SPECTRUM. Deterministic Signals with Finite Energy (l 2 ) Deterministic Signals with Infinite Energy N 1. n=0. N N X N(f) 2

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1 SPECTRUM Deterministic Signals with Finite Energy (l 2 ) Energy Spectrum: S xx (f) = X(f) 2 = 2 x(n)e j2πfn n= Deterministic Signals with Infinite Energy DTFT of truncated signal: X N (f) = N x(n)e j2πfn Power spectral density: P xx (f) = lim N N X N(f) 2 Digital Signal Processing 4 Lecture 5 SPECTRUM Stochastic Signals Power spectral density: { } P xx (f) = lim E N N X N(f) 2 = = k= r xx (k)e j2πfk wherer xx (k) = E{x(n)x (n k)} is the covariance sequence of the stationary stochastic process. Digital Signal Processing 5 Lecture 5

2 SPECTRUM OF FILTERED SIGNAL x(n) LTI:h(n) y(n) LTI system: y(n) = h(n) x(n) Frequency response: Y(f) = H(f)X(f) Energy spectral density: (l 2 -signals)s yy (f) = Y(f) 2 = S xx (f) H(f) 2 Power spectral density: (deterministic signals) P yy (f) = lim N N Y N(f) 2 = P xx (f) H(f) 2 Power spectral density: (stochastic signals) P yy (f) = E lim N { N Y N(f) 2 } = P xx (f) H(f) 2 Digital Signal Processing 6 Lecture 5 SAMPLING x c (t) X c (F) t = nt s x d (n) X d (f) Sampling frequency: F s = T s Poisson s summation formula (the sampling theorem): X d (f) = F s k= X c ((f k)f s ) = T s k= ( ) f k X c T s Digital Signal Processing 7 Lecture 5

3 SAMPLING STOCHASTIC SIGNALS Covariance sequence: r xd x d (k) = r xc x c (kt s ) Power Spectral Density: P xd x d (f) = T s k= ( ) f k P xc x c T s Digital Signal Processing 8 Lecture 5 NON-PARAMETRIC SPECTRAL ESTIMATION Infinite (true) stochastic process: x(n),n =..., 2,,0,,2,... Available data: N samples of a single realization: x N (n), n = 0,,2,...,N Problem formulation: Estimate the power spectral density ofx(n), givenx N (n) Digital Signal Processing 4 Lecture 5

4 NON-PARAMETRIC SPECTRAL ESTIMATION Stochastic signals Power spectral density: { P xx (f) = lim E XN (f) } 2 = = N N Estimate: ˆPxx (f) i) directly: from F{x N (n)} ii) indirectly: fromf{ˆr xx (k)} 2 k= r xx (k)e jfk Digital Signal Processing 42 Lecture 5 PERIODOGRAM Directly: ˆP xx (f) = N F{x N(n)} 2 = N N 2 x(n)e j2πfn Indirectly: ˆP xx (f) = F{ˆr xx (k)} = N k= N+ ˆr xx (k)e j2πfk if N k x(n+k)x (n) k = 0,,...,N ˆr xx (k) = N ˆr xx( k) k =, 2,..., N + Digital Signal Processing 43 Lecture 5

5 PERIODOGRAM, PROPERTIES Bias: E{ˆP xx (f)} = P xx (f) W R (f) 2 = P xx (f)+o( N ) biased but asymptotically unbiased Variance: E{(ˆP xx (f) P xx (f)) 2 } = P 2 xx(f)+o( N ) does not go to zero! = not a consistent estimate! Cross variance (different frequencies,f f 2 ) E{(ˆP xx (f ) P xx (f ))(ˆP xx (f 2 ) P xx (f 2 ))} = O( N ) estimates at different frequencies weakly correlated for largen = not a smooth estimator! Digital Signal Processing 44 Lecture 5 MODIFIED PERIODOGRAM Window the data: ˆP M xx(f) = NU F{w(n)x N(n)} 2 = NU N 2 w(n)x(n)e j2πfn Normalization: U = N N w(n) 2 Properties: Changes theo( N ) term of the bias and variance. Choice of windoww(n): Trade-off between resolution and leakage, see Table 8.2 in Hayes. Digital Signal Processing 45 Lecture 5

6 WINDOWING EFFECTS Side-lobes cause leakage, i.e. energy appears outside the main lobe. The width of main lobe determines the resolution capabilities. Time domain long window short window sharp edges Frequency domain narrow main-lobe wide main-lobe large side-lobes Different windows: trade-off between resolution and leakage. Digital Signal Processing 46 Lecture 5 RESOLUTION LIMITS W(f) Resolution if f 2 f < W! 0.5 W 3dB bandwidth 0 W 2 0 W 2 W(f-f ) W(f-f 2 ) Sum W(f-f ) W(f-f 2 ) Sum f f 2 0 f f 2 Digital Signal Processing 47 Lecture 5

7 BARTLETT METHOD Idea: Segment and average the data to decrease the variance. Number of segments: K Length of each segment: L Total data length: N = LK Segmentk: x k (n) = x(kl+n),n = 0,...,L,k = 0,...,K ˆP B xx(f) = K K k=0 L F {x k(n)} 2 }{{} Periodogram of segmentk Digital Signal Processing 48 Lecture 5 BARTLETT METHOD, PROPERTIES Compared to the periodogram: Variance: decrease by factork. Bias: L, increase by factork. Resolution: decrease by factork. Digital Signal Processing 49 Lecture 5

8 WELCH METHOD Idea: Allow overlapping segments and window the data. Number of segments: K Length of each segment: L,LK > N. Step between segment starts: D Segmentk: x k (n) = x(kd +n),n = 0,...,L,k = 0,...,K Temporal window: w(n),n = 0,...,L ˆP W xx(f) = K K k=0 LU F {w(n)x k(n)} 2 }{{} Modified Periodogram of segmentk Normalization: U = L w(n) 2 L Digital Signal Processing 50 Lecture 5 WELCH METHOD, PROPERTIES Window choice: Ordinary trade-off between resolution and leakage. Gives increased smoothness (averaging in the frequency domain). Variance: Slightly lower than for Bartlett when using 50% overlap between segments (L = 2D). Digital Signal Processing 5 Lecture 5

9 BLACKMAN-TUKEY METHOD Covariance sequence estimate: ˆr xx (k) Idea: ˆr xx (k) is less reliable for largek. Window the correlation sequence to put more emphasis on the most reliable values. M ˆP BT xx (f) = F{w(k)ˆr xx (k)} = Correlation window: k= M+ w(k)ˆr xx (k)e j2πfk w(k),n = M +,...,,0,,...,M w(k) = w( k) w(0) = Effective window length: M N. /2 /2 W(f)df = Digital Signal Processing 52 Lecture 5 BLACKMAN-TUKEY METHOD, PROPERTIES Bias: E{ˆP BT xx (f)} = E{ˆP xx (f)} W(f) = P xx (f) W triangle (f) W(f) Variance: Decreases by approximatelym/n, compared to the Periodogram. TypicallyM N. Smoothness: Windowing creates smooth estimate. Digital Signal Processing 53 Lecture 5

10 USING THE FFT In practice: Use the FFT to calculate ˆP xx (f) in all the methods. Frequency axis: ˆPxx (k) = ˆP xx (f) f= k M, wherem is the length of the DFT. Zero padding: Use zero padding,m > N, to get more points on the curve No zero padding 7N zeros Without zero padding With zero padding, M = 8N Digital Signal Processing 54 Lecture 5