1. 1 Spectrum Estimation Dr. Hassanpour Payam Masoumi Mariam Zabihi Advanced Digital Signal Processing Seminar Department of Electronic Engineering Noushirvani University

2. 2 Course Outlines Introduction Fourier Series and Transform Time/Frequency Resolutions Autocorrelation & spectrum estimation Non-parametric Methods Periodogram Modified Periodogram Bartlett’s Method Welch’s Method Blackman-Tukey Method Parametric Methods

3. 3 Fourier Series and Transform Fourier basis functions: real and imaginar parts of a complex sinusoid vector representation of a complex exponential. Re Im t

4. 4 Fourier Series : k=…,-1,0,1,… k t,

5. 5 tk, Fourier Transform:

6. 6 Discrete Fourier Transform (DFT) DFT

7. 7 Autocorrelation & Spectrum estimation Autocorrelation: Power spectrum : Spectrum estimation is a problem that involves estimating from finite number of noisy measurements of x(n).

8. 8 Nonparametric methods Peroidogram Modified periodogram Bartlett method Welch method Blackman-Tukey method

9. 9 The periodogram Estimated autocorrelation: Estimated power spectrum or periodogram:

10. 10 The periodogram cont.

11. 11 The periodogram of white noise : white noise with a variance, length N=32

12. 12 The estimated autocorrelation sequence White noise power spectrum The periodogram of white noise cont.

13. 13 Periodogram of sinusoid in noise

14. 14 Periodogram of sinusoid in noise cont.

15. 15 Periodogram Bias Thus, the bias is deference between estimated and actual Power spectrum.

16. 16 Periodogram of sinusoid in noise cont.

17. 17 Example:

18. 18 Periodogram Resolution Set equal to the width of main lobe of the spectral window at it’s half power or 6dB point.

19. 19 Example:

20. 20 Properties of the periodogram Bias: Resolution: Variance:

21. 21 Modified Periodogram Would there be any benefit in replacing the rectangular window with other windows? (for example triangular window)

22. 22 Example: N=128 Rectangular Window N=128 Hamming Window

23. 23 Properties of the M-periodogram Bias: Resolution: window dependent Variance:

24. 24 Bartlett’s method (periodogram averaging)....

25. 25 Properties of Bartlett’s method Bias: Resolution: Variance:

26. 26 Example:

27. 27 Example:

28. 28 Welch’s method (M-periodogram averaging).... Overlap = L-D

29. 29 Properties of Welch’s method Bias Resolution Window dependent Variance

30. 30 Example:

31. 31 Resolution:

32. 32 windowing:

33. 33 windowing:

34. 34 Blackman-Tukey’s method (Periodogram smoothing) Note: Bartlett & Welch are design to reduce the variance if the priodogram by averaging and modified it. Periodogram is computed by taking the Fourier transform of a consistent estimate of the auto correlation sequence. For any finite data record of length N, the variance of will be large for values of k that are close to N. for example: In Bartlett & Welch, the variance is decreased by reducing the variance of autocorrelation estimate by averaging.

35. 35 Blackman-Tukey’s method cont. In the Blackman-Tukey method, the variance is decreased by applying a window to in order to decrease the contribution of the unreliable estimates to the periodogram. Specifically, the Blackman-Tukey spectrum estimation is: For example, if w(k) is a rectangular window extending from –M to M with M<N-1, then having the largest variance are set to zero and consequently, the power spectrum estimation will have a smaller variance.

36. 36 Properties of B-T’s method Bias Resolution Window dependent Variance

37. 37 windowing:

38. 38 Performance comparisons We can summarized the performance of each technique in terms of two criteria. (I) Variability (which is a normalized variance) (II) Figure of merit That is approximately the same for all of the nonparametric methods

39. 39 Summery variabilityResolutionFigure of merit Periodogram Bartlett Welch Blackman Tukey ***50% overlap and the Bartlett window***