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ANALOG MODULATION
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The sources may be either
analog where the messages produced by the source are continuous in time and in
amplitude or digital that is discrete in time and has a finite number of
possible outputs.
We are now concerned with transmission of analog signals. The source output (i.e., the message signal) is denoted by m(t). We assume that the analog signal, m(t), is lowpass, and its bandwidth is W Hz. The message signal is modulated with a carrier signal of the form
_files/image002.gif){kind=small}
where _files/image004.gif){kind=small}
is the carrier
amplitude, _files/image006.gif){kind=small}
is the carrier
frequency, and _files/image008.gif){kind=small}
is the carrier phase.
Modulation converts the message signal m(t) from lowpass to bandpass, around
the center frequency _files/image009.gif){kind=small}
. The objective of using modulation is to achieve one or more
of the following;
(1)
to make the spectrum of the transmitted bandpass signal
match the passband characteristics of the channel because the channel may not
be suitable for transmitting the source output directly.
(2)
to accommodate simultaneous transmission of different
messages over the same channel.
(3)
to provide better noise immunity over a noisy channel by
expanding the bandwidth of the transmitted signal.
The third objective is met by
employing angle modulation.
In the
following labs, we consider the transmission and reception of analog signals by
amplitude modulation (AM), frequency modulation (FM) and phase modulation (PM).
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LAB 1. AMPLITUDE
MODULATION (AM)
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Objective
In this part, our objective is to
study AMPLITUDE MODULATION in detail. We will consider different methods of
amplitude modulation: double-sideband suppressed carrier (DSB-SC) AM,
conventional double-sideband AM, and single-sideband AM.
Introduction
In amplitude modulation (AM), the message signal m(t)
is impressed on the amplitude of the carrier signal c(t) using several
different methods resulting in double-sideband suppressed carrier AM,
conventional double-sideband AM, and single-sideband AM.
1-1 DSB-SC AMPLITUDE
MODULATION
A general sinusoidal carrier can
be represented mathematically as
_files/image011.gif){kind=small}
where _files/image012.gif){kind=small}
is known as the
carrier frequency. Since a sinusoid is completely specified by its amplitude,
phase, and frequency, it follows that once the carrier frequency is specified,
only two parameters are to be varied in the modulation process: the amplitude _files/image013.gif){kind=small}
and the phase _files/image015.gif){kind=small}
.
- Double
Sideband Suppressed Carrier AM (DSB-SC AM)
Notation:
m(t) : Message
signal.
_files/image017.gif){kind=small}
: Carrier signal
_files/image019.gif){kind=small}
: Modulated signal
*) Modulated signal:
_files/image021.gif){kind=small}
_files/image022.gif){kind=small}
Then,
_files/image024.gif){kind=small}
_files/image026.gif){kind=small}
By Fourier-transforming the
spectrum of modulated signal,
_files/image027.gif){kind=small}
_files/image029.gif){kind=small}
_files/image031.gif){kind=small}
If we
assume that the phase of carrier signal is zero, we obtain:
_files/image033.gif){kind=small}
Thus, the required bandwidth of the modulated signal
is equal to twice the bandwidth of the message signal as it is clear from the
expression.
EXAMPLE:
Assume m(t) is such
that:
|
|
|
_files/image034.gif)
_files/image035.gif)
=>_files/image037.gif){kind=small}
since _files/image039.gif){kind=small}
> 2W
|
|
_files/image040.gif)
=>_files/image042.gif){kind=small}
_files/image044.gif){kind=small}
|
|
_files/image046.gif)
If channel attenuation and noise
are ignored, the received signal can be represented as follows
_files/image048.gif){kind=small}
The receiver is composed of a PLL (Phase-Locked Loop)
and a LPF (Lowpass Filter).
|
|
_files/image050.gif)
_files/image052.gif){kind=small}
_files/image054.gif){kind=small}
At the output of LPF:
_files/image056.gif){kind=small}
When we assume that PLL works perfectly, the output
of the receiver would be
_files/image058.gif){kind=small}
.
Therefore the message signal is demodulated.
1-2
CONVENTIONAL
AMPLITUDE MODULATION
A conventional AM signal consists of a large carrier
component in addition to the DSB AM modulated signal. The transmitted signal is
expressed mathematically as
_files/image060.gif){kind=small}
_files/image062.gif){kind=small}
where _files/image064.gif){kind=small}
,
_files/image066.gif){kind=small}
: modulation index. _files/image068.gif){kind=small}
is chosen.
If _files/image069.gif){kind=small}
> 1, overmodulated. _files/image071.gif){kind=small}
.
Similar to the DSB-SC AM case, the
spectrum of a conventional AM signal occupies a bandwidth twice of the
bandwidth of the message signal.
We can demodulate the modulating
signal simply at the receiver by using an envelope detector. Envelope detector
consists of a diode and RC circuit. It is essentially a lowpass filter which charges
a capacitor to the peak amplitude of the incoming AM modulated signal as long
as the output of the envelope detector is larger than the input of the envelope
detector. When the amplitude of the incoming AM signal decreases, the capacitor
discharges. The major advantage of an envelope detector is very simple,
therefore cheap.
