Description
Problems
1. Random process v1[n] and v2[n] are independent and have the same correlation function
rv[n1, n0] = 0.5δ[n1 − n0].
(a) What is the correlation function of the random process
x[n] = v1[n] + 2v1[n + 1] + 3v2[n − 1]?
Is this random process WSS ?
(b) Find the correlation matrix for a random vector consisting of eight consecutive samples of x[n].
2. [∗]A real random process has the correlation function
Rx[l] = 2−|l| + δ[l].
(a) What is the power spectral density (PSD) of this random process ?
(b) Assume that the correlation function was estimated perfectly. A periodogram spectral estimate
is formed by using only the first three lag values, that is,
rˆx[l]
(
rx[l], l = 0, ±1, ±2
0, otherwise.
What is the expression for the periodogram estimate ? Write it in simplest form.
3. A continuous-time signal xa(t) is bandlimited to 5 kHz, i.e., has a spectrum Xa(f) that is zero for
|f| > 5 kHz. Only 10 seconds of the signal has been recorded and is available for processing. We would
like to estimate the power spectrum of xa(t) using the available data in a radix-2 FFT algorithm, and
it is required that the estimate have a resolution of at least 10 Hz. Suppose that we use Bartlett’s
method of periodogram averaging.
(a) If the data is sampled at Nyquist rate, what is the minimum section length that you may use to
get the desired resolution ?
(b) Using the minimum section length determined in part (a), with 10 seconds of data, how many
sections are available for averaging ?
(c) How does your choice of the sampling rate affect the resolution and variance of your estimate ?
Are there any benefits to sampling above the Nyquist rate ?
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4. Whenever the signal mean is unknown, a natural modification of the unbiased estimator of the autocorrelation function (ACF) is,
r˜x(k) = 1
N − k
N
X−1
n=k
(x(n) − x) (x(n − k) − x), k = 0, . . . , N − 1. (1)
and the biased estimator is,
rˆx(k) = 1
N
N
X−1
n=k
(x(n) − x) (x(n − k) − x), k = 0, . . . , N − 1. (2)
where x is the sample mean
x =
1
N
N
X−1
n=0
x(n).
Show that in the unknown mean case, the usual names, “unbiased” and “biased” sample covariance
associated with the above might no longer be appropriate. Indeed, both these estimators could be
biased; furthermore, ˆrx(k) could be less biased than ˜rx(k). To simplify calculations, assume that x(n)
is white noise.
5. [∗]Show that the following definitions of the periodogram are equivalent.
Pper(e
jω) = 1
N
N
X−1
n=0
x(n)(e
−jωn)
2
. (3)
Pper(e
jω) =
N
X−1
k=−(N−1)
rˆx(k)e
−jωk
, (4)
where ˆrx(k) is the biased autocorrelation estimate (assuming zero-mean x(n)).
6. Let x(n), n = 0, . . . , N − 1, be a real Gaussian process with zero mean and autocorrelation rx(k). The
autocorrelation can be estimated using
rˆx(k) = α(k)
N
X−1
n=k
(x(n) − x) (x(n − k) − x), k = 0, . . . , N − 1, (5)
where
α(k) = 1
N−k
, for unbiased ACF estimate
1
N
, for biased ACF estimate.
Using the moment factoring theorem for real Guassian RVs (which is different from that for complex
Gaussian RVs mentioned in class),
E {abcd} = E {ab} E {cd} + E {ac} E {bd} + E {ad} E {bc} − 2E {a} E {b} E {c} E {d} , (6)
show that
V ar {rˆx(k)} = α
2
(k)
N
X−k−1
m=−(N−k−1)
(N − k − |m|)
r
2
x
(m) + rx(m + k)rx(m − k)
.
Simulations For all simulation problems, you are advised to use Matlab’s report generation facility and
submit a html file. This should have proper comments and appropriately labelled plots. The analysis and
observations are as important as writing the code. If you are using scilab, then you can submit a pdf
file with appropriate comments and observation. A link will be provided in moodle to upload your final
report/code.
1. Computer power spectrum estimates for the real data sets S00 and S0X using the periodogram,
Bartlett, Blackman-Tukey, and Welch procedures. In all cases zero-pad your data appropriately before
giving it to the FFT routine so that you have computed enough points of the spectrum to give a smooth
plot. Plotting 256 points of the spectrum should be sufficient.
(a) For periodogram use N=16, 32, 128, and 512
(b) For Bartlett, number of subsequences K = 32, 16, 4 (or size of subsequences L = 16, 32, 128)
(c) For Welch, L = 16, 32, 128 and overlap L/2 (rectangular window)
(d) For Blackman-Tukey, M = 16, 32, 128 (rectangular, Bartlett windows)
Discuss the results of these experiments, especially trade-off between resolution and variance in the
spectral estimate. Which choice do you think provides the best estimate of the true spectrum ? What
can you say about the underlying processes ?
Note: For the data set S0X, use S01 if your month of birth is Jan-Apr, S02 if your month of birth is
May-Aug, and S03 if its Sep-Dec. Use the file getfile.m provided in moodle to read the data files.
References
1. Petre Stoica and Randolph Moses, “Spectral analysis of signals”, Prentice Hall, 2005. (Indian edition
available)
2. Monson H. Hayes, “Statistical signal processing and modeling”, Wiley India Pvt. Ltd., 2002. (Indian
edition available)
3. Charles W. Therrien, “Discrete random signals and statistical signal processing”, Charles W. Therrien,
2004.
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