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.

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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

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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.

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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

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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).

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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).

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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).

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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.

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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).

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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).

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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 = ?

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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 = ?

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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.

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