There is another demodulation
method that recovers the carrier signal from the received signal and uses
coherent demodulation. The modulated signal can be represented as follows
_files/image073.gif){kind=small}
.
When a narrow-bandpass filter that
is centered at _files/image074.gif){kind=small}
is used at the
receiver, the carrier signal can be recovered (i.e., _files/image076.gif){kind=small}
can be obtained from
the received signal). We then multiply the received signal with the carrier
signal and lowpass filter to demodulate the message signal (similar to the case
of DSB-SC AM).
1-3
SINGLE SIDEBAND
AMPLITUDE MODULATION
Single sideband amplitude
modulation scheme is similar to the double sideband-suppressed carrier AM
scheme. The spectrum of DSB-SC AM modulated signal has the scaled conjugate
symmetric spectrums of the message signal on both positive and negative
frequency ranges. The idea of single sideband AM is that only one side
spectrums on either the upper or the lower sideband from the carrier signal can
be used to demodulate the message signal. Only the difference between two
schemes is SSB (Single sideband) AM modulated signal requires half the
bandwidth of DSB-SC AM modulated signal. Therefore it increases the bandwidth
efficiency compared to DSB-SC AM on the bandwidth limited channel.
There exist two types of SSB-AM, USB-AM and LSB-AM, depending on either the upper or the lower sideband. The modulated signal is represented by
_files/image078.gif){kind=small}
where the minus sign determines
USB-AM and the plus sign determines LSB-AM. _files/image080.gif){kind=small}
is the Hilbert
transform of m(t), denoted by _files/image082.gif){kind=small}
.
In the frequency domain we have
_files/image084.gif){kind=small}
_files/image086.gif){kind=small}
otherwise
and
_files/image088.gif){kind=small}
_files/image089.gif){kind=small}
otherwise.
The
bandwidth of the SSB signal is half the bandwidth of the DSB and conventional
AM and it is equal to the bandwidth of the message signal.
The
demodulation is done in exactly the same way as DSB-SC AM, i.e., by a coherent
demodulator.
Problem 1-1. DSB-SC AM with a rectangular message
signal.
Let m(t)
be a rectangular signal having an amplitude of 1 and having a pulse-duration of
50ms. The carrier signal frequency is 3kHz.
Assume that the phase of the carrier signal is zero and the amplitude of the carrier signal is 1.
[Hint] In Analog Comm. on the drop-down menu, Cont. Sig. Gen. block is followed by modulation blocks.
a)
Show the spectrum of the message signal, M(f).
b) Show the spectrum of the DSB-SC AM modulated signal, U(f).
Problem 1-2. DSB-SC AM with a sinc function as a message
signal.
Let m(t) =
sinc(200t) for _files/image091.gif){kind=small}
= 0 for otherwise.
Assume that the phase of carrier signal is
zero and the amplitude of the carrier signal is 1, and the carrier signal
frequency is 1 kHz.
a)
Show the spectrum of the message signal, M(f).
b) Show the DSB-SC AM modulated signal both in time domain and in frequency domain.
Problem 1-3. DSB-SC AM with a sinusoidal message
signal having a single frequency component.
Let
_files/image093.gif){kind=small}
for _files/image095.gif){kind=small}
and 0 for otherwise,
where _files/image097.gif){kind=small}
is 1kHz.
Carrier
frequency is 10kHz.
Assume that the phase of carrier signal is
zero and the amplitude of the carrier signal is 1.
a) Show
the spectrum of the message signal, M(f).
b) Show the DSB-SC AM modulated signal both in
time domain and in frequency domain.
Problem 1-4. Single sideband AM with a sinusoidal
message signal.
Let _files/image099.gif){kind=small}
for _files/image100.gif){kind=small}
and 0 for otherwise,
where _files/image101.gif){kind=small}
is 200Hz.
Carrier frequency is 3kHz.
Assume that the phase of carrier signal is zero and
the amplitude of the carrier signal is 1.
a)
Show the spectrum of the message signal, M(f).
b)
Show the USB-AM modulated signal both in time domain and in
frequency domain.
c)
Show the LSB-AM modulated signal both in time domain and in
frequency domain.
Problem 1-5. Conventional AM with a sinusoidal message
signal.
Let _files/image103.gif){kind=small}
for _files/image104.gif){kind=small}
and 0 for otherwise,
where _files/image105.gif){kind=small}
is 400Hz.
Carrier
frequency is 5kHz.
Assume that the phase of carrier signal is
zero and the amplitude of the carrier signal is 1.
a) Find
the modulation index, a, mathematically.
b) Show the conventional AM modulated signal
both in time domain and in frequency domain.
Problem 1-6. Over modulated signal.
Let _files/image107.gif){kind=small}
for _files/image108.gif){kind=small}
and 0 for otherwise,
where _files/image109.gif){kind=small}
is 200Hz.
