ω

4
π

2
π
0
2
π
4
π
X
f
(
ω
)
1
140
The signal
x
(
t
) is used as the input to a continuoustime LTI system having the frequency response
H
f
(
ω
) shown.
ω

4
π

2
π
0
2
π
4
π
H
f
(
ω
)
1
1
Accurately sketch the spectrum
Y
f
(
ω
) of the output signal.
3.1.16 Two continuoustime LTI system are used in cascade. Their impulse responses are
h
1
(
t
) = sinc(3
t
)
h
2
(
t
) = sinc(5
t
)
.

h
1
(
t
)

h
2
(
t
)

Find the impulse response and sketch the frequency response of the total system.
3.1.17 Consider a continuoustime LTI system with the impulse response
h
(
t
) = 3 sinc(3
t
)
.
(a) Accurately sketch the frequency response
H
f
(
ω
).
(b) Find the output signal
y
(
t
) produced by the input signal
x
(
t
) = 2 + 5 cos(
πt
) + 7 cos(4
πt
)
.
(c) Consider a second continuoustime LTI system with impulse response
g
(
t
) =
h
(
t

2) where
h
(
t
) is as above.
For this second system, find the output signal
y
(
t
) produced by the input signal
x
(
t
) = 2 + 5 cos(
πt
) + 7 cos(4
πt
)
.
3.1.18 The signal
x
(
t
) is given the product of two sine functions,
x
(
t
) = sin(
π t
)
·
sin(2
π t
)
.
Find the Fourier transform
X
f
(
ω
).
3.1.19 A continuoustime signal
x
(
t
) has the spectrum
X
f
(
ω
),
ω

4
π

2
π
0
2
π
4
π
X
f
(
ω
)
1
(a) The signal
g
(
t
) is defined as
g
(
t
) =
x
(
t
) cos(4
πt
)
.
Accurately sketch the Fourier transform of
g
(
t
).
(b) The signal
f
(
t
) is defined as
f
(
t
) =
x
(
t
) cos(
πt
)
.
Accurately sketch the Fourier transform of
f
(
t
).
141
3.1.20 The signal
x
(
t
):
6
4
2
0
2
4
6
0
0.5
1
1.5
2
t
x(t)
has the Fourier transform
X
f
(
ω
):
3π
2π
π
0
π
2π
3π
0
1
2
3
4
5
ω
X
f
(
ω
)
Accurately sketch the signal
g
(
t
) that has the spectrum
G
f
(
ω
):
3π
2π
π
0
π
2π
3π
0
1
2
3
4
5
ω
G
f
(
ω
)
Note that the spectrum
G
f
(
ω
) is a sum of left and rightshifted copies of
X
f
(
ω
). Specifically,
G
f
(
ω
) =
X
f
(
ω

2
π
) +
X
f
(
ω
+ 2
π
)
.
In your sketch of the signal
g
(
t
) indicate its zerocrossings. Show and explain your work.
3.1.21 The lefthand column below shows four continuoustime signals. The Fourier transform of each signal appears
in the righthand column in mixedup order. Match the signal to its Fourier transform.
Signal
Fourier transform
1
2
3
4
142
5
0
5
0
0.5
1
1.5
t
x(t)
#1
5
0
5
1
0
1
2
ω
/
π
X
f
(
ω
)
C
5
0
5
0
0.5
1
1.5
t
x(t)
#2
5
0
5
0
2
4
ω
/
π
X
f
(
ω
)
D
5
0
5
0.5
0
0.5
1
1.5
t
x(t)
#3
5
0
5
0
0.5
1
1.5
ω
/
π
X
f
(
ω
)
B
5
0
5
1
0
1
2
t
x(t)
#4
5
0
5
0
0.5
1
1.5
ω
/
π
X
f
(
ω
)
A
143
3.1.22 The ideal continuoustime highpass filter has the frequency response
H
f
(
ω
) =
0
,

ω
 ≤
ω
c
1
,

ω

> ω
c
.
Find the impulse response
h
(
t
).
3.1.23 The ideal continuoustime bandstop filter has the frequency response
H
f
(
ω
) =
1

ω
 ≤
ω
1
0
ω
1
<

ω

< ω
2
1

ω
 ≥
ω
2
.
(a) Sketch
H
f
(
ω
).
(b) What is the impulse response
h
(
t
) of the ideal bandstop filter?
(c) Describe how to implement the ideal bandstop filter using only lowpass and highpass filters.
3.1.24 The ideal continuoustime bandpass filter has the frequency response
H
f
(
ω
) =
0

ω
 ≤
ω
1
1
ω
1
<

ω

< ω
2
0

ω
 ≥
ω
2
.
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 Fall '13
 Ivan
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Inverse Systems