those labeled T require a plan of attack, those labeled C usually have more than one
defensible answer.
1. E Show that if two discrete-time systems with frequency responses H1(ω) and
H2(ω) are connected in cascade, that the DTFT of the output is given by Y (ω) =
H1(ω)H2(ω)X(ω), where X(ω) is the DTFT of the input.
2. E This exercise verifies some of the relations in table 10.1.
(a) Let x be given by
∀ t ∈ R, x(t) = eiω0t,
where ω0 = 0. Use (10.2) to verify that its Fourier series coefficients are
∀
1
if m = 1
m ∈ Z,
Xm =
0
otherwise
(b) Let x be given by
∀ t ∈ R, x(t) = cos(ω0t),
where ω0 = 0. Use part (a) and properties of the Fourier series to verify that
its Fourier series coefficients are
∀
1/2 if |m| = 1
m ∈ Z,
Xm =
0
otherwise
(c) Let x be given by
∀ t ∈ R, x(t) = sin(ω0t),
where ω0 = 0. Use part (a) and properties of the Fourier series to verify that
its Fourier series coefficients are
1/2i
if m = 1
∀ m ∈ Z, Xm =
−1/2i if m = −1
0
otherwise
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10. THE FOUR FOURIER TRANSFORMS
x( t)
1
t
! p
! T T
p
Figure 10.17: A symmetric square wave.
(d) Let x be given by
∀ t ∈ R, x(t) = 1.
Use (10.2) to verify that its Fourier series coefficients are
∀
1
if m = 0
m ∈ Z,
Xm =
0
otherwise
Notice that the answer does not depend on your choice for p.
3. T Consider a symmetric square wave x ∈ ContPeriodicp, shown in Figure 10.17. It
is periodic with period p, and its fundamental frequency is ω0 = 2π/p radians/sec-
ond. Over one period, this is given by
∀
1
if t < T or t > p − T
t ∈ [0, p],
x(t) =
0
otherwise
(a) Show that its Fourier series coefficients are given by
∀
2T /p
if m = 0
m ∈ Z,
Xm =
sin(mω0T )/(mπ) otherwise
You can use l’Hôpital’s rule (see page 65) to verify that sin(x)/x = 1 when
x = 0, so this can be written more simply as
∀
sin(mω0T )
m ∈ Z,
Xm =
(10.31)
mπ
Note that this is real, as expected, since x is symmetric. Hint: The integration
formula (10.5) may be helpful.
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459
EXERCISES
x( n)
1
n
M
p
Figure 10.18: A symmetric discrete square wave.
(b) Let T = 0.5, and use Matlab to plot the Fourier series coefficients as a stem
plot (using the stem commmand) for m ranging from -20 to 20. Note that
you will have to be careful to avoid a divide by zero error at m = 0. Construct
plots for p = 2, p = 4, and p = 8.
4. E Let x be a periodic impulse train, given by
∞
∀ t ∈ R, x(t) = ∑ δ(t −np)
n=−∞
where p > 0 is the period. This signal has a Dirac delta function at all multiples of
p. Use (10.2) to verify that its Fourier series coefficients are
∀ m ∈ Z, Xm = 1/p.
Note that there is a subtlety in using (10.2) here. There are impulses at each end of
the integration interval. This subtlety can be avoided by observing that (10.2) can,
in fact, integrate over any interval that covers one period. Thus, it can equally well
be written
p/2
Z
∀
1
m ∈ Z,
Xm =
x(t)e−imω0t dt.
p−p/2
This simplifies the problem considerably.
5. E This exercise verifies some of the relations in table 10.2. In all cases, assume
that the frequency f is rational, and that it is related to the period p by f = m/p for
some integer m (see Section 7.6.1).
(a) Let x be given by
∀ n ∈ Z, x(n) = ei2πfn,
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10. THE FOUR FOURIER TRANSFORMS
where f = 0. Use (10.9) to verify that its Fourier series coefficients are
∀
p
if k ∈ {· · · m − 2p, m − p, m, m + p, m + 2p, · · · }
k ∈ Z,
X =
k
0
otherwise
(b) Let x be given by
∀ n ∈ Z, x(n) = cos(2π f n),
where f = 0. Use part (a) and properties of the DFT to verify that its Fourier
series coefficients are
p/2 if k ∈ {· · · m − 2p, m − p, m, m + p, m + 2p, · · · }
X =
p/
k
2
if k ∈ {· · · − m − 2p, −m − p, −m, −m + p, −m + 2p, · · · }
0
otherwise
(c) Let x be given by
∀ n ∈ Z, x(n) = sin(i2π f n),
where f = 0. Use part (a) and properties of the DFT to verify that its Fourier
series coefficients are
p/2i
if k ∈ {· · · m − 2p, m − p, m, m + p, m + 2p, · · · }
X =
−p/
k
2i
if k ∈ {· · · − m − 2p, −m − p, −m, −m + p, −m + 2p, · · · }
0
otherwise
(d) Let x be given by
∀ n ∈ Z, x(t) = 1.
