001package algs91; // section 9.9 002import stdlib.*; 003import algs12.Complex; 004/* *********************************************************************** 005 * Compilation: javac FFT.java 006 * Execution: java FFT N 007 * Dependencies: Complex.java 008 * 009 * Compute the FFT and inverse FFT of a length N complex sequence. 010 * Bare bones implementation that runs in O(N log N) time. Our goal 011 * is to optimize the clarity of the code, rather than performance. 012 * 013 * Limitations 014 * ----------- 015 * - assumes N is a power of 2 016 * 017 * - not the most memory efficient algorithm (because it uses 018 * an object type for representing complex numbers and because 019 * it re-allocates memory for the subarray, instead of doing 020 * in-place or reusing a single temporary array) 021 * 022 *************************************************************************/ 023 024public class FFT { 025 026 // compute the FFT of x[], assuming its length is a power of 2 027 public static Complex[] fft(Complex[] x) { 028 int N = x.length; 029 030 // base case 031 if (N == 1) return new Complex[] { x[0] }; 032 033 // radix 2 Cooley-Tukey FFT 034 if (N % 2 != 0) { throw new Error("N is not a power of 2"); } 035 036 // fft of even terms 037 Complex[] even = new Complex[N/2]; 038 for (int k = 0; k < N/2; k++) { 039 even[k] = x[2*k]; 040 } 041 Complex[] q = fft(even); 042 043 // fft of odd terms 044 Complex[] odd = even; // reuse the array 045 for (int k = 0; k < N/2; k++) { 046 odd[k] = x[2*k + 1]; 047 } 048 Complex[] r = fft(odd); 049 050 // combine 051 Complex[] y = new Complex[N]; 052 for (int k = 0; k < N/2; k++) { 053 double kth = -2 * k * Math.PI / N; 054 Complex wk = new Complex(Math.cos(kth), Math.sin(kth)); 055 y[k] = q[k].plus(wk.times(r[k])); 056 y[k + N/2] = q[k].minus(wk.times(r[k])); 057 } 058 return y; 059 } 060 061 062 // compute the inverse FFT of x[], assuming its length is a power of 2 063 public static Complex[] ifft(Complex[] x) { 064 int N = x.length; 065 Complex[] y = new Complex[N]; 066 067 // take conjugate 068 for (int i = 0; i < N; i++) { 069 y[i] = x[i].conjugate(); 070 } 071 072 // compute forward FFT 073 y = fft(y); 074 075 // take conjugate again 076 for (int i = 0; i < N; i++) { 077 y[i] = y[i].conjugate(); 078 } 079 080 // divide by N 081 for (int i = 0; i < N; i++) { 082 y[i] = y[i].times(1.0 / N); 083 } 084 085 return y; 086 087 } 088 089 // compute the circular convolution of x and y 090 public static Complex[] cconvolve(Complex[] x, Complex[] y) { 091 092 // should probably pad x and y with 0s so that they have same length 093 // and are powers of 2 094 if (x.length != y.length) { throw new Error("Dimensions don't agree"); } 095 096 int N = x.length; 097 098 // compute FFT of each sequence 099 Complex[] a = fft(x); 100 Complex[] b = fft(y); 101 102 // point-wise multiply 103 Complex[] c = new Complex[N]; 104 for (int i = 0; i < N; i++) { 105 c[i] = a[i].times(b[i]); 106 } 107 108 // compute inverse FFT 109 return ifft(c); 110 } 111 112 113 // compute the linear convolution of x and y 114 public static Complex[] convolve(Complex[] x, Complex[] y) { 115 Complex ZERO = new Complex(0, 0); 116 117 Complex[] a = new Complex[2*x.length]; 118 for (int i = 0; i < x.length; i++) a[i] = x[i]; 119 for (int i = x.length; i < 2*x.length; i++) a[i] = ZERO; 120 121 Complex[] b = new Complex[2*y.length]; 122 for (int i = 0; i < y.length; i++) b[i] = y[i]; 123 for (int i = y.length; i < 2*y.length; i++) b[i] = ZERO; 124 125 return cconvolve(a, b); 126 } 127 128 // display an array of Complex numbers to standard output 129 public static void show(Complex[] x, String title) { 130 StdOut.println(title); 131 StdOut.println("-------------------"); 132 for (Complex element : x) { 133 StdOut.println(element); 134 } 135 StdOut.println(); 136 } 137 138 139 /* ******************************************************************* 140 * Test client and sample execution 141 * 142 * % java FFT 4 143 * x 144 * ------------------- 145 * -0.03480425839330703 146 * 0.07910192950176387 147 * 0.7233322451735928 148 * 0.1659819820667019 149 * 150 * y = fft(x) 151 * ------------------- 152 * 0.9336118983487516 153 * -0.7581365035668999 + 0.08688005256493803i 154 * 0.44344407521182005 155 * -0.7581365035668999 - 0.08688005256493803i 156 * 157 * z = ifft(y) 158 * ------------------- 159 * -0.03480425839330703 160 * 0.07910192950176387 + 2.6599344570851287E-18i 161 * 0.7233322451735928 162 * 0.1659819820667019 - 2.6599344570851287E-18i 163 * 164 * c = cconvolve(x, x) 165 * ------------------- 166 * 0.5506798633981853 167 * 0.23461407150576394 - 4.033186818023279E-18i 168 * -0.016542951108772352 169 * 0.10288019294318276 + 4.033186818023279E-18i 170 * 171 * d = convolve(x, x) 172 * ------------------- 173 * 0.001211336402308083 - 3.122502256758253E-17i 174 * -0.005506167987577068 - 5.058885073636224E-17i 175 * -0.044092969479563274 + 2.1934338938072244E-18i 176 * 0.10288019294318276 - 3.6147323062478115E-17i 177 * 0.5494685269958772 + 3.122502256758253E-17i 178 * 0.240120239493341 + 4.655566391833896E-17i 179 * 0.02755001837079092 - 2.1934338938072244E-18i 180 * 4.01805098805014E-17i 181 * 182 *********************************************************************/ 183 184 public static void main(String[] args) { 185 int N = Integer.parseInt(args[0]); 186 Complex[] x = new Complex[N]; 187 188 // original data 189 for (int i = 0; i < N; i++) { 190 x[i] = new Complex(i, 0); 191 x[i] = new Complex(-2*Math.random() + 1, 0); 192 } 193 show(x, "x"); 194 195 // FFT of original data 196 Complex[] y = fft(x); 197 show(y, "y = fft(x)"); 198 199 // take inverse FFT 200 Complex[] z = ifft(y); 201 show(z, "z = ifft(y)"); 202 203 // circular convolution of x with itself 204 Complex[] c = cconvolve(x, x); 205 show(c, "c = cconvolve(x, x)"); 206 207 // linear convolution of x with itself 208 Complex[] d = convolve(x, x); 209 show(d, "d = convolve(x, x)"); 210 } 211 212}