信号与系统习题集
信号与系统练习题集
第一部分:信号与系统的时域分析
一、填空题 1. e
-(t -2)
u (t ) δ(t -3) = ( ).
2. The unit step response g (t ) is the zero-state response when the input signal is ( ). 3. Given two continuous – time signals x(t) and h(t), if their convolution is denoted by y(t), then the convolution of x (t -1) and h (t +1) is ( ). 4. The convolution x (t -t 1) *δ(t +t 2) =( ).
5. The unit impulse response h (t ) is the zero-state response when the input signal is ( ).
6. A continuous – time LTI system is stable if its unit impulse response satisfies the condition: ( ) .
7. A continuous – time LTI system can be completely determined by its ( ).
∞sin 2t
δ (t)dt= ( ). 8. ⎰ 2
-∞t 9. Given two sequences x [n ]={1, 2, 2, 1} and h [n ]={3, 6, 5}, their convolution x [n ]*h [n ]= ( ).
10. Given three LTI systems S1, S2 and S3, their unit impulse responses are h 1(t ) , h 2(t ) and
h 3(t ) respectively. Now, construct an LTI system S using these three systems: S1 parallel interconnected by S2, then series interconnected by S3. the unit impulse response of the system S is ( ).
11. It is known that the zero-stat response of a system to the input signal x(t) is y (t ) =⎰x (τ) d τ,
-∞t
then the unit impulse response h(t) is ( ).
12. The complete response of an LTI system can be expressed as a sum of its zero-state response and its ( ) response.
13. It is known that the unit step response of an LTI system is e
-2t
u (t ) , then the unit impulse response h(t)
is ( ).
∞π
14. x (t ) =⎰sin (δ(t -1) +δ(t +1)) dt = ( ).
02
15. We can build a continuous-time LTI system using the following three basic operations: ( ) , ( ), and ( ). 16. The zero-state response of an LTI system to the input signal x (t ) =u (t ) -u (t -1) is
s (t ) -s (t -1) , where s(t) is the unit step response of the system, then the unit impulse response
is h(t) = ( ).
17. The block diagram of a continuous-time LTI system is illustrated in the following figure. The differential
equation
describing
the input-output relationship
of the system
is
( ) .
(t )
18. The relationship between the unit impulse response h(t) and unit step response s(t) is s(t) = ( ), or h(t) = ( ).
二、选择题
1. For each of the following equations, ( ) is true.
A 、(t -1) δ(t ) =δ(t ) B 、(1+t ) δ(1-t ) =2δ(t ) C 、⎰(1+t ) δ(t ) dt =δ(t ) D 、⎰(1+t ) δ(1+t ) dt =1
-∞
-∞
∞
∞
2. Given two continuous-time signals x (t ) and h (t ) , if the convolution of x (t ) and h (t ) is denoted by y (t ) , then the convolution of signals x (t +1) and h (t -2) is ( ).
A. y (t ) B. y (t -1) C. y (t -2) D. y (t +1)
3. The unit impulse response of an LTI system is h(t) = e
-t
, this system is ( ).
A. causal and stable B. causal and unstable C. noncausal and unstable D. noncausal and stable
1
4. x (t ) =(2t 2+1) δ(t -2) dt = ( ).
-1
⎰
A. 1 B. 3 C. 9 D. 0
5. For an LTI system, if the input signal is x 1(t ) , the corresponding output response is y 1(t ) , if the input signal is x 2(t ) , the corresponding output response is y 2(t ) . And if the input signal is
ax 1(t ) +bx 2(t ) , the corresponding output response is ay 1(t ) +by 2(t ) ( a and b are arbitrary real numbers ). Then the system is a ( ) system.
A. linear B. causal C. nonlinear D. time – invariant
6. x (t -t 1) *δ(t -t 2) = ( ).
A. x (t -t 1-t 2) B. x (t -t 1+t 2) C. x (t +t 1-t 2) D. δ(t +t 1+t 2)
∞⎛π⎫
x (t ) =(t +sin t ) δ t -⎪dt = ( ). 7. ⎰-∞
⎝6⎭
A.
