**Question 1**

Part (a)

Generate and plot a sampled sine wave with f_{s }= 8kHz, of 4 seconds duration, with frequency ω_{0} = Π/10 rad/samp and amplitude A = 1:2. The waveform equation is

x(n) = A sin nω

Explain the role of each of the variables in this equation. What is the true (Hertz) frequency generated in this case?

Part (b)

Generate a Gaussian random signal vector, v(n), of the same length. Then generate a noisy signal of the form

y(n) = x(n) + αv(n)

Listen to the resulting signal y(n) for various values of . You will have to choose the value of experimentally { try both small and large, and investigate the dierences. Plot one of the waveforms, and comment briefly on your results.

(a) Plot clean sinusoidal waveform and comment

b) Plot waveform with noise amd comment

**Question 2**

**Part (a)**

A filter of the farm

G(z) = z^{2}/(z-p)(z-p*)

With r = 0:95 and ω_{n} = Π/10, plot the time response to the input sinusoidal waveform generated in the rst question. Show both the transient and steady-state response.**Part (b) **

Plot the frequency-domain response of the lter. Explain all your working, particularly how the z transfer function is converted to gain/phase plots.

**Part (c) **

Find the gain and phase from the time-domain response of part (a), and compare to that expected from the frequency response in part (b). Are the results the same?

(a) Time response (transient+steady-state)

(b) Frequency response (gain+phase)

(c) Compare gains and phases, explain results.

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