EE578/EE978 Assignment 1
Random Signals & Stochastic Processes
October 27, 2020
Please submit electronic copies of your answers (scanned handwriting is perfectly acceptable)
via MyPlace by Monday 23/11/2020. Your answers should contain appropriate plots of your results
with any annotations and brief discussions or justifications of your answers. If you submit Q1 one
week prior (by 16/11), I will give you written feedback and you will have the chance to update
your submission.
1. Random Signals.
(a) Characterise the PDF of the following signal:
x = round(rand(1000,1))*2-1;
What do you expect? What can you measure experimentally?
(b) You are given a system with impulse response
h[n] =
10∑
ν=0
(
4
5
)10−ν
δ[n− ν] +
20∑
ν=11
(
4
5
)
ν−10
δ[n− ν]
Sketch the impulse response.
(c) Produce an output
y = filter(h,1,x);
given the above x and h containing the coefficients of the impulse response h[n]. Estimate
the PDF of y. How can you enhance this estimate, and what can you say about the
shape of this PDF?
(d) Knowing that your input is uncorrelated, calculate the power of the output signal y both
analytically and experimentally.
2. Power Spectral Densities.
(a) A signal y[n] = h[n] ∗ x[n] emerges from a filter with transfer function H(z) = 1 + z−1
excited by an input x[n]. State the power spectral density (PSD) of y[n], Ry(e
jΩ), in
dependency of the input PSD, Rx(e
jΩ).
(b) The input x[n] is now modelled as a moving average filter G(z) = 1 − z−1, such that
x[n] = g[n] ∗ u[n], whereby u[n] is a zero-mean unit-variance uncorrelated Gaussian
process and g[n] ◦—• G(z). What is the output PSD Ry(ejΩ) now?
(c) With y[n] = h[n] ∗ g[n] ∗ u[n], determine the power of y[n]
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i. analytically based on the PSD Ry(e
jΩ);
ii. analytically based on the impulse response h[n] ∗ g[n];
iii. experimentally using Matlab.
3. Rate-Distortion Theory.
(a) You are measuring the PSD of a signal v[n]
Rv(e
jΩ) =
1
|(1−A(ejΩ)|2
where A(z) is a transfer function. What system excited by a zero-mean unit-variance
uncorrelated input could have caused such a PSD?
(b) State a filter w[n] (by reasoning, no calculation required) that, when applied to v[n],
produces an uncorrelated output z[n] = w[n] ∗ v[n].
(c) When having the option of quantising either v[n] or z[n], the coding gain G = σ2
v
/σ2
z
is
the ratio between the input and output power of w[n]. For A(z) =
√
0.99z−1, analytically
determine the input and output powers of w[n] and state the coding gain G.
S. Weiss, October 27, 2020
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