πππ1π1 B. -1 C. - D. +
626266
8. Given two sequences x 1[n ] and x 2[n ], their lengths are M and N respectively. The length of the
convolution of x 1[n ] and x 2[n ] is ( ).
A .M B .N C .M +N D .M +N -1
9. The unit impulse response of a continuous-time LTI system is h (t ) =2δ(t ) +
d δ(t )
, the differential dt
equation describing the input-output relation of this system is ( ).
dy (t ) dy (t )
=x (t ) B. y (t ) +2=x (t ) A. 2y (t ) +dt dt
dx (t ) dy (t ) dx (t )
=x (t ) +2C. y (t ) =2x (t ) + D.
dt dt dt
10. The input-output relation of a continuous-time LTI system is described by the differential
d 2y (t ) dy (t ) dx (t ) +2+3y (t ) =2x (t ) +equation: . The unit impulse response of the system h(t) 2
dt dt dt
( ).
A . does not include δ(t ) B. includes δ(t ) C. includes
d δ(t )
D. is uncertain dt
11. Signals x 1(t ) and x 2(t ) are shown in the following figures. The expression of the convolution x (t ) =x 1(t ) *x 2(t ) is ( ).
(1)
x 2(t )
(1)
A. u (t +1) -u (t -1) B. u (t +2) -u (t -2) C. u (t -1) -u (t +1) D. u (t -2) -u (t +2)
12. The following block diagram represents a continuous-time LTI system. The unit impulse
response h(t) satisfies ( ).
dy (t )
+y (t ) =x (t ) A. dt
B. h (t ) =x (t ) -y (t ) C.
dh (t )
+h (t ) =δ(t ) dt
x (t )
t )
D. h (t ) =δ(t ) -y (t )
13. The input-output relationship of a causal continuous-time system is described by the differential equation:
dy (t ) dx (t )
+3y (t ) =2, then the unit step response s (t ) = ( ). dt dt
11
A. 2e -3t u (t ) B. e -3t u (t ) C. 2e 3t u (t ) D. e 3t u (t )
22
三、综合题(分析、计算题)
1. The input-output relationship of a continuous-time LTI system is described by the equation:
y (t ) =⎰e -(t -τ) x (τ-2) d τ,
-∞
t
a. Determine the unit impulse response h(t) of the system.
b. Determine the system response y(t) to the input signal x (t ) =u (t +1) -u (t -2) .
2. Given an LTI system depicted in Figure 2. Assume that the impulse response of the LTI system is h(t) = e-t u(t), the input signal x(t) = u(t) - u(t-2). Determine and sketch the output response y(t) of the system by evaluating the convolution y(t) = x(t)*h(t).
3. Remember the following identities:
Figure 2
x (t ) =x (t ) *δ(t )
x (t -t 0) =x (t ) *δ(t -t 0)
δ(t +t 0) *δ(t -t 0) =δ(t )
dy (t ) dx (t ) dh (t )
=*h (t ) =x (t ) * dt dt dt
4. Consider an LTI system S and a signal x (t ) =2e -3t u (t -1) . If
x (t ) →y (t )
and
dx (t )
→-3y (t ) +e -2t u (t ) , dt
determine the impulse response h(t) of S.
5. Let x (t ) =u (t -3) -u (t -5) and h (t ) =e -3t u (t ) , as illustrated in the Figure 6.
(a). Compute y(t) = x(t)*h(t).
(b). Compute g(t) = dx(t)/dt * h(t).
(c). How is g(t) related to y(t)?
6. Let y (t ) =e u (t ) *
-t
∞
1
x (t )
t
31
5
h (t )
t
k =-∞
∑δ(t -3k )
Figure 6
Show that y (t ) =Ae -t for 0 ≤ t
d 2y (t ) dy (t ) dx (t )
+3+2y (t ) =+x (t ) dt 2dt dt
If the input signal is x (t ) =e -2t u (t ) , determine the zero-state response y(t) of the system.
