哈尔滨工程大学信号与系统试卷与答案

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1. (6 points) A continuous-time signal x(t) is shown in Fig.1. Sketch and
label each of the following signals:
(1) x(t/2) (2) x ' (t) (3) Єv{ x(t)- x(t+1) }

Figure 1 Figure 2
2. (12 points)
(1) An LTI system has the impulse response h(t) depicted in Fig.2.
Use linearity and time invariance to determine and sketch the system
output y(t) if the input x(t) is x(t) = 2δ(t+2) + δ(t-2).
(2) Evaluate the convolution integral: e-3tu(t)*u(t+2), and draw a
sketch of the result.
(3) Evaluate the discrete-time convolution sum: αnu[n]*u[n -2], and
draw a sketch of the result.

3. (6 points)
Determine whether the following signals are periodic. If they are
periodic, find the fundamental period.
(1) x[n] = (-1)n (2) cos[2n] (3) cos[2πn]

4. (6 points)
A continuous-time signal x(t) is defined by x(t) = { 2cos(100πt +
π/6) }2
(1) Specify the dc component of x(t).
(2) Specify the amplitude and fundamental frequency of the
sinusoidal component of x(t).
(3) Determine the Fourier series (FS) coefficients of x(t)

5. (10 points)
(1) Determine the time-domain signal represented by the Fourier

series coefficients:
X[k] = -jδ[k - 2] + jδ[k +2] + 2δ[k - 3] + 2δ[k + 3], ω0 = π
(2) Determine the discrete-time Fourier series (DTFS) coefficients of
the periodic signals depicted in Fig.3.

Figure 3 Figure 4
6. (10 points)
Determine and draw sketches of the Fourier series and Fourier
transform representation of the square wave depicted in Fig.4

7. (8 points)
(1) Compute the discrete-time Fourier transform (DTFT) of the
signal depicted in Fig.5.

Figure 5 Figure 6
(2) Draw the Fourier transform of a impulse-train sampled
version of the continuous-time signal having the Fourier
transform depicted in Fig.6 for (a) T = 1/2(s) and (b) T = 2(s),
where T is sampling period.

8. (6 points)
Shown in Fig.7 is the frequency response H(jω) of a
continuous-time filter. For each of the input signals x(t) below,
determine the filtered output signal y(t).
(1) x(t) = cos(2πt+θ) (2) x(t) = cos(4πt+θ)

哈尔滨工程大学试卷
考试科目: Signals and Systems
2006.7.19

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Figure 7
9. (8 points)
Find the inverse Fourier transform of the following spectra:
(1) X(jω) = 2δ(ω - 4)
(2) X(ejω) = 2cos(2ω)

(3) X(jω) = ,0,cos2

(4) X(ejω) = otherwise 02 4,/,je, on -π < ω < π
10. (16 points)
Consider a continuous-time LTI system for which the input x(t) and
output y(t) are related by the differential equation
y" (t) - y' (t) - 6y(t) = x' (t) + x(t)
(1) Determine the frequency response H(jω) of the system.
(2) Determine the system function H(s) of the system. Sketch the
pole-zero plot of H(s)
(3) Determine the system impulse response h(t) for each of the
following cases:
(a) The system is stable; (b) The system is causal.
(4) Let x(t) = e-2tu(t). Find the output y(t) of the causal system.

11. (6 points)
Consider a message signal m(t) with the spectrum shown in
Fig.8. The message bandwith ωm = 2π×103 rad/s. The signal is
applied to a product modulator, together with a carrier wave
Accos(ωct), producing the modulated signal s(t). The modulated
signal is next applied to a synchronous demodulator (shown in
Fig.9).
(1) Determine the spectrum of the demodulator output
when (a) the carrier frequency ωc = 2.5π×103 rad/s and (b) the
carrier frequency ωc = 1.5π×103 rad/s.
(2) What is the lowest carrier frequency for which each
component of the modulated signal s(t) is uniquely determined
by m(t).

Figure 8 Figure 9
12. (6 points)

(1) Draw a sketch of the spectrum of
x(t) = cos(50πt)sin(700πt)
Label the frequencies and complex amplitudes of each component.
(2) Determine the minimum sampling frequency that can be used to
sample x(t) without aliasing for any of the components.