Overview of Communication Topics
• Sinusoidal amplitude modulation
• Amplitude demodulation (synchronous and asynchronous)
• Double- and single-sideband AM modulation
• Pulse-amplitude modulation
• Pulse code modulation
• Frequency-division multiplexing
• Time-division multiplexing
• Narrowband frequency modulation
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
1
Handy Trigonometry Identities
J. McNames
cos(a + b)
=
cos(a) cos(b) − sin(a) sin(b)
sin(a + b)
=
sin(a) cos(b) + cos(a) sin(b)
cos(a) cos(b)
=
sin(a) sin(b)
=
sin(a) cos(b)
=
1
2
1
2
1
2
Portland State University
cos(a − b) +
cos(a − b) −
sin(a − b) +
ECE 223
1
2 cos(a + b)
1
2 cos(a + b)
1
2 sin(a + b)
Communications
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Introduction to Communication Systems
• Communications is a very active and large area of electrical
engineering
• Experienced a lot of growth through the nineties with the advent
of wireless cell phones and the internet
• Still an active area of research
• Fundamentals of signals and systems are essential to grasping
communications concepts
• The next two lectures will merely introduce some of the
fundamental concepts
• Will primarily focus on modulation and demodulation in
continuous-time
• Analogous concepts apply in discrete-time
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
3
Introduction to Amplitude Modulation
x(t)
y(t) = x(t) · c(t)
×
c(t)
y(t)
c(t) = cos(ωc t + θc )
• Modulation: the process of embedding an information-bearing
signal into a second signal
• Demodulation: extracting the information-bearing signal from
the second signal
• Sinusoidal Amplitude modulation: a sinusoidal carrier c(t) has
its amplitude modified by the information-bearing signal x(t)
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
4
Fourier Analysis of Sinusoidal Amplitude Modulation
For convenience, we will assume θc = 0.
c(t) = cos(ωc t)
C(jω) = π [δ(ω − ωc ) + δ(ω + ωc )]
y(t)
Y (jω)
X(jω) ∗ δ(ω − ωc )
Y (jω)
= x(t) · c(t)
=
1
2π
[X(jω) ∗ C(jω)]
= X (j(ω − ωc ))
= 12 X (j(ω − ωc )) + 12 X (j(ω + ωc ))
• Thus, sinusoidal AM shifts the baseband signal x(t) so that it is
centered at ±ωc
• Thus, x(t) can be recovered only if ωc > ωx so that the replicated
spectra don’t overlap
J. McNames
Portland State University
ECE 223
Communications
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5
Fourier Analysis of Sinusoidal Amplitude Modulation
X(jω)
1
−ωx
0
J. McNames
Portland State University
ω
ωc
0
Y (jω)
1
2
-ωc -ωx -ωc -ωc +ωx
ωx
C(jω)
π
-ωc
ω
ωc -ωx
0
ECE 223
ωc
ωc +ωx
Communications
ω
Ver. 1.11
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Example 1: Sinusoidal AM of a Random Signal
Example of Sinusoidal Amplitude Modulation
x(t)
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
c(t)
1
−3
x 10
0
−1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
−3
y(t)
0.2
x 10
0
−0.2
0
J. McNames
0.2
0.4
0.6
0.8
Portland State University
1
Time (s)
1.2
ECE 223
1.4
1.6
1.8
−3
x 10
Communications
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Example 1: MATLAB Code
function [] = AMTimeDomain();
close all;
N = 2000;
fc = 50e3;
fs = 1e6;
k
t
% No. samples
% Carrier frequency
% Sample rate
= 1:N;
= (k-1)/fs;
xh
[n,wn]
[b,a]
x
=
=
=
=
randn(1,N); % Random high-frequency signal
ellipord(0.01,0.02,0.5,60);
ellip(n,0.5,60,wn);
filter(b,a,xh); % Lowpass filter to create baseband signal
c = cos(2*pi*fc*t);
y = x.*c;
figure;
FigureSet(1,’LTX’);
subplot(3,1,1);
h = plot(t,x,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylim([-0.3 0.3]);
ylabel(’x(t)’);
title(’Example of Sinusoidal Amplitude Modulation’);
box off;
AxisLines;
subplot(3,1,2);
h = plot(t,c,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylim([-1.1 1.1]);
J. McNames
Portland State University
ECE 223
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ylabel(’c(t)’);
box off;
AxisLines;
subplot(3,1,3);
h = plot(t,y,’b’,t,x,’g’,t,-x,’r’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylim([-0.3 0.3]);
xlabel(’Time (s)’);
ylabel(’y(t)’);
box off;
AxisLines;
AxisSet(6);
print -depsc AMTimeDomain;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
9
Synchronous Sinusoidal Amplitude Demodulation
Transmitter
x(t)
×
c(t)
Receiver
y(t)
y(t)
×
Channel
w(t)
H(s)
x̂(t)
c(t)
y(t) = x(t) cos(ωc t)
w(t) = y(t) cos(ωc t)
= x(t) cos2 (ωc t)
1 1
= x(t) 2 + 2 cos(2ωc t)
= 12 x(t) + 12 x(t) cos(2ωc t)
• Synchronous demodulation assumptions
– The carrier c(t) is known exactly
– ω c > ωx
• The x(t) can be extracted by multiplying y(t) by the same carrier
and lowpass filtering the signal
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
10
Fourier Analysis of Sinusoidal AM Demodulation
Y (jω)
1
2
−ωc
C(jω)
π
−ωc
ω
ωc
0
W (jω)
1
2
−2ωc
ω
ωc
0
2ωc
0
H(jω)
2
ω
0
1
ω
R(jω)