8. In this problem, we illustrate one of the most important consequences of the properties of linearity and time invariance. Specifically, once we know the response of a linear system or a linear time-invariant system to a single input or responses to several inputs, we can directly compute the responses to many other input signals.
(a). Consider an LTI system whose response to the signal x1(t) in Figure 9(a) is the signal y1(t) illustrated in Figure 9(b). Determine and sketch carefully the response of the system to the input x2(t) depicted in Figure 9(c).
(b). Determine and sketch the response of the system considered in part (a) to the input x 3(t) shown in Figure 9(d).
t
t
第二部分:信号与系统的频域分析
一、填空题
⎧⎪2,
1. The frequency response of an ideal filter is given by H (j ω) =⎨
⎪⎩0,
ω≥100π
, if the input
signal is x (t ) =10cos(80πt ) +5cos(120πt ) , the corresponding output response y(t) = ( ).
2. The Fourier transform of signal x (t ) =cos(ω0t ) is ( ). 3. The Fourier transform of signal x (t ) =sin(ω0t +
π
6
) is ( ).
4. Assume that the Fourier transform of x (t ) is denoted as X (j ω) , then the Fourier transform of
y (t ) =e j ω0t x (t ) is Y (j ω) = ( ).
5. The Fourier transform of a continuous – time periodic signal x (t ) =( ).
6. It is known that the Fourier transform of x (t ) is X (j ω) =
tx (t ) is ( ).
k =-∞
∑a e
k
∞
jk ω0t
is X (j ω) =
1
, then the Fourier transform of j ω+1
7. The Fourier transform of signal x (t ) is denoted as X (j ω) , the Fourier transform of (t -1) x (t ) is ( ).
8. A time shifting leads to a ( ).
9. The frequency responses of two LTI systems are assumed to be H 1(j ω) and H 2(j ω) , the frequency response of the interconnection of H 1(j ω) cascaded by H 2(j ω) is H (j ω) = ( ).
10. A time-domain compression corresponds to a frequency-domain ( ). 11. For a signal x(t), if the condition
⎰
∞
-∞
x (t ) dt
exists, this condition is ( ) but not ( ).
12. Figure 12 shows a continuous-time signal x (t ) , its Fourier transform is denoted as X (j ω) , then X (0) = ( ). (Without evaluatingX (j ω) ).
13. For a continuous-time LTI system, if the zero-state response of the system to
the input signal
x (t ) =e -t u (t ) is
y (t ) =e -t u (t ) -e -2t u (t ) , then the frequency response of the system is H (j ω) = ( ). 14. The Fourier transform of signal x (t ) =( ).
15. The inverse Fourier transform of δ(ω) is x (t ) = ( ).
sin 4t
is X (j ω) = t
16. The frequency spectrum includes two parts, one is ( ), the other is ( ). 17. Let X (j ω) denote the Fourier transform of signal x (t ) , then the Fourier transform of signal
t
y (t ) =x (+3) *cos(4t ) is Y (j ω) =( ). (Expressed using X (j ω) ).
2
18. Let X (j ω) denote the Fourier transform of signal x (t ) , then the Fourier transform of signal
y (t ) =x (t ) cos(πt ) is Y (j ω) =( ). (Expressed using X (j ω) ).
19. The period of the periodic square wave increases, the space of the spectral lines ( ).
20. Consider a continuous-time ideal lowpass filter S whose frequency response is
⎧⎪1
H (j ω) =⎨
⎪⎩0
ω≤100
ω>100
When the input to this filter is a signal x(t) with fundamental period T = π/6 and Fourier series coefficients ak , it is found that
S
x (t ) −−→y (t ) =x (t )
For k ( ) it is guaranteed that ak = 0.
21. Consider a continuous-time LTI system whose frequency response is
+∞
H (j ω) =⎰h (t ) e -j ωt dt =
-∞
s i n 4(ω)
ω
⎧10≤t
⎩-14≤t
二、选择题
1. The frequency response of an ideal lowpass filter is
⎧⎪2,
H (j ω) =⎨
⎪⎩0,
≤120π>120π
.
