Problems 151

PROBLEMS

3.1 The message signal m (t) = 2 cos 400t + 4 sin(500t + } ) modulates the carrier sig­
nal c(t) = cos(8000nt), using DSB amplitude modulation. Find the time-domain
A
and frequency-domain representations of '1:he modulated signal and plot the spec­
trum (Fourier transform) of the modulated signal. What is the power content of the
modulated signal?

M In a DSB system, the carrier2 is c(t) = A cos 2nfct and the message signal is given
V by m (t) = sinc(t) + sinc (t) . Find the frequency-domain representation and the
bandwidth of the modulated signal.

3.3 The two signals (a) and (b), shown in Figure P-3.3, DSB modulate a carrier signal
A
c(t) = cos 2n
Jot. Precisely plot the resulting modulated signals as a function of
time and discuss their differences and similarities.

mi(t)

1 2
(a) (b)

Figure P-3.3

2nfct
1 x 2 (t) . Determine and sketch the spectrum of y (t) when
3.4 Suppose the signal x (t) = m (t) + cos is applied to a nonlinear system whose
output is y (t) = x (t) +
M (f) is as shown in Figure P-3.4 and W « fc.
M ( f)

-w w
f
Figure P-3.4

�-
f.;

!
i
1 52 Amplitude Modulation Chapter 3

3.5 The modulating signal

m (t) = 2 cos 4000nt + 5 cos 6000nt

is multiplied by the carrier

c (t) = 100 cos 2nlet,

where le = 50 kHz. Determine and sketch the spectrum of the DSB signal.

3.6 A DSB-modulated signal u (t) = Am(t) cos 2nlct is mixed (multiplied) with a local
carrier XL(t) = cos(2nlet + 8), and the output is passed through a lowpass filter
with a bandwidth equal to the bandwidth of the message m (t). The signal power at
the output of the lowpass filter is denoted by Pout· The modulated signal power is
denoted by Pu. Plot 1};;1
as a function of e for 0 :S e :S n .

3 .7 An AM signal has the form

u (t) = [20 + 2 cos 3000nt + 10 cos 6000nt] cos 2nlet,

where le = 105 Hz.

1. Sketch the (voltage) spectrum of u (t).

2. Determine the power in each of the frequency components.
3. Determine the modulation index.
4. Determine the sidebands' power, the total power, and the ratio of the sidebands '
power to the total power.

3.8 A message signal m (t) = cos 2000nt + 2 cos 4000nt modulates the carrier c (t) =
100 cos 2nlet, where le = 1 MHz to produce the DSB signal m (t) c (t) .

1. Determine the expression for the upper-sideband (USB) signal.

2. Determine and sketch the spectrum of the USB signal.

3.9 A DSB-SC signal is generated by multiplying the message signal m (t) with the
periodic rectangular waveform (shown in Figure P-3.9), then filtering the product
with a bandpass filter tuned to the reciprocal of the period Tp , with the bandwidth
2W, where W is the bandwidth of the message signal. Demonstrate that the output
u (t) of the bandpass filter (BPF) is the desired DSB-SC AM signal

u (t) = m (t) sin 2nlet,

where le = 1 / Tp .
3.10 Show that while generating a DSB-SC signal in Problem 3 .9, it is not necessary for
the periodic signal to be rectangular. This means that any periodic signal with the
period Tp can substitute for the rectangular signal in Figure P-3.9.

3.11 The message signal m(t) has the Fourier transform shown in Figure P-3. l l (a). This
signal is applied to the system shown in Figure P-3. 1 l (b) to generate the signal y(t).
1. Plot Y (f), the Fourier transform of y(t).
2. Show that if y(t) is transmitted, the receiver can pass it through a replica of
the system shown in Figure P-3 . l l (b) to obtain m(t)
back. This means that this
system can be used as a simple scrambler to enhance communication privacy.

M( f)

- w +W f

(a)

HPF LPF
!cutoff = fc [ -W, W]

A cos 2rrfct A cos2rr(fc + W)t

(b) Figure P-3.11

1 54 Amplitude Modulation Chapter 3

3.12 Show that in a DSB-modulated signal, the envelope of the resulting bandpass signal
is proportional to the absolute value
of the message signal. This means that an enve­
lope detector can be employed as a DSB demodulator if we know that the message
signal is always positive.

3.13 An AM signal is generated by modulating the carrier le = 800 kHz by the signal

m (t) = sin 2000nt + 5 cos 4000nt.

The AM signal
u(t) = 100 [ l + m (t)] cos 2nlct
is fed to a 50-Q load.

1. Determine and sketch the spectrum of the AM signal.

2. Determine the average power in the carrier and in the sidebands.
3. What is the modulation index?
4. What is the peak power delivered to the load?

3.14 The output signal from an AM modulator is

u (t) = 5 cos 1 800nt + 20 cos 2000nt + 5 cos 2200nt.

1. Determine the modulating signal m (t) and the carrier c (t).
2. Determine the modulation index.
3. Determine the ratio of the power in the sidebands to the power in the carrier.

