Matlab代写|Assignment代写 - EE578/EE978 Assignment Random Signals & Stochastic Processes
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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
45100ν 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 + zz1
excited by an input x[n]. State the power spectral density (PSD) of y[n], Ry(ejΩ), in
dependency of the input PSD, Rx(ejΩ).
(b) The input x[n] is now modelled as a moving average filter G(z) = 1 zz1
, 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] 1
i. analytically based on the PSD Ry(ejΩ);
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(ejΩ) = 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 = σv2/σz2
the ratio between the input and output power of w[n]. For A(z) = √0.99zz1
determine the input and output powers of w[n] and state the coding gain G.