If the input signal is x (t ) =10cos(100πt ) +5cos(200πt ) , the output response is y (t ) = ( ).
A. 10cos(100πt ) B. 10cos(200πt )
100πt ) D. 5cos(200πt ) C. 20cos(
2. The Fourier transform of the rectangular pulse x (t ) =u (t +1) -u (t -1) is ( ).
A. 4Sa (ω) B. 2Sa (ω) C. 2Sa (2ω) D. 4Sa (2ω)
3. Let X (j ω) denote the Fourier transform of a signal x (t ) , the Fourier transform of x (t ) e jt is ( ).
A. e -j ωX (j ω) B. e j ωX (j ω) C. X (j (ω-1)) D. X (j (ω+1))
4. Let X (j ω) denote the Fourier transform of signal x (t ) , the Fourier transform of x (t -1) is ( ).
A. e -j ωX (j ω) B. e j ωX (j ω) C. X (j (ω-1)) D. X (j (ω+1)) 5. The Fourier transform of the rectangular pulse x (t ) =u (t ) -u (t -1) is ( ).
A. sa () e
2
ω
-j
ω
2
B. sa () e
2
ω
j
ω
2
C. sa (ω) e -j ω D. sa (ω) e j ω
6. The condition for signal transmission with no distortion is that ( ). A. The magnitude response is a constant in the passband. B. The phase response is a line cross the origin in the passband.
C. The magnitude response is a constant and the phase response is a line cross the origin in the passband.
D. The phase response is a constant and the magnitude response is a line cross the origin. 7. The bandwidth of a signal x (t ) is 20KHz, the bandwidth of signal x (2t ) is ( ). A.20KHz B.40KHz C.10KHz
D.30KHz
8. Let X (j ω) denote the Fourier transform of signal x (t ) , the Fourier transform of t ( ).
A. X (j ω) -ω
dx (t )
is dt
dX (j ω)
d ωdX (j ω)
d ω
B. -X (j ω) +ω
dX (j ω)
d ω
C. -X (j ω) -ωD. X (j ω) +ω
dX (j ω)
d ω
9. Let X (j ω) denote the Fourier transform of signal x (t ) , the Fourier transform of
t
y (t ) =x (+b ) is ( ).
a
A. a X (j ω) e
jab ω
B. a X (ja ω) e
-jab ω
1ωj ω1ω-j ω
X (j ) e a D. X (j ) e a C. a a a a
b b
10. Let X (j ω) denote the Fourier transform of signal x (t ) =u (t +1) -u (t -1) , then X (0) = ( ).
A. 2 B. π C.
1
π D. 4 2
11. Let X (j ω) denote the Fourier transform of signal x (t ) , the Fourier transform of x (1-t ) is ( ).
A .-X (-j ω) e B .X (j ω) e
j ω
-j ω
C .X (-j ω) e
-j ω
D .X (-j ω) e j ω
12. Let X (j ω) denote the Fourier transform of signal x (t ) , the Fourier transform of
y (t ) =x (t ) δ(t -a ) is ( ).
A. X (j ω) e -ja ω B. x (a ) e -ja ω C. X (j ω) e ja ω D. x (a ) e ja ω
13. The Fourier transform of signal x (t ) =δ(t +τ) +δ(t -τ) is X (j ω) = ( ).
11
A. cos ωτ B. 2cos ωτ C. sin ωτ D. 2sin ωτ
22
14. Let x (t ) =e -t δ(t ), and y (t ) =⎰x (τ) d τ. The Fourier transform of y(t) is Y (j ω) =( ).
-∞t
A.
111
+πδ(ω) D. -+πδ(ω) B. j ω C. j ωj ωj ω
11
15. Consider the square wave x (t ) =u (t +τ) -u (t -τ) , as τ decreases, the width of the main
22
lobe of X (j ω) ( ).
