Digital Signal Processing (Lab 5)

Experiment 5: The Z-transform using MATLAB

Lab marking plan:

Sr. Instrument for CLO Maximum Marks

No. assessment
1 Lab Performance 1 5
2 Lab Report 2 5

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Problem 1
Analytically determine the Z-transform of the following signal using Z-transform tables.

1. x(n) = (0.8)nu(n − 2)

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2. Verify the Z-transform expression using MATLAB. You will first need to create a
symbolic variable ‘n’, then create the function above and then use the ‘ztrans’ function to compute
the Z-transform.

Problem 2
Consider the sequence x(n) = (0.9)n cos(πn/4)u[n]. Let y[n] = x[n/2], n = 0, ±2, ±4, ···; 0, otherwise.
1. Analytically determine Z{y[n]} = Y(z) using Z-transform tables along with properties of the
Z-transform.
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2. Verify the Z-transform expression using MATLAB.

Problem 3
If signals x1[n], x2[n], and x3[n] are related by x3[n] = x1[n] * x2[n], then
Z{x1[n]} = Z{x2[n]}Z{x3[n]}
i.e.

∑∞n=−∞ x [n] z 3
-n
=( ∑∞n=−∞ x [n] z )( ∑∞n=−∞ x [n] z )
1
-n
2
-n

1. Prove this result analytically by substituting the definition of convolution in the left-hand side of
x3[n] = x1[n] * x2[n].
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2. Let x1[n] = x2[n] = u[n]. Compute x3[n] = using the ‘conv’ function of MATLAB. Now
verify the result that you have proven above using the ‘ztrans’ function of MATLAB.
Problem 4
Consider a system with the following unit sample response system function. Using the z-transform
approach, show that the impulse response, h(n), of the overall system is given by

h[n] = (0.8)nsin(nπ/2) u[n − 1]

1. Using Z-transform tables analytically determine whether or not the system is stable.
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2. Using MATLAB compute the unit step response of the system. Using MATLAB plot the
unit step response for 0 ≤ n ≤ 20.
 Lab Summary
Summarize in your own words within one page what you learnt in this lab.
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