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Discrete Time Systems

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1. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T=0.2.

P(s) = (s+2)/(s+1)(s+5)

b) Given the z-transform of a sequence, U(z) = Z{u},

U(z) = z/(z^2 - 0.8z + 0.6)

i. Find the first 4 terms of u by long division.
ii. Find the underlying signal u(t) assuming that T=1 and no aliasing has occured.

2. a) The system (P_s)(s) = 4/(s^2 + 2s + 3) is subject to a staricase input with T = 0.2 seconds. Calculate the Z domain transfer function (P_z)(z).

b) The system, (P_s)(s) = 1/(s + 2) is subject to a staircase input with T = 0.1 seconds. Calculate the w-domain transfer function (P_z)(w) and comment on the relationship between s- and w-doman transfer functions.

3. a) Given (P_s)(s) = 4/(s + 3), find the w domain description for T = 0.2 and comment on the similarity and difference between (P_s)(s) and (P_w)(w). Use 3 decimal place in your calculations.

bi) Calculate the z-transform of the signal, y)t_ = e^(0t)sin(5t) sampled at T = 1.0.

bii) By finding the inverse z-transform, show that aliasing has occurred and explain why.

c) Find the first 4 terms in the inverse z-transform of Y(z) = 2z/(z^2 + 0.5z + 0.3) by means of long division. Use the final value of the signal to confirm that your results make sense.

4. a) The system, (P_s)(s) = 3/(s^2 + s + 2.5) is subject to a staircase input with T=0.2. Calculate the Z doman transfer function (P_z)(z).

b) A sequence, y_B, has a Z transform,

Y(z) = (z^2 + z)/(z^3 + 2z^2 + 1.4z + 0.3)

Find the first 5 terms in the sequence (y_0 ... y_4) by long division.

c) By considering the initial and final value theorems in the Z domain, find equivalent initial and final value theorems in the W domain and confirm that they are similar to the S domain theorems.

5. Show that aliasing will occur if the signal y(t) = e^-t(sin)(2t) is sampled with T=0.2 by finding the z transform of the sampled sequence and then finding the inverse z transform.

c) If a system has the transfer function P(s) = 1/(s/(3+1)) and T = 0.2, find (P_z)(w).

d) Show that the unit step signal, u(t) = sigma(t)

6. ai) The system, (P_s)(s) = (4s + 7)/(s^2 + 3s + 2) is subject to a staircase input with T = 0.5. Calculate the z domain transfer function, (P_z)(z).

aii) Use the final value theorem to find the final value of the output if the input is a step. Does your answer make sense with respect to the final value of the step response of the continuos system?

b) Given (P_s)(s) = 2/(s + 3), find the w domain description for T = 0.2 and comment of the similarity and difference between (P_s)(s) and (P_w)(w).

c) Given (P_z)(z) = (2z^2 - 0.5z + 1)/(z^2 + 0.2z + 0.5) and u_o = 1, u_1 = 2, u_n = 0, n =/ 0, 1, initioal y = 0.

i) Calculate y_n for n=0, 1, 2, and 3 by writing the difference equation and substituition.
ii) Does the oscillating behaviour make sense in terms of the pole locations in the z plane?

7. a) Find the z-transform of the following system transfer function assuming the input is a unit staircase (i.e. zero order held). The sampling rate is T = 0.2.

P(s) = 3/(s^2 + s + 1)

b) A signal y(t) = e^(-0.t)sin*2pi(t) is sampled at a rate T = 0.5, Show that the z-transform of the sampled sequence is Y(z) = (ze^-0.25)(sin(pi))/(z + e^-0.5)^2 = 0. Make a sketch to explain what has happened.

c) Given P(s) = 1/(s+1), find (P_z)(w) for T = 0.1.

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MATLAB Signals Theory- Linear Time - Inveriant Discrete Time System

A linear time-invariant discrete-time system has transfer function
h(z)=((z^2)-z-2)/((z^2) + 1.5z-1)

a. Use MATLAB to obtain the poles of the system. Is the system stable?
Explain.
b. Compute the step response. This should be done analytically, but you can
use MATLAB commands like conv and residue.
c. Plot the first seven values of the step response. Is the response increasing
or decreasing with time? Is this what you would expect, and why?

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