A. increases B. decreases C. does not change D. can not be determined
16. It is known that the bandwidth of x(t) is∆ω, the bandwidth of x (2t -1) is ( ).
A. 2∆ω B. ∆ω-1 C. 17. The inverse Fourier transform of X (j ω) =
11
∆ω D. ∆ω-1) 22
1
e j ωt 0 is x (t ) =( ). j ω+a
A. x (t ) =e -a (t +t 0) u (t ) B. x (t ) =e -a (t +t 0) u (t +t 0) C. x (t ) =e -a (t -t 0) u (t -t 0) D. x (t ) =e -a (t -t 0) u (t )
ω
18. The Fourier transform of signal x τ(t ) is X τ(j ω) =τSa (τ) , then the Fourier transform of
2signal y (t ) =x τ(t -1) is Y (j ω) =( ).
A. Y (j ω) =Sa (ω) e j ω B. Y (j ω) =Sa (ω) e -j ω C. Y (j ω) =2Sa (ω) e j ω D. Y (j ω) =2Sa (ω) e -j ω
19. Given an LTI system with its frequency response H (j ω) =
1
, it is known that the Fourier j ω+2
transform of the output response y(t) is Y (j ω) =
x (t ) =( ).
1
, the input signal
(j ω+2)(j ω+3)
A. x (t ) =e -2t u (t ) B. x (t ) =-e -3t u (-t ) C. x (t ) =e -3t u (t ) D. x (t ) =e 3t u (t )
⎧e -j ω
20. The frequency response of an ideal lowpass filter is H (j ω) =⎨
⎩0
ω
, its unit impulse ω≥2
response is h(t) = ( ).
A.
三、综合题(分析、计算)
1. Consider a continuous-time LTI system whose frequency response is
sin(4ω)
H (j ω) =
ω
sin 2t sin 2(t -1) sin t sin(t -1)
B. C. D.
π(t -1) π(t -1) π(t -1) π(t -1)
⎧10≤t
If the input to this system is a periodic signal x (t ) =⎨ with period T = 8, determine
⎩-14≤t
2. The fundamental frequency of a continuous-time periodic signal is ω0 = π, Figure 2 shows the spectral coefficients of x(t).
(a) Write out the expression of x(t).
(b) If x(t) is applied to an ideal highpass filter with
k
⎧1,
frequency response H (j ω) =⎨
⎩0,
determine the output signal y(t).
ω≥15π
otherwise
,
k
3. Figure 3.a illustrates a communication system. Let X1(jω) and X 2(jω) denote the Fourier transforms of x 1(t) and
Figure 2
x 2(t), respectively. It is known that ω1 = 4π, ω2 = 8π, and the frequency response of the ideal bandpass filter is H1(jω), the overall output response is y(t). (1). Plot the magnitude of the Fourier transform W(jω) of w(t).
(2). Choose an appropriate frequency ω3, so that the output response is y(t) = x1(t); (3). Plot the magnitude responses of H1(jω) and H2(jω).
x 1
t )
x (t )
ω2t )
cos(ω3t )
(a)
Figure 3
4. Figure 4 shows the Fourier transform X (j ω) of a periodic continuous-time signal x (t ) . (1). Write out the expression of x (t ) .
⎧1,
(2). Let H (j ω) =⎨
⎩0,
be the
ω≤12π
Otherwise
frequency response of an ideal lowpass filter, and x (t ) is applied to the filter, determine the output response y(t) of the filter.。
Figure 4
5. For a causal LTI system, the input and output signals are x (t ) =e -t u (t ) +e -3t u (t ) ,
y (t ) =(2e -t -2e -4t ) u (t ) , respectively. (1). Determine the frequency response H (j ω) . (2). Determine the unit impulse response h (t ) .
(3). Determine the differential equation describing the input-output relationship of the system.
⎧1, 1≤ω≤3
6. The frequency response of an ideal bandpass filter is H (j ω) =⎨, the unit impulse
0, Otherwise ⎩
response is denoted as h(t), we now have that h (t ) =
sin t
g (t ) , determine g (t ) . πt
7. A continuous-time signal x (t ) =cos(πt ) is sampled by the impulse train p (t ) =getting x p (t ) , where, T = 0.5s.
(1). Plot the Fourier transform X (j ω) of x (t ) . (2). Plot the Fourier transform X p (j ω) of x p (t ) .