ω
J. McNames
Portland State University
ECE 223
Communications
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11
Synchronous AM Demodulation Observations
Transmitter
x(t)
×
Receiver
y(t)
y(t)
×
Channel
c(t)
w(t)
H(s)
x̂(t)
c(t)
• The lowpass filter H(s) should have a passband gain of 2
• The transition band is very wide so the filter does not need to be
close to ideal (e.g. it can be low order)
• We learned how to design this type of filter in ECE 222
• We assumed the signal spectrum X(jω) was real
• The same ideas hold if X(jω) is complex
• Called synchronous demodulation because we assumed the
transmitter and receiver carrier signals c(t) were in phase
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
12
Synchronous AM Demodulation Carrier Phase Analysis
Suppose the transmitter and receiver carrier signals differ by a phase
shift:
cT (t)
cR (t)
=
=
cos(ωc t + θ)
cos(ωc t + φ)
w(t) = y(t) cR (t)
=
x(t) cT (t) cR (t)
= x(t) cos(ωc t + θ) cos(ωc t + φ)
1
1
= x(t) 2 cos(θ − φ) + 2 cos(2ωc t + θ + φ)
=
J. McNames
1
2 x(t) cos(θ
Portland State University
− φ) + 12 x(t) cos(2ωc t + θ + φ)
ECE 223
Communications
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Synchronous AM Demodulation Carrier Phase Comments
w(t) = 12 x(t) cos(θ − φ) + 12 x(t) cos(2ωc t + θ + φ)
• If θ = φ then we have the same case as before and we recover
x(t) exactly after a lowpass filter with a passband gain of 2
• If |θ − φ| =
π
2,
we lose the signal completely
• Otherwise, the received signal is attenuated
• The phase relationship of the oscillators must be maintained over
time
• This type of careful synchronization is difficult to maintain
• Phase-locked loops (PLL) can be used to solve this problem
• In ECE 323 you will design and build PLL’s
• The carrier frequency ωc of the transmitter and receiver must also
be the same and remain so over time
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
14
Introduction to Asynchronous AM Demodulation
• Asynchronous modulation does not require the carrier signal
c(t) be available in the receiver
• Thus, there is no need for synchronization
• Asynchronous Modulation Assumptions:
ωc ωx
x(t) > 0 for all t
• However, it does require that the baseband signal x(t) be positive
• In this case, the envelope of the modulated signal y(t) is
approximately the same as the baseband signal x(t)
• Thus, we can recover a good approximation of x(t) with an
envelope detector
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
15
Example 2: Asynchronous Amplitude Modulation
Example of Asynchronous Sinusoidal AM Modulation
0.4
x(t)
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
c(t)
1
1.8
−3
x 10
0
−1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
y(t)
−3
x 10
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
1.4
1.6
1.8
−3
x 10
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
16
Example 2: MATLAB Code
function [] = AAMTimeDomain();
close all;
N = 2000;
fc = 50e3;
fs = 1e6;
k
t
% No. samples
% Carrier frequency
% Sample rate
= 1:N;
= (k-1)/fs;
xh
[n,wn]
[b,a]
x
x
=
=
=
=
=
rand(1,N)-0.5;
% Random high-frequency signal limited to [-0.5 0.5]
ellipord(0.02,0.03,0.5,60);
ellip(n,0.5,60,wn);
filter(b,a,xh);
% Lowpass filter to create baseband signal
x + 0.2;
% Convert to positive signal
c = cos(2*pi*fc*t);
y = x.*c;
figure;
FigureSet(1,’LTX’);
subplot(3,1,1);
h = plot(t,x,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylim([-0.1 0.4]);
ylabel(’x(t)’);
title(’Example of Asynchronous Sinusoidal AM Modulation’);
box off;
AxisLines;
subplot(3,1,2);
h = plot(t,c,’r’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
J. McNames
Portland State University
ECE 223
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ylim([-1.1 1.1]);
ylabel(’c(t)’);
box off;
AxisLines;
subplot(3,1,3);
h = plot(t,y,’g’,t,x,’b’,t,-x,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylim([-0.39 0.39]);
xlabel(’Time (s)’);
ylabel(’y(t)’);
box off;
AxisLines;
AxisSet(8);
print -depsc AAMTimeDomain;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
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Diodes
I
+
Ideal Model
Real Model
I
I
V
-
0
0.7
V
0
0.7
V
• Diodes can be used as a key component in envelope detectors
• Like resistors, diodes do not have memory and the relationship
between voltage and current does not depend on time
• Key idea: diodes only allow current to flow in one direction
• When they are on (V ≥ 0.7 V), they act like an ideal voltage
source
• When they are off (V < 0.7 V), they act like an open circuit
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
19
Example 3: Full Wave Rectifier
vs
R
+
vo
-
Draw the equivalent circuits for vs > 0 and vs < 0 assuming an ideal
model of the diode with a threshold voltage of 0 V.
