1. Spectral Leakage Pp289 jdsp

2. Freq of kth sample, No centering

3. Centering and the PSD PSDs are centered if the DC component is at N/2, where N= number of samples. Before and after centering

4. How do I center the FFT? UlawCodec ulc = new UlawCodec(); final double[] doubleArray = ulc.getDoubleArray(); double d[] = Mat1.truncateToIntegralPowerOfTwo(doubleArray); FFT1dDouble f = new FFT1dDouble(); double in_im[] = new double[d.length]; Mat1.centering(d); f.computeForwardFFT(d, in_im); f.normalize(); final double[] psd = f.getPSDNotNormalized(); Graph.graph(psd, "", "", "psd"); f.computeBackwardFFT(d, in_im); Mat1.centering(d); UlawCodec ulc2 = new UlawCodec(f.getRealData()); ulc2.play();

5. How do I implement centering? public static void centering(double r[]) { int s = 1; for (int i = 0; i < r.length; i++) { s = -s; r[i] *= s; }

6. How do I compute freqs?

7. Computing Freqs The center freq should be zero. But what about 4001 and 3999?

8. Thus every bin gives one freq Bins give a unique freq. K=n/2=0 hz K=n = 4000 hz K=0= -4000 hz

9. Real signals Real signals have zero complex components Real signals have symmetrical PSDs Real signals are positive on the right Real signals are negative on the left One bin maps to one frequency

10. Spectral leakage frequency analysis effectfrequency analysis Applies to: – finite-length signals –finite-length segments energy has "leaked" out of the original signal spectrum into other buckets.

11. Windowing window function (also known as an apodization function or tapering function) a function that is zero-valued outside of some chosen intervalfunctioninterval

12. Windowing windowing is the root cause of spectral leakage

13. Does Windowing Really Help? Non-rectangular window functions actually increase the total leakagewindow functions they can also redistribute it to places where it does the least harm, depending on the application windowing predistorts the input samples so that the spectral leakage is evened out

14. For example

15. Frequency Bin Often the kth array element in the frequency domain is referred to as a frequency bin

16. The 400 Hz spectrum we expect the maximum amplitude to occur at. k=102, and 103<- energy spreads around two bins

17. Spectral Leakage When a single freq leaks into more than one bucket in the fft. This occurs because of rectangular windowing function use in a STFT It smearing the PSD across multiple buckets.

18. To limit spectral leakage Use a non-rectangular windowing function A function that is limited in time and zero everywhere else. Finite support. An a-periodic function

19. Select a windowing function Use either speed Or accuracy Or ease of implementation Do a straight multiply of the window and your input signal. You may need to do this several times, if you don’t want to loose data…

20. But I need speed! If you don’t overlap, you will loose data, but you will gain speed. You need to know the nature of the input signal!

21. There are many windows Hanning Blackman Bartlet Hamming Etc.

22. Windows

23. Code for Hanning public static double[] makeHanning(int n) { double window[] = new double[n]; double arg = 2.0 * Math.PI / (n - 1); for (int i = 0; i < n; i++) window[i] = 0.5 - 0.5 * Math.cos(arg * i); return window; }

24. Lyon Window

25. Lyon Window Code public static double[] makeLyon(int n) { double window[] = new double[n]; double u = 0.0; double du = 2.0 / n; for (int i = 0; i < n / 2; i++) { window[i] = y5(0, 1.0, u); u += du; } u = 0; for (int i = n / 2; i < n; i++) { window[i] = y5(1.0, 0.0, u); u += du; } return window; }

26. y5 public static double y5(double y0, double y1, double u) { double t2 = u * u; double t3 = t2 * t2; return (6 * y1 - 6 * y0) * t3 * u + (-15 * y1 + 15 * y0) * t3 + (10 * y1 - 10 * y0) * t2 * u + y0; }

27. Lyon vs Hanning Window