green">186</FONT> }<a name="line.186"></a> <FONT color="green">187</FONT> for (int i = halfN; i < n; ++i) {<a name="line.187"></a> <FONT color="green">188</FONT> // D<sub>bottom</sub><a name="line.188"></a> <FONT color="green">189</FONT> // The bottom part works with subtraction<a name="line.189"></a> <FONT color="green">190</FONT> final int twoI = 2 * i;<a name="line.190"></a> <FONT color="green">191</FONT> yCurrent[i] = yPrevious[twoI - n] - yPrevious[twoI - n + 1];<a name="line.191"></a> <FONT color="green">192</FONT> }<a name="line.192"></a> <FONT color="green">193</FONT> }<a name="line.193"></a> <FONT color="green">194</FONT> <a name="line.194"></a> <FONT color="green">195</FONT> // return the last computed output vector y<a name="line.195"></a> <FONT color="green">196</FONT> return yCurrent;<a name="line.196"></a> <FONT color="green">197</FONT> <a name="line.197"></a> <FONT color="green">198</FONT> }<a name="line.198"></a> <FONT color="green">199</FONT> /**<a name="line.199"></a> <FONT color="green">200</FONT> * The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.<a name="line.200"></a> <FONT color="green">201</FONT> * @param x input vector<a name="line.201"></a> <FONT color="green">202</FONT> * @return y output vector<a name="line.202"></a> <FONT color="green">203</FONT> * @exception IllegalArgumentException if input array is not a power of 2<a name="line.203"></a> <FONT color="green">204</FONT> */<a name="line.204"></a> <FONT color="green">205</FONT> protected int[] fht(int x[]) throws IllegalArgumentException {<a name="line.205"></a> <FONT color="green">206</FONT> <a name="line.206"></a> <FONT color="green">207</FONT> // n is the row count of the input vector x<a name="line.207"></a> <FONT color="green">208</FONT> final int n = x.length;<a name="line.208"></a> <FONT color="green">209</FONT> final int halfN = n / 2;<a name="line.209"></a> <FONT color="green">210</FONT> <a name="line.210"></a> <FONT color="green">211</FONT> // n has to be of the form n = 2^p !!<a name="line.211"></a> <FONT color="green">212</FONT> if (!FastFourierTransformer.isPowerOf2(n)) {<a name="line.212"></a> <FONT color="green">213</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.213"></a> <FONT color="green">214</FONT> "{0} is not a power of 2",<a name="line.214"></a> <FONT color="green">215</FONT> n);<a name="line.215"></a> <FONT color="green">216</FONT> }<a name="line.216"></a> <FONT color="green">217</FONT> <a name="line.217"></a> <FONT color="green">218</FONT> // Instead of creating a matrix with p+1 columns and n rows<a name="line.218"></a> <FONT color="green">219</FONT> // we will use two single dimension arrays which we will use in an alternating way.<a name="line.219"></a> <FONT color="green">220</FONT> int[] yPrevious = new int[n];<a name="line.220"></a> <FONT color="green">221</FONT> int[] yCurrent = x.clone();<a name="line.221"></a> <FONT color="green">222</FONT> <a name="line.222"></a> <FONT color="green">223</FONT> // iterate from left to right (column)<a name="line.223"></a> <FONT color="green">224</FONT> for (int j = 1; j < n; j <<= 1) {<a name="line.224"></a> <FONT color="green">225</FONT> <a name="line.225"></a> <FONT color="green">226</FONT> // switch columns<a name="line.226"></a> <FONT color="green">227</FONT> final int[] yTmp = yCurrent;<a name="line.227"></a> <FONT color="green">228</FONT> yCurrent = yPrevious;<a name="line.228"></a> <FONT color="green">229</FONT> yPrevious = yTmp;<a name="line.229"></a> <FONT color="green">230</FONT> <a name="line.230"></a> <FONT color="green">231</FONT> // iterate from top to bottom (row)<a name="line.231"></a> <FONT color="green">232</FONT> for (int i = 0; i < halfN; ++i) {<a name="line.232"></a> <FONT color="green">233</FONT> // D<sub>top</sub><a name="line.233"></a> <FONT color="green">234</FONT> // The top part works with addition<a name="line.234"></a> <FONT color="green">235</FONT> final int twoI = 2 * i;<a name="line.235"></a> <FONT color="green">236</FONT> yCurrent[i] = yPrevious[twoI] + yPrevious[twoI + 1];<a name="line.236"></a> <FONT color="green">237</FONT> }<a name="line.237"></a> <FONT color="green">238</FONT> for (int i = halfN; i < n; ++i) {<a name="line.238"></a> <FONT color="green">239</FONT> // D<sub>bottom</sub><a name="line.239"></a> <FONT color="green">240</FONT> // The bottom part works with subtraction<a name="line.240"></a> <FONT color="green">241</FONT> final int twoI = 2 * i;<a name="line.241"></a> <FONT color="green">242</FONT> yCurrent[i] = yPrevious[twoI - n] - yPrevious[twoI - n + 1];<a name="line.242"></a> <FONT color="green">243</FONT> }<a name="line.243"></a> <FONT color="green">244</FONT> }<a name="line.244"></a> <FONT color="green">245</FONT> <a name="line.245"></a> <FONT color="green">246</FONT> // return the last computed output vector y<a name="line.246"></a> <FONT color="green">247</FONT> return yCurrent;<a name="line.247"></a> <FONT color="green">248</FONT> <a name="line.248"></a> <FONT color="green">249</FONT> }<a name="line.249"></a> <FONT color="green">250</FONT> <a name="line.250"></a> <FONT color="green">251</FONT> }<a name="line.251"></a> </PRE> </BODY> </HTML>