For a simple fft-experiment the speed of the routine is not so important. But for e.g. parallel-code-search a fast FFT-implementation is necessary. I compared three implementations.

FFT1 is a previous described implementation. I just simply coded the formulas for a regular Fourier transformation. It worked like a charm!

the code:

 ```package FFT1; public class Fourier { public Fourier() { } public Complex[] dft(double[] x) { int n = x.length; Complex[] xX = new Complex[n]; // FFT-bins for (int k = 0; k < n; k++) { Complex accum = new Complex(0.0, 0.0); for (int nn = 0; nn < n; nn++) { Complex wnkN = Complex.expjphi(-2 * Math.PI * nn * k / n); Complex prod = wnkN.multiply(x[nn]); accum = accum.add(prod); } xX[k] = accum; } return xX; } public Complex[] idft(Complex[] X) { int n = X.length; Complex[] x = new Complex[n]; for (int k = 0; k < n; k++) { Complex accum = new Complex(0.0, 0.0); for (int nn = 0; nn < n; nn++) { Complex wnkN = Complex.expjphi(2 * Math.PI * nn * k / n); Complex prod = wnkN.multiply(X[nn]); accum = accum.add(prod); } x[k] = accum.multiply((double) (1.0 / n)); } return x; } } ```

And the runtime results:

```running straight-forward FFT
numSam = 1024
status = OK
status = OK
outputFileName = data\output
start 	 7-feb-2011 16:44:35
stop 	 7-feb-2011 16:44:35
runtime (msec): 228.0
```
 ```package FFT1; import java.text.DateFormat; import java.util.Calendar; import java.util.Date; import util.FileIO; import util.ShowPlot2d; public class Engine { DateFormat f = DateFormat.getDateTimeInstance(DateFormat.MEDIUM, DateFormat.MEDIUM); DateFormat ff = DateFormat.getDateTimeInstance(DateFormat.MEDIUM, DateFormat.MEDIUM); double time1; int numSam = (int) Math.pow(2, 10); double[] inBuffer; public Engine() { } public void startTimer(){ Calendar cal = Calendar.getInstance(); time1 = cal.getTimeInMillis(); Date now = cal.getTime(); System.out.println("start " + "\t" + " " + f.format(now)); } public void stopTimer(){ Calendar cal = Calendar.getInstance(); double time2 = cal.getTimeInMillis(); System.out.println("stop " + "\t" + " " + ff.format(time2)); double dif = time2 - time1; System.out.println("runtime (msec): " + dif + "\n"); } public double[] readSamples(int number) { FileIO inputfile = new FileIO(); inputfile.FileSetup("output"); int kar; double inputSamples[] = new double[number]; for (int k = 0; k < number; k++) { kar = inputfile.readByte(); if (kar > 127) { kar = kar - 256; } inputSamples[k] = kar; } return inputSamples; } public void testFFT1(String title) { System.out.println(title); System.out.println("numSam = " + numSam); inBuffer = readSamples(numSam); Fourier myFourier = new Fourier(); startTimer(); Complex[] outBuffer = myFourier.dft(inBuffer); stopTimer(); double[] plotBuffer = new double[numSam]; for (int i = 0; i < outBuffer.length; i++) { plotBuffer[i] = outBuffer[i].magnitude(); } ShowPlot2d myPlot = new ShowPlot2d(1); myPlot.add(plotBuffer, 1, "myPlot"); myPlot.showFigure(); } } ```