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
20
Example 3: Workspace
J. McNames
Portland State University
ECE 223
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21
Envelope Detectors
+
y(t)
-
i(t)
+
R e(t)
-
+
i(t)
+
y(t)
C
-
R
r(t)
-
• A diode can be used to extract the upper half of the modulated
signal
• This is called a half-wave rectifier
• Roughly speaking, when y(t) > 0, e(t) ≈ y(t)
• By connecting a capacitor in parallel with the resistor, the RC
circuit acts like a first-order lowpass filter
• This smoothes the received waveform
• A full-wave rectifier that recovers both the negative and positive
peaks has better performance
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
22
Example 4: Envelope Tracking
Example of Envelop Tracking of an Asynchronous AM Signal
y(t)
0.2
0
−0.2
Full−wave Rectifier
Half−wave Rectifier
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
−3
x 10
0.3
0.2
0.1
0
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
−3
x 10
0.3
0.2
0.1
0
0.8
J. McNames
0.9
1
1.1
1.2
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1.3
Time (s)
ECE 223
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1.5
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1.8
−3
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Example 4: MATLAB Code
function [] = EnvelopeTracking();
close all;
N
fc
fs
al
=
=
=
=
2000;
50e3;
1e6;
0.95;
% No. samples
% Carrier frequency
% Sample rate
% First-order filter parameter
k
t
= 1:N;
= (k-1)/fs;
rand(’state’,10);
xh
= rand(1,N)-0.5;
% Random high-frequency signal limited to [-0.5 0.5]
[n,wn] = ellipord(0.01,0.02,0.5,60);
[b,a] = ellip(n,0.5,60,wn);
x
= filter(b,a,xh);
% Lowpass filter to create baseband signal
x
= x + 0.2;
% Convert to positive signal
c
= cos(2*pi*fc*t);
y
= x.*c;
eh
= y.*(y>0);
rh
= filter(1-al,[1 -al],eh-mean(eh))+mean(eh);
ef
= abs(y);
rf
= filter(1-al,[1 -al],ef-mean(ef))+mean(ef);
figure;
FigureSet(1,’LTX’);
subplot(3,1,1);
h = plot(t,y,’g’,t,x,’b’,t,-x,’b’);
set(h,’LineWidth’,0.2);
xlim(1e-3*[0.8 1.8]);
ylim([-0.39 0.39]);
ylabel(’y(t)’);
title(’Example of Envelop Tracking of an Asynchronous AM Signal’);
box off;
J. McNames
Portland State University
ECE 223
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Ver. 1.11
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AxisLines;
subplot(3,1,2);
h = plot(t,eh,’b’,t,rh,’r’);
set(h,’LineWidth’,0.2);
xlim(1e-3*[0.8 1.8]);
ylim([-0.02 0.39]);
ylabel(’Half-wave Rectifier’);
box off;
AxisLines;
subplot(3,1,3);
h = plot(t,ef,’b’,t,rf,’r’);
set(h,’LineWidth’,0.2);
xlim(1e-3*[0.8 1.8]);
ylim([-0.02 0.39]);
ylabel(’Full-wave Rectifier’);
xlabel(’Time (s)’);
box off;
AxisLines;
AxisSet(6);
print -depsc EnvelopeTracking;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
25
Asynchronous Amplitude Modulation Terminology
x(t)
+
×
A
c(t)
y(t)
• Most baseband signals will not be positive
• We can make amplitude-limited signals, |x(t)| ≤ xmax , positive by
adding a constant A such that A > xmax
• The envelope detector then approximates x(t) + A
• x(t) can then be extracted with a highpass filter to remove A (the
DC component)
• The ratio m = xmax /A is called the modulation index
• If expressed in percentage, 100xmax /A, it is called the percent
modulation
• The spectrum of y(t) contains impulses to account for A
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
26
Spectrum of Asynchronous Amplitude Modulation
X(jω)
1
−ωx
0
ωx
C(jω)
π
-ωc
ω
ω
ωc
0
Y (jω)
Aπ
1
2
-ωc -ωx -ωc -ωc +ωx
J. McNames
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ECE 223
ωc -ωx
ωc
ωc +ωx
Communications
ω
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Asynchronous Amplitude Modulation Tradeoffs
x(t)
+
×
A
c(t)
y(t)
• In most applications, the FCC limits the transmission power
• For asynchronous AM, transmitting the carrier component requires
a portion of this power
• Thus, asynchronous AM is less efficient than synchronous AM
• However, the receiver is easier and cheaper to build
• As m → 1, more of the transmitter power is used for the baseband
signal x(t)
• As m → 0, the signal is easier to demodulate with an envelope
detector
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
28
Single Sideband AM
Y (jω)
1
2
-ωc -ωx -ωc -ωc +ωx
ωc -ωx
0
ωc
ωc +ωx
ω
• Let us define the bandwidth of the signal as ωx , the highest
frequency component of the signal
• The signal transmitted requires twice the bandwidth, 2ωx
• Near ωc , the signal content for both negative and positive
frequencies is transmitted
• We don’t need this much information to reconstruct X(jω)
• If we know X(jω) for either positive or negative frequencies, we
can use symmetry to construct the other part
• Thus, we only need to transmit one of the sidebands
J. McNames
Portland State University
ECE 223
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29
Single Sideband AM Continued
Y (jω)
1
2
-ωc -ωc +ωx
ωc -ωx
0
ωc
ω
• What we have discussed so far uses double-sideband modulation
• We can use single-sideband modulation by removing the upper or
lower sidebands
• Requires only half the bandwidth!