Use (10.9) to verify that its Fourier series coefficients are
p
if k ∈ {· · · − 2p, −p, 0, p, 2p, · · · }
Xm =
0
otherwise
Notice that this result depends on your choice for p. This is a consequence of
the unfortunate scaling that is used for the DFT, vs. the discrete Fourier series
(DFS).
6. T Consider a symmetric discrete square wave x ∈ DiscPeriodicp, shown in Figure
10.18. It is periodic with period p, and its fundamental frequency is ω0 = 2π/p
radians/sample. Over one period, this is given by
∀
1
if n ≤ M or n ≥ p − M
n ∈ {0, 1, · · · , p − 1},
x(n) =
0
otherwise
where, in the figure, M = 2 and p = 8. For this problem, however, assume that M
and p can be any positive integers where p > 2M.
Lee & Varaiya, Signals and Systems
461
EXERCISES
(a) Show that the DFT is given by
∀
2M + 1,
if k is a multiple of p
k ∈ Z,
X =
k
sin(k(M + 0.5)ω0)/ sin(kω0/2),
otherwise
You can use l’Hôpital’s rule (see page 65) to verify that
sin(Kx)/ sin(x) = K
when x = 0, so this can be written more simply as
∀
sin(k(M + 0.5)ω0)
k ∈ Z,
X =
k
(10.32)
sin(kω0/2)
Note that this is real, as expected, since x is symmetric. Hint: The summation
identity formula (10.10) may be helpful.
(b) Let M = 2 and use Matlab to plot the Fourier series coefficients as a stem plot
(using the stem commmand) for m ranging from −p + 1 to p − 1. Note that
you will have to be careful to avoid a divide by zero error at k = 0. Construct
plots for p = 8, p = 16, and p = 32.
7. E Let x be a discrete periodic impulse train, given by
∞
∀ n ∈ Z, x(n) = ∑ δ(n−mp)
m=−∞
where p > 0 is the integer period. This signal has a Kronecker delta function at all
multiples of p. Use (10.9) to verify that its DFT coefficients are
∀ k ∈ Z, X =
k
1.
8. T Consider a symmetric discrete rectangle x ∈ DiscSignals, shown in Figure 10.19.
This is given by
∀
1
if |n| ≤ M
n ∈ Z,
x(n) =
0
otherwise
(a) Show that its DTFT is given by
∀
sin(ω(M + 0.5))
ω ∈ R,
X (ω) =
sin(ω/2)
Hint: The summation identity formula (10.10) may be helpful.
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10. THE FOUR FOURIER TRANSFORMS
x( n)
1
n
! M
M
Figure 10.19: A symmetric discrete rectangle signal.
(b) Let M = 3 and use Matlab to plot the DTFT with ω ranging from −π to pi.
9. E Consider x given by
∀ t ∈ R, x(t) = sin(ω0t).
Show that the CTFT is given by
∀ ω ∈ R, X(ω) = (π/i)δ(ω − ω0) − (π/i)δ(ω + ω0).
10. C In Section 10.6 we explored the relationship between the CTFT of a periodic
continuous-time signal and its Fourier series. In this problem we do the same for
discrete-time signals.
(a) Consider a discrete-time signal x with DTFT given by
∀ ω ∈ [−π,π], X(ω) = 2πδ(ω − ω0)
where ω0 = 2π/p for some integer p. The DTFT above is given over one
cycle only, but note that since it is a DTFT, it is periodic. Use the inverse
DTFT to determine x.
(b) Is x in part (a) periodic? If so, what is the period, and what is its DFT?
(c) More generally, consider a discrete-time signal x with DTFT given by
p/2
∀ ω ∈ [−π,π], X(ω) = 2π ∑ Xkδ(ω−kω0)
k=− p/2
where ω0 = 2π/p for some integer p. For simplicity, assume that p is odd,
and let p/2 be the largest integer less than p. Use the inverse DTFT to
determine x.
Lee & Varaiya, Signals and Systems
463
EXERCISES
(d) Is x in part (b) periodic? If so, what is the period, and what is its DFT? Give
the DTFT in terms of the DFT coefficients.
11. T Consider a continuous-time signal x with Fourier transform X . Let y be such that
∀ t ∈ R, y(t) = x(at),
for some real number a. Show that its Fourier transform Y is such that
∀
1
ω ∈ R,
Y (ω) = |a|X(ω/a).
12. E Consider the discrete-time signal y given by
∀ n ∈ Z, y(n) = sin(ω1n)x(n).
