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New Mexico State UniversityKlipsch School of Electrical Engineering
Signals and Systems IPractice Final Exam B
Name:
Solve problems 1–3 and two from problems 4 – 7.Circle below which two of problems 4 – 7 you wish to be graded.
Prob. 1 / 20 pointsProb. 2 / 20 pointsProb. 3 / 20 pointsProb. 4 / 20 pointsProb. 5 / 20 pointsProb. 6 / 20 pointsProb. 7 / 20 pointsTotal / 100 points
EE312 Signals and Systems I Final Exam
Prob. 1
The linear, constant-coefficient differential equation (LCCDE) of a continuous-time (CT),linear, time-invariant (LTI) system is given by
1
10000
d2y(t)
d2t+
101
1000
dy(t)
dt+ y(t) =
1
100
dx(t)
dt+ x(t).
Assume at rest conditions.
(a) Draw a block diagram of the system using differentiators and not integrators.
(b) Determine the frequency response, H(jω) of the system.
1
EE312 Signals and Systems I Final Exam
Prob. 1 (cont.)
(c) Determine the impulse response, h(t) of the system.
(d) Let the input to the system be, x(t) = ej100t. Use any method∗ you wish to determinethe output, y(t).
∗ Solve LCCDE for y(t), convolve h(t)∗x(t), transform-domain F−1 {H(jω)X(jω)}, or eigenfunction theory
2
EE312 Signals and Systems I Final Exam
Prob. 2
The linear, constant-coefficient difference equation (LCCDE) of a discrete-time (DT) LTIsystem is given by
y[n]− 1
4y[n− 2] = x[n]− 2x[n− 1].
Assume at rest conditions.
(a) Draw a block diagram of the system.
(b) Determine the frequency response, H(ejω) of the system.
3
EE312 Signals and Systems I Final Exam
Prob. 2 (cont.)
(c) Determine the impulse response, h[n] of the system.
(d) Let the input to the system be, x[n] = ej2n. Use any method∗ you wish to determine theoutput, y[n].
∗ Solve LCCDE for y[n], convolve h[n] ∗ x[n], transform-domain F−1{H(ejω)X(ejω)
}, or eigenfunction
theory
4
EE312 Signals and Systems I Final Exam
Prob. 3
A CT LTI system with frequency response, H(jω) is constructed by cascading two CTLTI systems with frequency responses, H1(jω) and H2(jω) as depicted below; obviouslyH(jω) = H1(jω)H2(jω).
x(t) y(t)H1(jω) H2(jω)
H(jω)
The two figures below show the straight-line approximations of the Bode magnitude plots ofH1(jω) and H(jω). Complete the following parts to determine H2(jω).
20log10|H1
(jω)|
ω (rads/s)1 10
6 dB
24 dB
8 40 100
+20 dB/decade –20 dB/decade
(a) H1(jω)
20log10|H(jω)|
ω (rads/s)1 10
–20 dB
8 100
–40 dB/decade
(b) H(jω)
(a) Determine the three break frequencies of H1(jω).
5
EE312 Signals and Systems I Final Exam
Prob. 3 (cont.)
(b) Determine the constant factor for H1(jω), A1 such that
H1(jω) =A1(jω + ω1)
(jω + ω2)(jω + ω3)
where ω1, ω2, and ω3 are the break frequencies from (a).
Hint: From the plot, 20 log10 |H1(j0)| = 6 or |H1(j0)| = 2.
(c) Determine the break frequency(ies) and constant factor for H(jω).
(d) Determine H2(jω) = H(jω)/H1(jω) using your results in (b) and (c).
6
EE312 Signals and Systems I Final Exam
Prob. 4
Part I:
Let
x(t) =
t2, 0 ≤ t < 1−t2, −1 ≤ t < 00, otherwise
F↔ X(jω)
For this problem, the actual value of X(jω) does not matter. The first solution is given asan example.
(a) Graph x1(t) = x(t− 1.6) and write X1(jω) in terms of X(jω).
Solution: X1(jω) = e−j1.6ωX(jω)
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x1(t
)
t (s)
(b) Graph x1(t) = x
(−1
2t
)and write X1(jω) in terms of X(jω).
(c) Graph x1(t) = x (3t+ 1) and write X1(jω) in terms of X(jω).
7
EE312 Signals and Systems I Final Exam
Prob. 4 (cont.)
Part II:
Let
x[n] =sin(πn/4)
πn
F↔ X(ejω) =
{1, 0 ≤ |ω| ≤ π/40, π/4 < |ω| ≤ π
X(ejω) is periodic with a period of 2π.
(d) x1[n] = x[n− 3]. Graph the magnitude and phase response of X1(ejω) for −π ≤ ω ≤ π.
(e) x1[n] = x[n]∗ejπn/2. Graph the magnitude and phase response of X1(ejω) for−π ≤ ω ≤ π.
(f) X1(ejω) = e−j4ωX(ejω). Write x1[n] in terms of x[n] and graph the magnitude and phase
response of X1(ejω) for −π ≤ ω ≤ π.
8
EE312 Signals and Systems I Final Exam
Prob. 5
(a) Let x(t) = u(t) and h(t) = e−1000tu(t) + e−10tu(t). Determine y(t) = h(t) ∗ x(t) for−∞ < t <∞ using graphical convolution(s).
9
EE312 Signals and Systems I Final Exam
Prob. 5 (cont.)
(b) Let x[n] = u[n] and h[n] =
(1
2
)nu[n] +
(−1
2
)nu[n]. Determine y[n] = h[n] ∗ x[n] for
−∞ < n <∞ using graphical convolution(s).
10
EE312 Signals and Systems I Final Exam
Prob. 6
(a) Let
x(t) =
{1− e−t, −1 ≤ t < 0et − 1, 0 ≤ t < 1
be a periodic signal with a fundamental period T = 2 s. Graph x(t) over the interval−3 ≤ t ≤ 3 and determine the Fourier Series (FS) coefficients, ak.
ak = ?
11
EE312 Signals and Systems I Final Exam
Prob. 6 (cont.)
(b) Letx[n] = δ[n]− 2δ[n− 1] + 4δ[n− 2]− 2δ[n− 3] + δ[n− 4]
be a periodic signal with a fundamental period N = 5. Graph x(t) over the interval −5 ≤n ≤ 9 and determine the Discrete-Time Fourier Series (DTFS) coefficients, ak.
ak = ?
12
EE312 Signals and Systems I Final Exam
Prob. 7
Let the input signal be
x(t) = cos(20πt) + cos(200πt) + cos(2000πt)
and the impulse response beh(t) = 10000te−100tu(t).
Use eigenfunction theory (not phasors or convolution) to determine the output signal, y(t).Express y(t) in terms of cosines.
13
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