• An obvious approach: lowpass (to retain lower sidebands) or
highpass filter (to retain upper sidebands)
• Requires a nearly ideal high-frequency filter
• SSB modulation increases the cost of the transmitter
• If asynchronous modulation is used, it also increases the cost and
complexity of the receiver
J. McNames
Portland State University
ECE 223
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30
Frequency-Division Multiplexing
1
X1 (jω)
1
X2 (jω)
ω
1
X3 (jω)
ω
1
2
ω
Y (jω)
ω
• We can transmit multiple signals using a single transmitting
antenna with frequency-division multiplexing (FDM)
• Each baseband signal is shifted to a different frequency band
• Thus, multiple baseband signals can be transmitted
simultaneously over a single wideband channel
• The different modulated signals y1 (t), y2 (t), and y3 (t) are simply
summed before sending to the antenna
• To recover a specific signal, the corresponding frequency band
usually is extracted with a bandpass filter
J. McNames
Portland State University
ECE 223
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Time-Division Multiplexing
• The sampling theorem tells us we can represent any bandlimited
signals by its samples x(nT ) as long as ωs > 2ωx
• Thus we can convert multiple bandlimited signals into
discrete-time signals: x1 (t) → x1 [n], x2 (t) → x2 [n],
x3 (t) → x3 [n]
• Time-Division multiplexing interleaves these signals to form a
composite signal
. . . x1 [0], x2 [0], x3 [0], x1 [1], x2 [1], x3 [1], x1 [2], x2 [2], x3 [2] . . .
• A different time interval is assigned to each signal
• We could then form a continuous-time signal using bandlimited
interpolation
• If M signals are multiplexed and each signal has a bandwidth of
ωx , the multiplexed signal y(t) will require a bandwidth of M × ωx
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
32
Example 5: Time-Division Multiplexing
Example of Time−Division Multiplexing
1
x [n]
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
2
x [n]
1
0.5
0
3
x [n]
1
0.5
0
y[n]
1
0.5
0
0
J. McNames
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10
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Ver. 1.11
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Example 5: MATLAB Code
function [] = TDMultiplexing();
close all;
N
= 10;
% No. samples
k
x1
x2
x3
=
=
=
=
1:N;
rand(N,1);
rand(N,1);
rand(N,1);
y
k1
k2
k3
=
=
=
=
zeros(3*N,1);
1:3:3*N;
2:3:3*N;
3:3:3*N;
y(k1) = x1;
y(k2) = x2;
y(k3) = x3;
wc
T
t
n
=
=
=
=
pi;
1; % Sample rate
0:0.01:3*N+1;
1:3*N;
yr = zeros(size(t)); % Reconstructed signal
for cnt = 1:length(n),
yr = yr + (wc*T/pi)*y(cnt)*sinc(wc*(t-n(cnt)*T)/pi);
end;
figure;
FigureSet(1,’LTX’);
subplot(4,1,1);
h = stem(k,x1,’g’);
set(h(1),’MarkerSize’,2);
J. McNames
Portland State University
ECE 223
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34
set(h(1),’MarkerFaceColor’,’g’);
hold off;
xlim([0 N+1]);
ylim([0 1.05]);
ylabel(’x_1[n]’);
title(’Example of Time-Division Multiplexing’);
box off;
subplot(4,1,2);
h = stem(k,x2,’b’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’b’);
hold off;
xlim([0 N+1]);
ylim([0 1.05]);
ylabel(’x_2[n]’);
box off;
subplot(4,1,3);
h = stem(k,x3,’r’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’r’);
hold off;
xlim([0 N+1]);
ylim([0 1.05]);
ylabel(’x_3[n]’);
box off;
subplot(4,1,4);
h = plot(t,yr,’k’);
hold on;
h = stem(k1,y(k1),’g’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’g’);
set(h(3),’Visible’,’Off’);
h = stem(k2,y(k2),’b’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’b’);
set(h(3),’Visible’,’Off’);
h = stem(k3,y(k3),’r’);
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
35
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’r’);
set(h(3),’Visible’,’Off’);
hold off;
xlim([0 3*N+1]);
ylim([-0.3 1.3]);
ylabel(’y[n]’);
box off;