Show that the DTFT is
∀ ω ∈ R, Y(ω) = (X(ω − ω1) − X(ω + ω1))/2i,
where X = DTFT(x).
13. E In this exercise, you verify various properties of the Fourier series in table 10.6.
In all parts below, x is a periodic continuous-time signal with period p and funda-
mental frequency ω0 = 2π/p.
(a) Let y be given by
∀ t ∈ R, y(t) = x(t − τ)
for some real number τ. Show that the Fourier series coefficients of y are
given by
∀ m ∈ R, Ym = e−imω0τXm,
where Xm are the Fourier series coefficients of x.
(b) Let y be given by
∀ t ∈ R, y(t) = eiω1tx(t)
where ω1 = Mω0, for some M ∈ Z. Show that the Fourier series coefficients
of y are given by
∀ m ∈ Z, Ym = Xm−M,
where Xm are the Fourier series coefficients of x.
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10. THE FOUR FOURIER TRANSFORMS
(c) Let y be given by
∀ t ∈ R, y(t) = cos(ω1t)x(t)
where ω1 = Mω0, for some M ∈ Z. Show that the Fourier series coefficients
of y are given by
∀ m ∈ Z, Ym = (Xm−M + Xm+M)/2,
where Xm are the Fourier series coefficients of x.
(d) Let y be given by
∀ t ∈ R, y(t) = sin(ω1t)x(t)
where ω1 = Mω0, for some M ∈ Z. Show that the Fourier series coefficients
of y are given by
∀ m ∈ Z, Ym = (Xm−M − Xm+M)/2i,
where Xm are the Fourier series coefficients of x.
14. T Consider a discrete-time signal x with Fourier transform X . For each of the new
signals defined below, show that its Fourier transform is as shown.
(a) If y is such that
∀
x(n/N) if n is an integer multiple of N
n ∈ Z,
y(n) =
0
otherwise
for some integer N, then its Fourier transform Y is such that
∀ ω ∈ R, Y(ω) = X(ωN).
(b) If w is such that
∀ n ∈ Z, w(n) = x(n)eiαn,
for some real number α, then its Fourier transform W is such that
∀ ω ∈ R, W(ω) = X(ω − α).
(c) If z is such that
∀ n ∈ Z, z(n) = x(n)cos(αn),
for some real number α, then its Fourier transform Z is such that
Z(ω) = (X (ω − α) + X(ω + α))/2.
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EXERCISES
15. T Consider the FIR system described by the following block diagram:
x( n)
unit
x( n-1)
unit
x( n-2)
unit
x( n-3)
delay
delay
delay
b
b
0
1
b 2
b 3
y( n)
The notation here is the same as in Figure 9.15. Suppose that this system has
frequency response H. Define a new system with the identical structure as above,
except that each unit delay is replaced by a double delay (two cascaded unit delays).
Find the frequency response of that system in terms of H.
16. T This exercise discusses amplitude modulation or AM. AM is a technique that
is used to convert low frequency signals into high frequency signals for transmis-
sion over a radio channel. Conversion of the high frequency signal back to a low
frequency signal is called demodulation. The system structure is depicted below:
transmission medium (air)
x( t)
y( t)
y( t)
w( t)
z( t)
H(!)
cos (! c t)
cos (! c t)
The circular components multiply their input signals. The transmission medium
(air, for radio signals) is approximated here as a medium that passes the signal y
unaltered.
Suppose your AM radio station is allowed to transmit signals at a carrier frequency
of 740 kHz (this is the frequency of KCBS in San Francisco). Suppose you want to
send the audio signal x : R → R. The AM signal that you would transmit is given
by, for all t ∈ R,
y(t) = x(t) cos(ωct),
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10. THE FOUR FOURIER TRANSFORMS
where ωc = 2π × 740, 000 is the carrier frequency (in radians per second). Sup-
pose X (ω) is the Fourier transform of an audio signal with magnitude as shown
below:
| X(!)|
#
!
$2"10,000
0
2"10,000
(a) Show that the Fourier transform Y of y in terms of X is
Y (ω) = (X (ω − ωc) + X(ω + ωc))/2.
(b) Carefully sketch |Y (ω)| and note the important magnitudes and frequencies
on your sketch.
Note that if X (ω) = 0 for |ω| > 2π × 10, 000, then Y (ω) = 0 for ||ω| − |ωc|| >
2π × 10, 000. In words, if the signal x being modulated is bandlimited to less
than 10 kHz, then the modulated signal is bandlimited to frequencies that are
withing 10 kHz of the carrier frequency. Thus, an AM radio station only needs
to occupy 20 kHz of the radio spectrum in order to transmit audio signals up
to 10 kHz.
