EE531 (Semester II, 2010) 6. Spectral analysis. power spectral density. periodogram analysis. window functions 6-1

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1 6. Spectral analysis EE531 (Semester II, 2010) power spectral density periodogram analysis window functions 6-1

2 Wiener-Khinchin theorem: Power Spectral density if a process is wide-sense stationary, the autocorrelation function and the power spectral density form a Fourier transform pair: Continuous S(ω) = e iωτ R(τ)dτ R(t) = 1 2π S(ω)e iωt dω Discrete S(ω) = k= k= R(k)e iωk R(k) = 1 2π π π S(ω)e iωk dω (under a condition for the existence of the Fourier transform, e.g., R(t) is absolutely integrable or R(k) is absolutely summable) Spectral analysis 6-2

3 Properties of PSD S(ω) is self-adjoint, i.e., S(ω) = S (ω), ω S(ω) 0 for all ω S(ω)dω = R(0) = Ex(t)x(t) 0 (average power) for real processes, S( ω) = S(ω) T for discrete-time processes, S(ω) is a periodic function of period 2π Spectral analysis 6-3

4 Cross-power spectral density the cross-power spectrum of x(t) and y(t) is the Fourier transform of the cross correlation R xy (τ): Continuous S xy (ω) = Discrete e iωτ R xy (τ)dτ R xy (t) = 1 2π S xy (ω)e iωt dω S xy (ω) = k= k= R xy (k)e iωk R xy (k) = 1 2π π π S xy (ω)e iωk dω It follows from R xy ( τ) = R yx(τ) that S xy (ω) = S yx(ω) Spectral analysis 6-4

5 LTI systems with random inputs u y H Fact: if u(t) is wide-sense stationary, y(t) is also wide-sense stationary the mean is constant for all t Ey(t) = h(s)eu(t s) = µ u h(s) s= s= R y (t 1,t 2 ) depends only on the time shift t 1 t 2 R y (t 1,t 2 ) = = h(s)e[u(t 1 s)u(t 2 v) ]h (v) s= v= s= v= h(s)r u (t 1 t 2 +v s)h (v) Spectral analysis 6-5

6 Fact: y(t), u(t) are jointly wide-sense stationary the input-output cross correlation is R yu (t 1,t 2 ) = E h(k)u(t 1 k)u(t 2 ) it also follows that = k= k= R yu (τ) = k= h(k)r u (t 1 t 2 k) h(k)r u (τ k) R y (τ) = k= h(k)r uy (τ k) conclusion: the correlations are in the form of convolution sum Spectral analysis 6-6

7 Spectral relations for LTI systems Using the convolution property of the Fourier transform of R yu (τ),r y (τ), we have the relations: S yu (ω) = H(ω)S u (ω), S y (ω) = H(ω)S uy (ω) With S uy (ω) = S yu(ω), we have S y (ω) = H(ω)S u (ω)h(ω) In terms of z-transform, this could be written as S y (z) = H(z)S u (z)h(z) where H(z) = H( z) T and we should be aware that z = e iω in the analysis Spectral analysis 6-7

8 Example 1 suppose the covariance function of a staionary process is given by R(k) = λ2 a k 1 a 2, a < 1, λ R the spectral density can be obtained via z-transform S(z) = λ2 1 a 2 a k z k = λ2 1 1 a 2 a k z k + = λ2 1 a 2 k= ( az 1 az + z ) z a = k= λ 2 (1 az)(1 az 1 ) a k z k k=0 substituting z = e iω gives S(ω) = λ 2 (1 ae iω )(1 ae iω ) = λ 2 1+a 2 2acosω Spectral analysis 6-8

9 Example 2 a linear system given in a state-space form y(t) = ay(t 1)+e(t) where e(t) is a white noise with variance λ 2 the transfer function is given by H(z) = 1 1 az 1 therefore the spectral density of y is S y (ω) = λ 2 (1 ae iω )(1 ae iω ) = λ 2 1+a 2 2acosω Spectral analysis 6-9

10 Spectral analysis use the same model as in correlation analysis: R yu (τ) = h(k)r u (τ k) k=0 taking DFT gives the spectral representation S yu (ω) = H(ω)S u (ω) if S u (ω) 0 for all ω, then we can estimate Ĥ(ω) = Ŝyu(ω)Ŝu(ω) 1, where Ŝyu,Ŝu can be computed via DFT Spectral analysis 6-10

11 Periodogram analysis an infinite-length discrete-time signal y(t) is windowed by a length-n window w(t), 1 t N ỹ(t) = w(t)y(t) define a function Y N (ω) given by Y N (ω) = 1 N N t=1 w(t)y(t)e iωt the periodogram, an estimate of S y (ω), is defined by Ŝ y (ω) = 1 C Y N(ω) 2, where C = 1 N N t=1 w(t) 2 is a normalization factor Spectral analysis 6-11

12 Periodogram analysis Ŝ y (ω) is called periodogram when w(t) is rectangular, and modified periodogram for other types of windows, e.g., Hamming, Barlett, etc. in practice, the periodogram is evaluated at a finite number of frequencies ω k = 2πk/R, 0 k R 1 by replacing Ŝy(ω) with the length-r DFT Y[k] of the length-n sequences y[k]: Ŝ y (ω k ) = Ŝy[k] = 1 C Y[k] 2 usually R > N to provide a finer resolution of the periodogram C = (1/N) N t=1 w(t) 2 is a normalization factor Spectral analysis 6-12

13 Window functions suppose we use a rectangular window of length N Ŝ y (ω) = 1 N = 1 N = N N n=1m=1 N 1 k= N +1n=1 k N 1 k= N +1 y(m)y (n)e iω(m n) N k ˆR y (k)e iωk y(n+k)y (n)e iωk the periodogram is the Fourier transform of ˆR y (k) a few samples of y(n) is used in estimating ˆR y (k) when k is large, yielding a poor estimate of R y (k) Spectral analysis 6-13

14 Window functions use the window functions that vanish for τ > M to weight out the estimated correlation for large τ Rectangular Barlett Hamming w(τ) = 1, τ M w(τ) = 1 τ /M, τ M w(τ) = cos ( ) 2πτ, τ M 2M +1 M should be small compared to N to reduce the fluctuations of the periodogram Spectral analysis 6-14

15 Example 10 2 Window length = Power/frequency (db/hz) Frequency (Hz) y(t) = cos(400πt)+ν(t), with N = 301 Spectral analysis 6-15

16 10 2 Window length = Window length = Power/frequency (db/hz) Power/frequency (db/hz) Frequency (Hz) Frequency (Hz) Spectral analysis 6-16

17 References Chapter 2,6 in L. Ljung, System Identification: Theory for the User, Prentice Hall, Second edition, 1999 Chapter 3 in T. Söderström and P. Stoica, System Identification, Prentice Hall, 1989 Chapter 11 in R.D. Yates, D.J. Goodman, Probability and Stochastic Processes: a friendly introduction for Electrical and Computer engineers, John Wiley & Sons, Inc., Second edition, 2005 Chapter 9 in A. Papoulis and S. U. Pillai, Probability Random Variables and Stochastic Processes, McGraw-Hill, Fourth edition, 2002 Chapter 3,15 in S. K. Mitra, Digital Signal Processing, McGraw-Hill, International edition, 2006 Spectral analysis 6-17