FFT2 is a more efficient implementation that I found on the internet:

 ```package FFT2; /************************************************************************* * Compilation: javac FFT.java * Execution: java FFT N * Dependencies: Complex.java * * Compute the FFT and inverse FFT of a length N complex sequence. * Bare bones implementation that runs in O(N log N) time. Our goal * is to optimize the clarity of the code, rather than performance. * * Limitations * ----------- * - assumes N is a power of 2 * * - not the most memory efficient algorithm (because it uses * an object type for representing complex numbers and because * it re-allocates memory for the subarray, instead of doing * in-place or reusing a single temporary array) * *************************************************************************/ public class FFT { // compute the FFT of x[], assuming its length is a power of 2 public static Komplex[] fft(Komplex[] x) { int N = x.length; // base case if (N == 1) return new Komplex[] { x[0] }; // radix 2 Cooley-Tukey FFT if (N % 2 != 0) { throw new RuntimeException("N is not a power of 2"); } // fft of even terms Komplex[] even = new Komplex[N/2]; for (int k = 0; k < N/2; k++) { even[k] = x[2*k]; } Komplex[] q = fft(even); // fft of odd terms Komplex[] odd = even; // reuse the array for (int k = 0; k < N/2; k++) { odd[k] = x[2*k + 1]; } Komplex[] r = fft(odd); // combine Komplex[] y = new Komplex[N]; for (int k = 0; k < N/2; k++) { double kth = -2 * k * Math.PI / N; Komplex wk = new Komplex(Math.cos(kth), Math.sin(kth)); y[k] = q[k].plus(wk.times(r[k])); y[k + N/2] = q[k].minus(wk.times(r[k])); } return y; } // compute the inverse FFT of x[], assuming its length is a power of 2 public static Komplex[] ifft(Komplex[] x) { int N = x.length; Komplex[] y = new Komplex[N]; // take conjugate for (int i = 0; i < N; i++) { y[i] = x[i].conjugate(); } // compute forward FFT y = fft(y); // take conjugate again for (int i = 0; i < N; i++) { y[i] = y[i].conjugate(); } // divide by N for (int i = 0; i < N; i++) { y[i] = y[i].times(1.0 / N); } return y; } // compute the circular convolution of x and y public static Komplex[] cconvolve(Komplex[] x, Komplex[] y) { // should probably pad x and y with 0s so that they have same length // and are powers of 2 if (x.length != y.length) { throw new RuntimeException("Dimensions don't agree"); } int N = x.length; // compute FFT of each sequence Komplex[] a = fft(x); Komplex[] b = fft(y); // point-wise multiply Komplex[] c = new Komplex[N]; for (int i = 0; i < N; i++) { c[i] = a[i].times(b[i]); } // compute inverse FFT return ifft(c); } // compute the linear convolution of x and y public static Komplex[] convolve(Komplex[] x, Komplex[] y) { Komplex ZERO = new Komplex(0, 0); Komplex[] a = new Komplex[2*x.length]; for (int i = 0; i < x.length; i++) a[i] = x[i]; for (int i = x.length; i < 2*x.length; i++) a[i] = ZERO; Komplex[] b = new Komplex[2*y.length]; for (int i = 0; i < y.length; i++) b[i] = y[i]; for (int i = y.length; i < 2*y.length; i++) b[i] = ZERO; return cconvolve(a, b); } // display an array of Complex numbers to standard output public static void show(Komplex[] x, String title) { System.out.println(title); System.out.println("-------------------"); for (int i = 0; i < x.length; i++) { System.out.println(x[i]); } System.out.println(); } } ```

I had to change the name of the class Complex into Komplex to avoid problems in other java-code.

Running the program yields:

```running faster FFT
numSam = 1024
status = OK
status = OK
outputFileName = data\output
start 	 7-feb-2011 16:55:29
stop 	 7-feb-2011 16:55:29
runtime (msec): 9.0
```

This is fast! But with inplace-FFT it can even be faster:

 ```package DSP; import src_old.Complex; /************************************************************************* * Compilation: javac InplaceFFT.java * Execution: java InplaceFFT N * Dependencies: Complex.java * * Compute the FFT of a length N complex sequence in-place. * Uses a non-recursive version of the Cooley-Tukey FFT. * Runs in O(N log N) time. * * Reference: Algorithm 1.6.1 in Computational Frameworks for the * Fast Fourier Transform by Charles Van Loan. * * * Limitations * ----------- * - assumes N is a power of 2 * * *************************************************************************/ public class InplaceFFT { // compute the FFT of x[], assuming its length is a power of 2 public static void fft(Complex[] x) { // check that length is a power of 2 int N = x.length; if (Integer.highestOneBit(N) != N) { throw new RuntimeException("N is not a power of 2"); } // bit reversal permutation int shift = 1 + Integer.numberOfLeadingZeros(N); for (int k = 0; k < N; k++) { int j = Integer.reverse(k) >>> shift; if (j > k) { Complex temp = x[j]; x[j] = x[k]; x[k] = temp; } } // butterfly updates for (int L = 2; L <= N; L = L+L) { for (int k = 0; k < L/2; k++) { double kth = -2 * k * Math.PI / L; Complex w = new Complex(Math.cos(kth), Math.sin(kth)); for (int j = 0; j < N/L; j++) { Complex tao = w.times(x[j*L + k + L/2]); x[j*L + k + L/2] = x[j*L + k].minus(tao); x[j*L + k] = x[j*L + k].plus(tao); } } } } public static void ifft(Complex[] x) { // inverse FFT using above FFT for(int i=0;i ```

Running this implementation yields:

```running inplace FFT
numSam = 1024