AxisLines;
AxisSet(6);
print -depsc TDMultiplexing;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
36
Pulse Amplitude Modulation
T
x(t)
×
y(t)
+∞
y(t) =
x(nT ) p(t − nT )
n=−∞
p(t)
• In modern communication systems, the baseband signal x(t) is
first sampled to form x(nT ) in accord with the sampling theorem
• In a pulse-amplitude modulation (PAM) system, each sample is
multiplied by a pulse p(t)
• Time-division multiplexing can be easily combined with PAM
• Thus we could use p(t) = sinc( tw
π ) to ensure y(t) is bandlimited
to w
• We require w > 2ωx =
2π
T
to satisfy the sampling theorem
• AM could then be used to shift this to any frequency band
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
37
Pulse-Code Modulation
T
x[n]
Sign
×
x(t)
Modulation/
Demodulation
T
r(t)
r[n]
Sign
p(t)
• In practice, digital systems encode discrete-time signals with
discrete amplitudes
• Most digital signal processing (DSP) uses discrete-valued signals
• Continuous-valued signals are converted to discrete-valued signals
using analog-to-digital (ADC) converters
• Discrete-valued signals can be encoded using binary 1’s and 0’s
• These discrete signals can be transmitted over a communications
channel by transmitting
– 0: Transmit −p(t)
– 1: Transmit +p(t)
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
38
Example 6: PCM
Create a random digital (discrete-time and discrete-valued) signal
consisting of fifty 0’s and 1’s. Encode the baseband signal x(t) such
that the bandwidth is limited to 100 Hz. What is the minimum time
required to transmit the signal? Plot the discrete-time signal x[n], the
baseband encoded signal x(t), and an “eye” diagram of the
overlapping received pulses. Assume that the channel does not cause
any distortion and that the receiver and transmitter sampling times are
synchronized. Hint: recall that
W
tW
⇔ PW (jω)
sinc
π
π
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
39
Example 6: Workspace
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
40
Example 6: Plot of x[n] and x(t)
Example of Pulse−Code Modulation
1
0.6
1
x [n]
0.8
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
11
2
x(t)
1
0
−1
−2
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
41
Example 6: Eye Diagram
Eye Diagram
2
1.5
1
x(t)
0.5
0
−0.5
−1
−1.5
−2
−0.01
J. McNames
−0.008
−0.006
−0.004
−0.002
Portland State University
0
0.002
Time (sec)
ECE 223
0.004
0.006
Communications
0.008
0.01
Ver. 1.11
42
Example 6: MATLAB Code
function [] = PCMEx();
close all;
N = 50;
% No. samples
n = 1:N;
% Discrete-time index
xd = (rand(N,1)>0.5); % Digital signal
wc
T
Ts
t
=
=
=
=
2*pi*50;
% Limit pulse bandwidth to 100 Hz (-50 to 50)
pi/wc;
% Sample period
0.0005;
0:Ts:(N+1)*T;
figure;
FigureSet(1,’LTX’);
subplot(2,1,1);
h = stem(n,xd,’b’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’b’);
hold off;
xlim([0 11]);
ylim([0 1.05]);
ylabel(’x_1[n]’);
title(’Example of Pulse-Code Modulation’);
box off;
subplot(2,1,2);
xc = zeros(size(t)); % Modulated signal x(t)
for cnt = 1:length(n),
s = -1*(xd(cnt)==0) + 1*(xd(cnt)==1);
p = s*sinc(wc*(t-n(cnt)*T)/pi);
plot(t,p,’b’);
hold on;
xc = xc + p;
end;
plot(t,xc,’g’);
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
43
plot(n*T,-1*(xd==0)+1*(xd==1),’ro’,’MarkerSize’,2,’MarkerFaceColor’,’r’);
hold off;
xlim([0 11*T]);
ylabel(’x(t)’);
box off;
AxisLines;
AxisSet(6);
print -depsc PCMSignals;
figure;
FigureSet(2,’LTX’);
for cnt = 1:length(n),
k = -round(T/Ts):round(T/Ts);
plot(k*Ts,xc(round(n(cnt)*T/Ts)+k+1));
hold on;
end;
hold off;
xlim([min(k*Ts) max(k*Ts)]);
ylabel(’x(t)’);
xlabel(’Time (sec)’);
title(’Eye Diagram’);
box off;
AxisSet(6);
AxisLines;
print -depsc PCMEyeDiagram;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
44
Example 6: MATLAB Code
function [] = PCMEx();
close all;
N = 50;
