1 Non-Parametric Power Spectrum Estimation Methods Eric Hui SYDE 770 Course Project November 28, 2002

2 Introduction Applications of Power Spectrum Estimation (PSE): Applications of Power Spectrum Estimation (PSE): Wiener Filter Wiener Filter Feature Extraction Feature Extraction Non-parametric PSE does NOT assume any data-generating process or model (e.g. autoregressive model). Non-parametric PSE does NOT assume any data-generating process or model (e.g. autoregressive model).

3 Motivation Ideal autocorrelation: Ideal autocorrelation: Actual autocorrelation: Actual autocorrelation: Limited (finite length of) data due to: Limited (finite length of) data due to: Availability of data Availability of data Assumption of stationary Assumption of stationary

4 Periodogram Method n x(n)N0 DTFT redefined as n x N (n) N0

5 Periodogram Method DTFT N0-N k DTFT 0 k

6 “Good” Method? Necessary conditions for mean-square convergence: Necessary conditions for mean-square convergence: Asymptotically Unbiased Asymptotically Unbiased Zero Variance Zero Variance k PSD k as N ↑ k k PSD

7 Evaluation of Methods Resolution Resolution How much “blurring” effect is there on the power spectrum? How much “blurring” effect is there on the power spectrum? Bias (Asymptotic) Bias (Asymptotic) Does the estimation approach the true value with more data (i.e. as N increases)? Does the estimation approach the true value with more data (i.e. as N increases)? Variance Variance Does the amount of deviation from the true value depend on the data length (i.e. N)? Does the amount of deviation from the true value depend on the data length (i.e. N)? k PSD k as N ↑ k k PSD k

8 Different PSE Methods Periodogram Method Periodogram Method Apply rectangular window to x(n) to get x N (n). Apply rectangular window to x(n) to get x N (n). Modified Periodogram Method Modified Periodogram Method Apply non-rectangular window to x(n) to get x N (n). Apply non-rectangular window to x(n) to get x N (n). Bartlett’s Method Bartlett’s Method Average the Periodogram estimate of non-overlapping sub-intervals of x(n). Average the Periodogram estimate of non-overlapping sub-intervals of x(n). Welch’s Method Welch’s Method Average the Modified Periodogram estimate of overlapping sub-intervals of x(n). Average the Modified Periodogram estimate of overlapping sub-intervals of x(n). Blackman-Turkey Method Blackman-Turkey Method Apply non-triangular window to r(x). Apply non-triangular window to r(x). k N0-N DTFT k 0 k as N ↑ k PSD

9 Application: Feature Extraction PSD linearize repeat for whole image

10 Questions or Comments? …