3.15 A DSB-SC AM signal is modulated by the signal

m (t) = 2 cos 2000nt + cos 6000nt.

The modulated signal is

u(t) = l OOm (t) cos 2nlct,

where le = 1 MHz.

1. Determine and sketch the spectrum of the AM signal.

2. Determine the average power in the frequency components.

3.16 An SSB-AM signal is generated by modulating an 800 kHz carrier by the signal
m (t) = cos 2000nt + 2 sin 2000nt. The amplitude of the carrier is Ac = 100.
1. Determine the signal m (t).
2. Determine the (time-domain) expression for the lower sideband of the SSB-AM
signal.
3. Determine the magnitude spectrum of the lower-sideband-SSE signal.
Problems 1 55

3.17 Weaver's SSB modulator is illustrated in Figure P-3. 17. By taking the input signal as
m (t) = cos 2nfmt where fm < W, demonstrate that by proper choice of Ji and Ji,
the output is an SSB signal.

LPF
BW = W

m(t) + SSB
90° 90° + 1--signal
---;11o­

LPF
BW = W

Figure P-3.17

3.18 The message signal m(t),

whose spectrum is shown in Figure P-3. 18, is passed
through the system shown in that figure.

cos (2rcf0t)

x2(t)
Square law Yi (t) =
i t)
Bandpass Lowpass __Y_.,.
device filter filter

cos (2rcf0t) M (f)

-w w f

Figure P-3.18

2
The bandpass filter has a bandwidth of W centered at f0, and the lowpass filter has
a bandwidth of W. Plot the spectra of the signals x(t), y1 (t), y2 (t), y3(t), and y4 (t).
What are the bandwidths o f these signals?
1 56 Amplitude Modulation Chapter 3

3.19 The system shown in Figure P-3 . 1 9 is used to generate an AM signal. The modulat­
ing signal m (t)
has zero mean and its maximum (absolute) value is Am = max I m (t) I .
The nonlinear device has the input-output characteristic

y (t) = ax (t) + bx 1 (t) .

1. Express y (t ) in terms of the modulating signal m (t) and the carrier c (t) =
cos 2 nfct .
2. What is the modulation index?
3. Specify the filter characteristics that yield an AM signal at its output.

Nonlinear y(t) u(t)

Linear
memoryless
filter AM signal
system

c(t) = cos (2nfot) Figure P-3.19

3.20 The signal m(t), whose Fourier transform M (f) is shown in Figure P-3.20, is to be
transmitted from point A to point B. We know that the signal is normalized, meaning
mt
that - 1 :::; ( ) :::; 1 .

M ( f)

-10,000 10,000 f
Figure P-3.20

1. If USSB is employed, what is the bandwidth of the modulated signal?

2. If DSB is employed, what is the bandwidth of the modulated signal?
3. If an AM scheme with a = 0.8 is used, what is the bandwidth of the modulated
signal?

3.21 A vestigial-sideband modulation system is shown in Figure P-3.2 1 . The bandwidth

of the message signal m (t) is W, and the transfer function of the bandpass filter is
shown in the figure.
Problems 1 57

1. Determine h1(t),
which is the lowpass equivalent of
the impulse response of t�e bandpass filter.
h(t), h(t)
where represents

2. Derive an expression for the modulated signal u(t).

VSB signal
BPF
H( f ) u (t)

H (f)
1

Figure P-3.21

3.22 Find expressions for the in-phase and quadrature components, Xc(t) Xs(t),
and as
well as the envelope and phase,
USSB, and lower SSB (LSSB).
V(t) E>(t),and for DSB, SSB, conventional AM ,

3.23 The normalized signal m n (t) has a bandwidth of 10,000 Hz, and its power content is
0.5 Watts. The carrier A cos 2nfot has a power content of 200 Watts.
1. If mn (t)
modulates the carrier using SSB amplitude modulation, what will be
the bandwidth and the power content of the modulated signal?
.
.

2. If the modulation scheme is DSB SC, what is the answer to Part 1 ?

3. If the modulation scheme is AM with a modulation index of 0.6, what is the
answer to Part 1 ?

3.24 We wish to transmit 60 voice-band signals by SSB (upper-sideband) modulation and

frequency-division multiplexing (FDM). Each of the 60 signals has a spectrum as
shown in Figure P-3.24. Note that the voiceband signal is band limited to 3 kHz.
If each signal is frequency translated separately, we require a frequency synthesizer
that produces 60 carrier frequencies to perform the FDM. On the other hand, if we
subdivide the channels into L groups of K subchannels each, such that LK = 60,
we may reduce the number of frequencies from the synthesizer to L + K.

1. Illustrate the spectrum of the SSB signals in a group of K subchannels. Assume

that a 1 kHz guard band separates the signals in adjacent frequency subchannels
and that the carrier frequencies are fci = 1 0 kHz, = 14 kHz, . . . , etc.
fc2