⎧1,
(3). Let H (j ω) =⎨
⎩0,
4π≤≤8πotherwise
k =-∞
∑δ(t -kT )
∞
be the frequency response of an ideal bandpass filter. If
x p (t ) is applied to the filter, the output response is denoted as y(t), plot the Fourier transform
Y (j ω) of y(t).
(4). By observingY (j ω) , write out the expression of y (t ) .
8. Figure 8 illustrates a communication system, The Fourier transforms of the input and output signals x (t ) and y (t ) are denoted as X (j ω) and Y (j ω) , respectively. If x(t) = cos(0.5πt), determine y(t) and plot Y (j ω) .
x (t )
cos(3π
t )
cos(5πt )
9. Let X(jω) denote the Fourier transform of the signal x(t) depicted in Figure P4.25.
x 1(t )
(a). Find ∠X (j ω) .
(b). Find X (j 0) .
∞
1
t
(c). Evaluate
-∞
⎰X (j ω) d ω.
Figure P4.25.a.
∞
(d). Evaluate
-∞∞
⎰
X (j ω)
2sin ω
ω
2
e j 2ωd ω.
(e). Evaluate
-∞
⎰
X (j ω) d ω.
(f). Sketch the inverse Fourier transform of Re{ X(jω)}.
Note: You should perform all these calculations without explicitly evaluating X(jω). 10. Consider an LTI system whose response to the input
x (t ) =e -t +e -3t u (t )
is y (t ) =2e -t -2e -4t u (t ) (a). Find the frequency response of this system. (b). Determine the system’s impulse response.
(c). Find the differential equation relating the input and the output of this system.
sin t
11. Let g (t ) =x (t ) cos 2t *
πt
Assuming that x(t) is real and X(jω) = 0 for |ω| ≥ 1, show that there exists an LTI system S such that x(t)→g(t). (Note: find the relationship between x(t) and g(t))
S
()
()
第三部分:信号与系统的s 域分析
一、填空题
1. The ROC of the Laplace transform X (s ) =transform is X (s ) = ( ).
2. It is known that the LTI system described by H (s ) =H(s) is ( ).
3. The system function of a causal LTI system is H (s ) =
s +2
, then the differential equation
s 2+4s +3
11+ is Re{s }
1
is stable, then the ROC of
(s +2)(s +3)
describing the input-output relationship of the system is ( ). 4. The Laplace transform of signal e -2t u (t ) is ( ), the associated ROC is ( ). 5. Assume that the system function of an anticausal LTI system is H (s ) =
1
, then the frequency s -2
response H (j ω) = ( ), the unit impulse response h (t ) = ( ).
6. It is known that a causal continuous-time LTI system H(s) is stable, then all of the poles of H(s) are located in ( ).
7. It is known that the Laplace transform of signal x(t) is X (s ) =
x (t ) *δ(t -1) = ( ).
d 2y (t ) dy (t ) dx (t )
8. The differential equation +2+2y (t ) =+3x (t ) describes the input-output
dt dt dt 2
1
, Re{s }>-1, then s +1
relationship of an LTI system, then the system function is H(s) = ( ).
9. The Laplace transform of a causal signal x(t) is denoted as X(s), then the Laplace transform of signal
t
⎰
-∞
x (τ-1) d τis ( ).
10. Given a continuous-time LTI system, it is known that the zero-state response to arbitrary input x(t) is x(t-t0), t0 > 0, then the system function of the system is H(s) = ( ). 11. The Laplace transform of a continuous-time signal x(t) is X (s ) =x(t) = ( ).
12. Figure 1.12 illustrates an LTI system, its system function is H(s) = ( ).
1
e -s , Re{s }>0, then
s (2s +1)
(t )
Figure 1.12
13. It is known that a causal continuous-time LTI system with system function H (s ) =stable, then the coefficient a must satisfy: ( ).
1
is s -a
14. Given two signals x 1(t ) =e -2t u (t ) and x 2(t ) =e -3t u (t ) , and y (t ) =x 1(t -2) *x 2(-t +3) , then the Laplace transform of y(t) is Y(s) = ( ).