% No. samples
n = 1:N;
% Discrete-time index
xd = (rand(N,1)>0.5); % Digital signal
wc
T
Ts
t
=
=
=
=
2*pi*50;
% Limit pulse bandwidth to 100 Hz (-50 to 50)
pi/wc;
% Sample period
0.0005;
0:Ts:(N+1)*T;
figure;
FigureSet(1,’LTX’);
subplot(2,1,1);
h = stem(n,xd,’b’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’b’);
hold off;
xlim([0 11]);
ylim([0 1.05]);
ylabel(’x_1[n]’);
title(’Example of Pulse-Code Modulation’);
box off;
subplot(2,1,2);
xc = zeros(size(t)); % Modulated signal x(t)
for cnt = 1:length(n),
s = -1*(xd(cnt)==0) + 1*(xd(cnt)==1);
p = s*sinc(wc*(t-n(cnt)*T)/pi);
plot(t,p,’b’);
hold on;
xc = xc + p;
end;
plot(t,xc,’g’);
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
45
plot(n*T,-1*(xd==0)+1*(xd==1),’ro’,’MarkerSize’,2,’MarkerFaceColor’,’r’);
hold off;
xlim([0 11*T]);
ylabel(’x(t)’);
box off;
AxisLines;
AxisSet(6);
print -depsc PCMSignals;
figure;
FigureSet(2,’LTX’);
for cnt = 1:length(n),
k = -round(T/Ts):round(T/Ts);
plot(k*Ts,xc(round(n(cnt)*T/Ts)+k+1));
hold on;
end;
hold off;
xlim([min(k*Ts) max(k*Ts)]);
ylabel(’x(t)’);
xlabel(’Time (sec)’);
title(’Eye Diagram’);
box off;
AxisSet(6);
AxisLines;
print -depsc PCMEyeDiagram;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
46
Example 7: Noise Tolerance of PCM
Repeat the previous example, but this time assume that the channel
adds noise that is uniformly distributed between -0.5 and 0.5. Can you
still accurately receive the signal?
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
47
Example 7: Plot of x[n] and x(t)
Example of Pulse−Code Modulation
1
x [n]
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
11
x(t)
2
0
−2
4
r(t)
2
0
−2
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
48
Example 7: Eye Diagram
Eye Diagram
3
2
x(t)
1
0
−1
−2
−3
−0.01
J. McNames
−0.008
−0.006
−0.004
−0.002
Portland State University
0
0.002
Time (sec)
ECE 223
0.004
0.006
0.008
Communications
0.01
Ver. 1.11
49
Example 7: MATLAB Code
function [] = PCMNoisEx();
close all;
N = 50;
% No. samples
n = 1:N;
% Discrete-time index
xd = (rand(N,1)>0.5); % Digital signal
wc
T
Ts
t
nt
=
=
=
=
=
2*pi*50;
% Limit pulse bandwidth to 100 Hz (-50 to 50)
pi/wc;
% Sample period
0.0002;
0:Ts:(N+1)*T;
n*(T/Ts);
figure;
FigureSet(1,’LTX’);
subplot(3,1,1);
h = stem(n,xd,’b’);
set(h(1),’MarkerSize’,2);
set(h(1),’MarkerFaceColor’,’b’);
hold off;
xlim([0 11]);
ylim([0 1.05]);
ylabel(’x_1[n]’);
title(’Example of Pulse-Code Modulation’);
box off;
subplot(3,1,2);
xc = zeros(size(t)); % Modulated signal x(t)
for cnt = 1:length(n),
s = -1*(xd(cnt)==0) + 1*(xd(cnt)==1);
p = s*sinc(wc*(t-n(cnt)*T)/pi);
plot(t,p,’b’);
hold on;
xc = xc + p;
end;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
50
plot(t,xc,’g’);
plot(n*T,-1*(xd==0)+1*(xd==1),’ro’,’MarkerSize’,2,’MarkerFaceColor’,’r’);
hold off;
xlim([0 11*T]);
ylabel(’x(t)’);
box off;
AxisLines;
subplot(3,1,3);
r = xc + (rand(size(xc))-0.5); % Add noise to the received signal
plot(t,r,’b’);
hold on;
plot(n*T,r(1+n*round(T/Ts)),’ro’,’MarkerSize’,2,’MarkerFaceColor’,’r’);
plot(t,xc,’g’);
hold off;
xlim([0 11*T]);
ylabel(’r(t)’);
box off;
AxisLines;
AxisSet(6);
print -depsc PCMNoiseSignals;
figure;
FigureSet(2,’LTX’);
for cnt = 1:length(n),
k = -round(T/Ts):round(T/Ts);
plot(k*Ts,r(n(cnt)*round(T/Ts)+k+1));
hold on;
end;
hold off;
xlim([min(k*Ts) max(k*Ts)]);
ylabel(’x(t)’);
xlabel(’Time (sec)’);
title(’Eye Diagram’);
box off;
AxisSet(6);
AxisLines;
print -depsc PCMNoiseEyeDiagram;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
51
Example 8: Communication System
Design a system (transmitter and receiver) that transmits a stereo
audio signal through a channel in the frequency band of 1.2 MHz
±40 kHz. Discuss the design tradeoffs of different approaches to this
problem and sketch the spectrum of signals at each stage of the
process.