(c) At the receiver, the problem is to recover the audio signal x from y. One way
is to demodulate by multiplying y by a sinewave at the carrier frequency to
obtain the signal w, where
w(t) = y(t) cos(ωct).
What is the Fourier transform W of w in terms of X ? Sketch |W (ω)| and note
the important magnitudes and frequencies.
(d) After performing the demodulation of part (c), an AM receiver will filter the
received signal through a low-pass filter with frequency response H(ω) such
that H(ω) = 1 for |ω| ≤ 2π × 10, 000 and |H(ω)| = 0 for |ω| > 2π × 20, 000.
Let z be the filtered signal, as shown in the figure above. What is the Fourier
transform Z of z? What is the relationship between z and x?
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EXERCISES
17. T In the following parts, assume that x is a discrete-time signal given by
∀ n ∈ Z, x(n) = δ(n + 1) + δ(n) + δ(n − 1),
and that S is an LTI system with frequency response H given by
∀ ω ∈ R, H(ω) = e−iω.
(a) Find X = DTFT(x) and make a well-labeled sketch for ω ∈ [−π, π] in radian-
s/sample. Check that X is periodic with period 2π.
(b) Let y = S(x). Find Y = DTFT(y).
(c) Find y = S(x).
(d) Sketch x and y and comment on what the system S does.
18. T Consider a causal discrete-time LTI system S with input x and output y such that
∀ n ∈ Z, y(n) = x(n) + ay(n − 1)
where a ∈ R is a given constant such that |a| < 1.
(a) Find the impulse response h of S.
(b) Find the frequency response of S by letting the input be eiωn and the output be
H(ω)eiωn, and solving for H(ω).
(c) Use your results in parts (a) and (b) and the fact that the DTFT of an impulse
response is the frequency response to show that h given by
∀ n ∈ Z, h(n) = anu(n)
has the discrete-time Fourier transform H = DTFT(h) given by
1
H(ω) =
,
1 − ae−iω
where u(n) is the unit step, given by (2.16).
(d) Use Matlab to plot h(n) and |H(ω)| for a = −0.9 and a = 0.9. You may
choose the interval of n for your plot of h, but you should plot |H(ω)| in the
interval ω ∈ [−π, π]. Discuss the differences between these plots for the two
different values of a.
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10. THE FOUR FOURIER TRANSFORMS
19. T Suppose a discrete-time signal x has DTFT given by
X (ω) = i sin(Kω)
for some positive integer K. Note that X (ω) is periodic with period 2π, as it must
be to be a DTFT.
(a) Determine from the symmetry properties of X whether the time-domain signal
x is real.
(b) Find x. Hint: Use Euler’s relation and the linearity of the DTFT.
20. T Consider a periodic continuous-time signal x with period p and Fourier series
X : Z → C. Let y be another signal given by
y(t) = x(t − τ)
for some real constant τ. Find the Fourier series coefficients of y in terms of those
of X .
21. T Consider the continuous-time signal given by
sin(πt/T )
x(t) =
.
(πt/T )
Show that its CTFT is given by
T,
if |ω| ≤ π/T
X (ω) =
0,
if |ω| > π/T
The fact from calculus (10.5) may be useful.
22. T If x is a continuous-time signal with CTFT X , then we can define a new time-
domain function y such that
∀ t ∈ R, y(t) = X(t).
That is, the new time domain function has the same shape as the frequency domain
function X . Then the CTFT Y of y is given by
∀ ω ∈ R, Y(ω) = 2πx(−ω).
That is, the frequency domain of the new function has the shape of the time domain
of the old, but reversed and scaled by 2π. This property is called duality because it
shows that time and frequency are interchangeable. Show that the property is true.
Lee & Varaiya, Signals and Systems
469
EXERCISES
23. T Use the results of exercises 21 and 22 to show that a continuous time signal x given by
π/a,
if |t| ≤ a
x(t) =
0,
if |t| > a
where a is a positive real number, has CTFT X given by
sin(aω)
X (ω) = 2π
.
(aω)
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11
Sampling and Reconstruction
Contents
11.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
11.1.1 Sampling a sinusoid . . . . . . . . . . . . . . . . . . . . . . 472
Basics: Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
11.1.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
11.1.3 Perceived pitch experiment . . . . . . . . . . . . . . . . . . . 475
11.1.4 Avoiding aliasing ambiguities . . . . . . . . . . . . . . . . . 479
11.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
Probing Further: Anti-Aliasing for Fonts . . . . . . . . . . . . . . . . 480
11.2.1 A model for reconstruction . . . . . . . . . . . . . . . . . . . 481
11.3 The Nyquist-Shannon sampling theorem
. . . . . . . . . . . . . . 486
Probing Further: Sampling . . . . . . . . . . . . . . . . . . . . . . . 490
Probing Furt