15. A causal continuous-time LTI system has rational system function H(s). The sufficient and necessary condition for which the system is stable is that all of the poles of H(s) are located in the ( ) of the s-plane.
二、选择题
1. The Laplace transform of signal x (t ) =u (t ) -u (t -1) is ( ).
A. (1-e -s ) /s B. (1-e s ) /s C. s (1-e -s ) D. s (1-e s )
2. The system function H(s) of an LTI system has two poles at p1 = -3, p2 = -1, and one zero at z = -2 in the finite s-plane. It is known that H(0) = 2, then the system function is ( ).
A. H (s ) =
2(s +2) 2(s +3)
B. H (s ) =
(s +1)(s +3) (s +2)(s +1) 3(s +2) (s +2)
D. H (s ) =
(s +1)(s +3) (s +1)(s +3)
11
+, then the s +2s +3
C. H (s ) =
3. The Laplace transform of signal x (t ) =e -3t u (t ) -e -2t u (-t ) is X (s ) =ROC of X(s) is ( ).
A. Re{s }>-2 B. Re{s }>-3 C. -3
11
with its ROC: Re{s }>-1, then the inverse Laplace transform is x(t) +2
s +2(s +1)
= ( ).
A. e -t u (t ) +e -2t u (t ) B. te -t u (t ) +e -2t u (t )
C. e -t u (t ) +te -2t u (t ) D. e -t u (t ) +te -t u (t )
5. The system function of an LTI system is H (s ) =( ).
A. causal and stable C. anticausal and stable
B. causal and unstable D. anticausal and unstable
11+, then the s +2s +1
s +2
, Re{s }>-1, then the system is
s 2+4s +3
6. The Laplace transform of signal x (t ) =e -2t u (t ) -e -t u (-t ) is X (s ) =ROC of X(s) is ( ).
A. R {s }>-2 B. R {s }>-1 C. -2
A. e -t u (t ) +e -2t u (t ) B. te -t u (t ) +e -2t u (t ) C. e -t u (t ) +te -2t u (t ) D. e -t u (t ) +te -t u (t ) 8. The system function of a causal LTI system is H (s ) =
A. stable
C. critical stable (临界稳定)
B. unstable D. indeterminacy
s +6
, then the system is ( ).
s 2-5s -6
11
, Re{s }>-1, then impulse +2
s +1(s +1)
9. The unit step response of a continuous-time LTI system is s (t ) =(1+te -2t ) u (t ), then the system function of the system is H(s) = ( ).
A .1+
s
(s +2) 2
B .
1s + s (s +2) 2
111
++C . s s +2(s +2) 2
D .1+
1
(s +2) 2
1
, then x(t) = ( ). 2
(s +a )
10. The Laplace transform of a causal signal x(t) is X (s ) =
A. e -at u (t ) C. t 2e -at u (t )
B. te at u (t ) D. te -at u (t )
11. A continuous-time LTI system can be described by its system function H(s), and it is known that the frequency response H(jω) exists, then the system must be ( ).
A. time-invariant B. causal C. stable D. linear
12. Let y (t ) =u (t ) *(δ(t ) -δ(t -4)) , then the Laplace transform of y(t) is Y(s) = ( ).
111
A. Y (s ) =(1-e 4s ) B. Y (s ) =-
s s s +4111
C. Y (s ) =(1-e -4s ) D. Y (s ) =+
s s s +41-(s -2)
e , Re{s }>-2 be the Laplace transform of signal x(t), then ( ). 13. Let X (s ) =
s +2
A. x (t ) =e -2t u (t -1) B. x (t ) =e -2(t -2) u (t -1) C. x (t ) =e -2t u (t -2) D. x (t ) =e -2(t -1) u (t -1)
14. Let X(s) denote the Laplace transform of x(t), then the Laplace transform of signal x(2t - 5) is ( ).
1s -5s 1s 5s 1s 2s 1s -2s
A. X () e B. X () e C. X () e D. X () e
22222222
5
5
15. H(s) is the system function of an LTI system, then mathematical form of the unit impulse response h(t) can only be determined by ( ).
