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# EEL 3135 – Lab #07 solution

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Full Name:
EEL 3135 – Lab #07
Question #1: (Difference Equations and Pole-Zeros Plots)
with the corresponding descriptions. This is designed to show you how to visualize the output
impulse response, filter output, pole-zero response for a given transfer function (in the Z-domain).
comments. You will use elements of this MATLAB code for the rest of the lab assignment.
Question #2: (Z-Transform)

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Full Name:
EEL 3135 – Lab #07
Question #1: (Difference Equations and Pole-Zeros Plots)
with the corresponding descriptions. This is designed to show you how to visualize the output
impulse response, filter output, pole-zero response for a given transfer function (in the Z-domain).
comments. You will use elements of this MATLAB code for the rest of the lab assignment.
Question #2: (Z-Transform)
For the following Z-transforms, plot the corresponding impulse response and the pole-zero plot.
Use Lab Question #1 as a guide.
(a) H(z) = 1 − (0.2)z
−1
(b) H(z) = 1 − 1.5z
−1
(c) H(z) = 1 − 2z
−1 + 0.5z
−2
(d) H(z) = (1 − e
j3π/4
z
−1
)(1 − e
−j3π/4
z
−1
)
(e) H(z) = 1
1 − (0.8)z−1
(f) H(z) = 1
1 − (1.4)z−1
(g) H(z) = 1
(1 − e
j3π/4z−1)(1 − e−j3π/4z−1)
(h) H(z) = 1
(1 − (0.8)e
jπ/4z−1)(1 − (0.8)e−jπ/4z−1)
Question #3: (More Z-Transform)
(a) For what pole-zero conditions is the impulse response unstable (i.e., goes to ∞ as n → ∞)?
(b) For what pole-zero conditions is the impulse response stable (i.e., goes to zero as n → ∞)?
(c) For what pole-zero conditions is the impulse response critically stable (i.e., steady amplitude as n → ∞)?
(d) For what pole-zero conditions is the impulse response finite in length?
(e) For what pole-zero conditions is the impulse response infinite in length?
(f) For what pole-zero conditions is the impulse response periodic (with a frequency 0)?
1
Question #4: (Loan Difference Equations)
In this problem, we will study a simplified difference equation interest model. Hence, consider the
following difference equation model for your student loan:
y[n] = (1 + α)y[n − 1] − βy[n − R] − γy[n − Q] + x[n]
where α is the yearly loan interest rate, β is the percentage of the loan you pay per year after
graduation, R is the number of years till graduation, γ is the percentage of the loan you pay after
you get rich from your awesome invention, and Q is the number of years until you create your
invention. The output y[n] is the loan amount. The time parameter n is in years. When positive,
the input x[n] is the initial loan amount.
(a) Assume you never pay your loan (β = 0, γ = 0 and R, Q does not matter). Assume we start
with a loan \$120, 000 at time n = 0, i.e., x[n] = 120000 δ[n]. Let α = 0.09. Plot y[n] for 50
years, 0 ≤ n ≤ 50, and the pole-zero plot for the system. Use Lab Question #1 as a guide.