EE6140: Spectral Estimation

Assignment 3

(August 24, 2010)

1.∗ Consider a zero-mean complex AR(1) process with parameters a1 and σ2

u > 0. Find thevariance of the process when (i) |a1| < 1 and (ii) |a1| ≥ 1. Where does the pole of theAR filter have to lie for the process to be WSS?

2.∗ Our goal in this exercise is to come up with an example R′xx ma-

trix that is singular. Recall that the first column of the Toeplitz ma-trix R′

xx is (rxx[q], rxx[q + 1], . . . , rxx[q + p− 1])T , while its first row is(rxx[q], rxx[q − 1], . . . , rxx[q − p+ 1]). To come up with an example of R′

xx that issingular, let p = q = 1, i.e., consider an ARMA(1,1) process. Comment on the result.

3. Show that the ACF for an AR(2) process is given by

rxx[k] = σ2

u

1 + r2

1 − r2

√

1 +

(

1 − r2

1 + r2

)2

cot2 2πf0

1 − 2r2 cos(4πf0) + r4r|k| cos(2πf0 − ψ)

where

ψ = arctan

[

1 − r2

1 + r2cot 2πf0

]

and a1 = −2r cos 2πf0 and a2 = r2.

Hint: Note that

rxx[k] =1

2πj

∮

C

σ2

u zk−1

A(z)A∗(1/z∗)dz

where C is the unit circle; now use Cauchy’s residue theorem and simplify.

4. Computer Simulation In this exercise we will look at how well the sample autocorre-lation performs in estimating the autocorrelation of white noise.

(a) Generate 1000 samples of zero-mean unit variance white Gaussian noise.

(b) Estimate the first 100 lags of the autocorrelation sequence using the sample auto-correlation

rxx[k] =1

1000

999∑

n=0

x[n] x[n − k]

How close is your estimate to the true autocorrelation sequence rxx[k] = δ[k] ?

EE6140 Assignment 3 CSR

(c) Segment your white noise sequence into ten different segments each having 100samples, and estimate the autocorrelation by averaging the sample autocorrelationsof each subsequence, i.e.,

rxx[k] =1

1000

9∑

m=0

99∑

n=0

x[n+ 100m] x[n− k + 100m] k = 0, 1, . . . , 99

How does your estimate compare to that in part (b) ? How does it compare withthe true autocorrelation sequence rxx[k] = δ[k] ?

(d) Generate 10000 samples of zero-mean unit variance white noise and estimate thefirst 100 lags as in part (b). How does this estimate compare with that in part (b) ?What conclusions can you draw from these experiments?

(e) Repeat for uniform noise.

5. Computer Simulation The AR process x[n] is given by

x[n] = a1 x[n− 1] + a2 x[n− 2] + b0 u[n]

where u[n] is unit variance white noise. First, choose u[n] to be Gaussian; repeat foruniform noise.

(a) With a1 = 0, a2 = −0.81, and b0 = 1 generate 24 samples of the process x[n]. Makesure that the initial transients have died down.

(b) Estimate the autocorrelation sequence using (a) the biased autocorrelation estima-tor, and (b) the unbiased autocorrelation estimator. Compare these with the trueACF.

(c) Using your estimated ACF, compute the PSD by computing its Fourier transform.

(d) Using the estimated ACF, solve the Yule-Walker equations and estimate the valueof b0, a1, and a2. Comment on the accuracy of the estimates.

(e) Using the estimates obtained earlier, compute the following:

Pxx(f) =b20

|1 + a1e−j2πf + a2ej4πf |2

(f) Compare the estimates using the two different approaches with the true powerspectrum.

Submit starred problems on 31.08.2010

EE6140 Page 2 of 2 24.08.2010