A. the zeros of H (s ) B. the poles of H (s )
C. the input signal D. the input signal and the poles of H (s ) 三、综合题(分析、计算)
1. Given a causal LTI system, if the input signal is x (t ) =e -t u (t ) , the output response is
111
y (t ) =(e -t -e -2t +e -4t ) u (t ) ,
326
(1). Determine the system function H(s); (2). Is the system stable? Why?
(3). If we apply the signal x (t ) =e -2t u (t ) to the system, determine the corresponding output response y(t).
2. Consider a causal LTI system described by the differential equation:
d 2y (t ) dy (t ) dx (t )
+3+2y (t ) =+3x (t ) 2
dt dt dt
(1). Determine the system function H(s); (2). Plot the pole-zero diagram. (3). Is the system stable? Why?
(4). If we apply the signal x (t ) =e -t u (t ) to the system, determine the corresponding output response y(t).
s
is the Laplace transform of a continuous-time signal x(t).
s 2+3s +2
Determine the inverse Laplace transforms for each of the following ROCs:
3. It is known that X (s ) =
(1). Re{s} > -1; (2). -2
s +2
be the Laplace transform of signal x(t). Determine all possible
s 2+7s +12
expressions of x(t) corresponding to X(s) using the partial-fraction expansion technique.
s +4dg (t )
5. It is known that the system function of an LTI system is H (s ) =2, and h (t ) =,
dt s +3s +2
where, h(t) and g(t) are the unit impulse and unit step responses of the system. Determine the unit step responses for each of the following ROCs of H(s).
4. Let X (s ) =
(1). Re{s} > -1; (2). -2
6. The block diagram of a causal LTI system is illustrated in Figure 3.6. (1). Determine the system function H(s); (2). Determine the unit impulse response h(t); (3). Plot the pole-zero diagram of the system; (4). Is the system stable? Why?
7. The block diagram of a causal LTI system is shown in Figure 3.7.
x (t (t )
Figure 3. 6
X (s Y (s )
(1). Determine the unit impulse response h(t);
(2). Determine the differential equation describing the input-output relationship of the system; (3). If the input signal isx (t ) =e u (t ) , determine the output response y(t); (4). Is the system stable? Why?
8. The input-output relationship of a causal LTI system is described by the differential equation:
d 2y (t ) dy (t ) -t
+5+6y (t ) =e u (t ) *x (t ) +x (t ) 2
dt dt
-3t
(1). Determine the system function H(s); (2). Determine the unit impulse response h(t); (3). Is the system stable? Why?
9. Suppose that we are given the following information about an LTI system: (1). The system is causal;
(2). The system function is rational and has only two poles, at s = -2 and s = 4; (3). If x(t) = 1, then y(t) = 0.
(4). The value of the impulse response at t = 0+ is 4.
Determine the system function H(s), the unit impulse response h(t), interpret the stability of the system.
10. Using geometric evaluation of the magnitude of the Fourier transform from the corresponding pole-zero plot, determine, for each of the following Laplace transforms, whether the magnitude of the corresponding Fourier transform is approximately lowpass, highpass, or bandpass: (1). H 1(s ) =
1
,
(s +1)(s +3)
Re{s }>-1; 1
Re{s }>-;
2
-2
-1
j ω
σ 2
s
, (2). H 2(s ) =2
s +s +1
1
s 2
, (3). H 3(s ) =2
s +2s +1
Re{s }>-1
Figure 3.11
11. Consider an LTI system for which the system function H(s) has the pole-zero pattern shown in Figure 3.11.
(1). Indicate all possible ROCs that can be associated with this pole-zero pattern;
(2). For each ROC identified in part (1), specify whether the associated system is stable and/or causal.
12. (a) Determine the differential equation relating vi (t) and vo (t) for the RLC circuit of Figure 3.12. (b) Suppose that vi (t) = e-3t u(t). Using the Laplace transform, determine vo (t) for t > 0.
(t )
Figure 3.12 RLC circuit