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
52
Example 8: Workspace 1
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
53
Example 8: Workspace 2
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
54
Sinusoidal Angle Modulation
c(t) = A cos (ωc t + θc )
c(t) = A cos (θ(t))
• So far we have discussed different types of amplitude modulation
• Angle Modulation alters the angle of the carrier signal rather
than the amplitude
• Define the instantaneous angle of the carrier signal c(t) as θ(t)
• There are two forms of angle modulation
– Phase modulation (PM): θ(t) = ωc t + θ0 + kp x(t)
– Frequency modulation (FM):
dθ(t)
dt
= ωc + kf x(t)
• Note that for FM, θ(t) = (ωc + kf x(t)) t
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
55
Angle Modulation Versus Amplitude Modulation
Angle Modulation (say FM) versus Amplitude Modulation (AM)
+ One advantage of FM is that the amplitude of the signal
transmitted can always be at maximum power
+ FM is also less sensitive to many common types of noise than AM
- However, FM generally requires greater bandwidth than AM
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
56
Example 9: Angle Modulation
Create a random signal bandlimited to ±1 Hz and amplitude limited
to one (e.g. |x(t)| ≤ 1). Modulate the signal use PM and FM with a
carrier frequency of 3 Hz. Use kp = 3 and kf = 4π. Plot the baseband
signal, the carrier signal, and the modulated signals.
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
57
Example 9: Signal Plot
Example of Sinusoidal AM Modulation
x(t)
1
0
−1
c(t)
1
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
Time (s)
6
7
8
9
0
1
2
3
4
5
Time (s)
6
7
8
9
0
−1
PM
1
0
−1
FM
1
0
−1
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
58
Example 9: MATLAB Code
%function [] = AngleModulation();
close all;
N
fc
fs
fx
kp
kf
=
=
=
=
=
=
500;
3;
50;
1;
3;
2*pi*2;
%
%
%
%
%
%
No. samples
Carrier frequency (Hz)
Sample rate (Hz)
Bandlimit of baseband signal
PM scaling coefficient
FM scaling coefficient
wc = 2*pi*fc;
k
t
= 1:N;
= (k-1)/fs;
xh
[n,wn]
[b,a]
x
x
=
=
=
=
=
randn(1,N); % Random high-frequency signal
ellipord(0.95*fx/(fs/2),fx/(fs/2),0.5,60);
ellip(n,0.5,60,wn);
filter(b,a,xh); % Lowpass filter to create baseband signal
x/max(abs(x)); % Scale so maximum amplitude is 1
c = cos(wc*t); % Carrier signal
yp = cos(wc*t + kp*x);
theta = cumsum(wc + kf*x)/fs; % Approximate integral of angle
yf = cos(theta);
figure;
FigureSet(1,’LTX’);
subplot(4,1,1);
h = plot(t,x,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylabel(’x(t)’);
title(’Example of Sinusoidal AM Modulation’);
J. McNames
Portland State University
ECE 223
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59
box off;
AxisLines;
subplot(4,1,2);
h = plot(t,c,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
ylabel(’c(t)’);
box off;
AxisLines;
subplot(4,1,3);
h = plot(t,yp,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
xlabel(’Time (s)’);
ylabel(’PM’);
box off;
AxisLines;
subplot(4,1,4);
h = plot(t,yf,’b’);
set(h,’LineWidth’,0.2);
xlim([0 max(t)]);
xlabel(’Time (s)’);
ylabel(’FM’);
box off;
AxisLines;
AxisSet(6);
print -depsc AngleModulation;
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
60
Relationship of Angle and Frequency Modulation
dθ(t)
= ωc + kf x(t)
dt
θ(t) = ωc t + θ0 + kp x(t)
• These two forms are easily related.