Carrier frequency is 10kHz.
Assume that the phase of carrier signal is zero and
the amplitude of the carrier signal is 1.
a) Find the
modulation index, a, mathematically.
b)
When A is equal to 0.5, show the modulated signal in time
domain with your modulation index used.
c)
When A is equal to 2, show the modulated signal in time
domain with your modulation index used.
d) Explain the effect of overmodulation.
Problem 1-7. Demodulation of DSB-SC AM modulated
signal.
Let us
consider signals in Problem 1-2._files/image111.gif){kind=small}
Assume
that there is no noise.
[Hint] After connecting Channel block at
the end of modulation block, Channel block is followed by Coherent
Detection block. And Demod. block is
connected at the end of Coherent Detection block.
a)
Show the demodulated message signal, m’(t) with your
specific bandwidth of LPF.
b)
Show both the spectrum of the demodulated message signal, M’(f),
and the demodulated message signal, m’(t) at several bandwidths of LPF, such as
_files/image113.gif){kind=small}
, where W is the bandwidth of the message signal.
c)
Find the minimum required bandwidth of LPF for reasonable
resolution of the demodulated signal.
Problem 1-8. Demodulation of DSB-SC AM modulated
signal (2).
Let us
consider signals in Problem 1-3.
Assume
that there is no noise.
a)
Show the demodulated message signal, m’(t).
b)
Show the spectrum of the demodulated message signal, M’(f) at
several bandwidths of LPF, such as _files/image115.gif){kind=small}
and_files/image117.gif){kind=small}
.
c)
Find the minimum required bandwidth of LPF for reasonable
resolution of the demodulated signal.
Problem 1-9. Demodulation of SSB-AM modulated signal.
Let us consider
signals in Problem 1-4.
Assume
that there is no noise.
[Hint] After
connecting Channel block at the end of modulation block, Channel block is
followed by Coherent Detection block. And Demod. block is
connected at the end of Coherent Detection block.
a)
Show the demodulated message signal, m’(t) with your
specific bandwidth of LPF. Assumed that USB is used.
b)
Show both the spectrum of the demodulated message signal, M’(f),
and the demodulated signal, m’(t) at several bandwidths of LPF, such as _files/image119.gif){kind=small}
and_files/image120.gif){kind=small}
.
c)
Find the minimum required bandwidth of LPF for reasonable
resolution of the demodulated signal.
Problem 1-10. Envelope Detector for Conventional AM
modulated signal.
Let us
consider signals in Problem 1-6.
Let us
replace the time duration with _files/image122.gif){kind=small}
and max|m(t)| is equal to 1.
Assume
that there is no noise.
[Hint] After connecting Channel block at
the end of modulation block, Channel block is followed by Envelope
Detection block.
a)
Show both the demodulated message signal, m’(t) and the
spectrum of the demodulated signal, M’(f) at several time constant values
between _files/image124.gif){kind=small}
and_files/image126.gif){kind=small}
.
b)
Explain the relationship between the resolution of the
demodulated signal and a time constant.
Quiz.
1.
Let the DSB-SC AM modulated signal be u(t) and let the
message signal be m(t).
Show the relationship between U(f)
and M(f) mathematically.
2.
In DSB-SC AM, assume the modulating signal, m(t), is a
sinusoidal signal. Show the plots of U(f) on the cases of both _files/image128.gif){kind=small}
and _files/image130.gif){kind=small}
with J-DSP and
explain the results.
3.
How can we recover m(t) from the DSB-SC AM modulated signal,
u(t) ?
And determine the demodulated
signal.
Assumed that the phase of the carrier
at the receiver is known as _files/image132.gif){kind=small}
by PLL and r(t) is
equal to u(t) in the absence of noise.
Show a block diagram if necessary.
4.
What is the bandwidth of the DSB-SC AM modulated signal?
Assumed W is equal to the bandwidth of the message signal.
1)
W/2
2)
W
3)
3W/2
4)
2W
5.
When we look at the spectral characteristics of the
conventional AM modulated signal, explain the difference of the conventional AM
with DSB-SC AM.
6.
How can we recover m(t) from the conventional AM modulated
signal, u(t) ?
Assumed that envelope detector is
used to demodulate the message signal in the absence of noise.
7.
How can we recover m(t) from the conventional AM modulated
signal, u(t), with a pilot tone?
Assumed that noise is absence in
channel.
Show a block diagram if necessary.
8. What is the bandwidth of the conventional AM
modulated signal? Assumed W is equal to the bandwidth of the message signal.
1)
W/2
2)
W
3)
3W/2
4)
2W
9. Let the USSB-AM modulated signal be u(t) and
let the message signal be m(t).
Show the relationship between U(f)
and M(f) mathematically.
10. What is the bandwidth of the SSB-AM modulated
signal? Assumed W is equal to the bandwidth of the message signal.
1) W/2
2)
W
3)
3W/2
4)
2W