• PM with x(t) is equivalent to FM with
• FM with x(t) is equivalent to PM with
dx(t)
dt
t
x(τ ) dτ
0
• For c(t), instantaneous frequency is defined as
ωi (t) ≡
dθ(t)
dt
• Frequency modulation: ωi (t) = ωc + kf x(t)
• Phase modulation: ωi (t) = ωc + kp dx(t)
dt
J. McNames
Portland State University
ECE 223
Communications
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61
Frequency Modulation
• Consider a sinusoidal baseband signal x(t) = A cos(ωx t)
• This models a bandlimited signal limited to ±ωx
ωi (t) = ωc + kf A cos(ωx t)
• Then
• The instantaneous frequency varies between ωc + kf A and
ωc − kf A
• The modulated signal is then of the form
t
ω
x(τ ) dτ = cos ωc t +
sin(ωx t) + θ0
y(t) = cos ωc t + kf
ωx
−∞
• Define the following variables
ω
m≡
ωx
ω ≡ kf A
• m is called the modulation index
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
62
Narrowband Frequency Modulation
y(t)
=
cos(ωc t + m sin(ωm t))
= cos(ωc t) cos(m sin(ωm t)) − sin(ωc t) sin(m sin(ωm t))
When m is small (m π2 ), this is called narrowband FM modulation
and we may use the following approximations.
cos(m sin(ωm t)) ≈ 1
sin(m sin(ωm t)) ≈ m sin(ωm t)
Thus
y(t)
J. McNames
= cos(ωc t) − m sin(ωc t) sin(ωm t)
= cos(ωc t) − m
2 cos(ωc t − ωm t) +
Portland State University
ECE 223
m
2
cos(ωc t + ωm t)
Communications
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Narrowband Frequency Modulation Continued
Y (jω)
A
2
-ωc -ωx -ωc -ωc +ωx
π
-ωc +ωx
-ωc -ωx -ωc
ωc -ωx
0
ωc
ωc +ωx
ω
Y (jω)
mπ/2
ωc -ωx
ωc
−mπ/2
ωc +ωx
ω
• Like AM, spectrum contains sidebands
• Unlike AM, sidebands are out of phase by 180◦
• Bandwidth is twice that of x(t) like double-sideband AM
• Carrier frequency is present and strong
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
64
Example 10: Angle Modulation
Create a sinusoidal baseband signal with a fundamental frequency of
1 Hz and a carrier sinusoidal signal at 12 Hz. Plot these signals and
the modulated signals after applying amplitude modulation and
frequency modulation. Use a scaling factor kf = 1 and a modulation
index of m = 0.5. Solve for the baseband signal amplitude A.
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
65
Example 10: Signal Plot
Example of Sinusoidal AM and FM Modulation
x(t)
5
0
−5
c(t)
1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.6
−0.4
−0.2
0
Time (s)
0.2
0.4
0.6
−0.6
−0.4
−0.2
0
Time (s)
0.2
0.4
0.6
0
−1
AM
5
0
−5
FM
1
0
−1
J. McNames
Portland State University
ECE 223
Communications
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Example 10: Relevant MATLAB Code
%function [] = AMFM();
close all;
fx
fc
fs
kf
m
=
=
=
=
=
1;
15;
100;
1;
0.5;
%
%
%
%
%
Signal frequency (Hz)
Carrier frequency (Hz)
Sampling frequency
FM scaling coefficient
Modulation index
wx = 2*pi*fx; % Signal frequency (rad/s)
wc = 2*pi*fc; % Carrier frequency (rad/s)
A = m*wx/kf; % Modulating amplitude
t = -0.75:0.001:0.75;
x
c
ya
yf
= A*cos(wx*t); % Baseband signal
= cos(wc*t);
% Carrier signal
= x.*c;
% Amplitude modulated signal
= cos(wc*t + m*sin(wx*t));
figure;
FigureSet(1,’LTX’);
subplot(4,1,1);
h = plot(t,x,’b’);
set(h,’LineWidth’,0.2);
xlim([min(t) max(t)]);
ylabel(’x(t)’);
title(’Example of Sinusoidal AM and FM Modulation’);
box off;
AxisLines;
subplot(4,1,2);
h = plot(t,c,’b’);
set(h,’LineWidth’,0.2);
xlim([min(t) max(t)]);
J. McNames
Portland State University
ECE 223
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ylabel(’c(t)’);
box off;
AxisLines;
subplot(4,1,3);
h = plot(t,ya,’b’,t,x,’r’,t,-x,’g’);
set(h,’LineWidth’,0.2);
xlim([min(t) max(t)]);
xlabel(’Time (s)’);
ylabel(’AM’);
box off;
AxisLines;
subplot(4,1,4);
h = plot(t,yf,’b’);
set(h,’LineWidth’,0.2);
xlim([min(t) max(t)]);
xlabel(’Time (s)’);
ylabel(’FM’);
box off;
AxisLines;
AxisSet(6);
print -depsc AMFM;
J. McNames
Portland State University
ECE 223
Communications
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Summary
• Modulation is the process of embedding one signal in another with
desirable properties for communication
• Most forms of modulation are nonlinear
• Sinusoidal AM is relatively simple and inexpensive
• Synchronous AM is more efficient than asynchronous AM, but is
also more expensive
• FM is more tolerant of noise than FM, but requires more
bandwidth and cost
• Filters and frequency analysis using the Fourier transform have a
crucial role in communication systems
• Frequency- (FDM) and time-division (TDM) multiplexing can be
used to merge multiple bandlimited signals into a single composite
signal with a larger bandwidth
J. McNames
Portland State University
ECE 223
Communications
Ver. 1.11
69