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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
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) = 12 cos(a − b) + 1
2 cos(a + b)
sin(a) sin(b) = 12 cos(a − b) − 1
2 cos(a + b)
sin(a) cos(b) = 12 sin(a − b) + 1
2 sin(a + b)
J. McNames Portland State University ECE 223 Communications Ver. 1.11 2
Introduction to Communication Systems
• Communications is a very active and large area of electricalengineering
• Experienced a lot of growth through the nineties with the adventof wireless cell phones and the internet
• Still an active area of research
• Fundamentals of signals and systems are essential to graspingcommunications concepts
• The next two lectures will merely introduce some of thefundamental concepts
• Will primarily focus on modulation and demodulation incontinuous-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)
c(t)
y(t)×
y(t) = x(t) · c(t) c(t) = cos(ωct + θc)
• Modulation: the process of embedding an information-bearingsignal into a second signal
• Demodulation: extracting the information-bearing signal fromthe second signal
• Sinusoidal Amplitude modulation: a sinusoidal carrier c(t) hasits 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
xh = randn(1,N); % Random high-frequency signal[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
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;
xh = rand(1,N)-0.5; % Random high-frequency signal limited to [-0.5 0.5][n,wn] = ellipord(0.02,0.03,0.5,60);[b,a] = ellip(n,0.5,60,wn);x = filter(b,a,xh); % Lowpass filter to create baseband signalx = 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;
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 signalx = x + 0.2; % Convert to positive signalc = 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 Communications Ver. 1.11 24
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)
c(t)
y(t)
A
×+
• Most baseband signals will not be positive
• We can make amplitude-limited signals, |x(t)| ≤ xmax, positive byadding 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 (theDC component)
• The ratio m = xmax/A is called the modulation index
• If expressed in percentage, 100xmax/A, it is called the percentmodulation
• 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
0
0
0
1
ω
ω
ω
π
12
Aπ
ωx−ωx
ωc
ωc
ωc-ωx ωc+ωx-ωc
-ωc
-ωc-ωx -ωc+ωx
X(jω)
C(jω)
Y (jω)
J. McNames Portland State University ECE 223 Communications Ver. 1.11 27
Asynchronous Amplitude Modulation Tradeoffs
x(t)
c(t)
y(t)
A
×+
• In most applications, the FCC limits the transmission power
• For asynchronous AM, transmitting the carrier component requiresa 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 basebandsignal x(t)
• As m → 0, the signal is easier to demodulate with an envelopedetector
J. McNames Portland State University ECE 223 Communications Ver. 1.11 28
Single Sideband AM
0ω
12
ωcωc-ωx ωc+ωx-ωc-ωc-ωx -ωc+ωx
Y (jω)
• Let us define the bandwidth of the signal as ωx, the highestfrequency component of the signal
• The signal transmitted requires twice the bandwidth, 2ωx
• Near ωc, the signal content for both negative and positivefrequencies is transmitted
• We don’t need this much information to reconstruct X(jω)
• If we know X(jω) for either positive or negative frequencies, wecan use symmetry to construct the other part
• Thus, we only need to transmit one of the sidebands
J. McNames Portland State University ECE 223 Communications Ver. 1.11 29
Single Sideband AM Continued
0ω
12
ωcωc-ωx-ωc -ωc+ωx
Y (jω)
• What we have discussed so far uses double-sideband modulation
• We can use single-sideband modulation by removing the upper orlower sidebands
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
x(t)
p(t)
y(t)×T
y(t) =+∞∑
n=−∞x(nT ) p(t − nT )
• In modern communication systems, the baseband signal x(t) isfirst sampled to form x(nT ) in accord with the sampling theorem
• In a pulse-amplitude modulation (PAM) system, each sample ismultiplied by a pulse p(t)
• Time-division multiplexing can be easily combined with PAM
• Thus we could use p(t) = sinc( twπ ) to ensure y(t) is bandlimited
to w
• We require w > 2ωx = 2πT 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
p(t)
Sign Modulation/Demodulation Signx[n]
x(t) r(t)r[n]×
TT
• In practice, digital systems encode discrete-time signals withdiscrete amplitudes
• Most digital signal processing (DSP) uses discrete-valued signals
• Continuous-valued signals are converted to discrete-valued signalsusing 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 communicationschannel 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) signalconsisting of fifty 0’s and 1’s. Encode the baseband signal x(t) suchthat the bandwidth is limited to 100 Hz. What is the minimum timerequired to transmit the signal? Plot the discrete-time signal x[n], thebaseband encoded signal x(t), and an “eye” diagram of theoverlapping received pulses. Assume that the channel does not causeany distortion and that the receiver and transmitter sampling times aresynchronized. Hint: recall that
W
πsinc
(tW
π
)⇔ PW (jω)
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
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),
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),
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 channeladds noise that is uniformly distributed between -0.5 and 0.5. Can youstill accurately receive the signal?
J. McNames Portland State University ECE 223 Communications Ver. 1.11 47
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),
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 stereoaudio signal through a channel in the frequency band of 1.2 MHz±40 kHz. Discuss the design tradeoffs of different approaches to thisproblem and sketch the spectrum of signals at each stage of theprocess.
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 (ωct + 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 ratherthan 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) = ωct + θ0 + kpx(t)
– Frequency modulation (FM): dθ(t)dt = ωc + kfx(t)
• Note that for FM, θ(t) = (ωc + kfx(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 signaltransmitted 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 limitedto one (e.g. |x(t)| ≤ 1). Modulate the signal use PM and FM with acarrier frequency of 3 Hz. Use kp = 3 and kf = 4π. Plot the basebandsignal, the carrier signal, and the modulated signals.
J. McNames Portland State University ECE 223 Communications Ver. 1.11 57
Example 9: Signal Plot
0 1 2 3 4 5 6 7 8 9−1
0
1
x(t)
Example of Sinusoidal AM Modulation
0 1 2 3 4 5 6 7 8 9−1
0
1
c(t)
0 1 2 3 4 5 6 7 8 9−1
0
1
Time (s)
PM
0 1 2 3 4 5 6 7 8 9−1
0
1
Time (s)
FM
J. McNames Portland State University ECE 223 Communications Ver. 1.11 58
Example 9: MATLAB Code%function [] = AngleModulation();close all;
N = 500; % No. samplesfc = 3; % Carrier frequency (Hz)fs = 50; % Sample rate (Hz)fx = 1; % Bandlimit of baseband signalkp = 3; % PM scaling coefficientkf = 2*pi*2; % FM scaling coefficient
wc = 2*pi*fc;
k = 1:N;t = (k-1)/fs;
xh = randn(1,N); % Random high-frequency signal[n,wn] = ellipord(0.95*fx/(fs/2),fx/(fs/2),0.5,60);[b,a] = ellip(n,0.5,60,wn);x = filter(b,a,xh); % Lowpass filter to create baseband signalx = x/max(abs(x)); % Scale so maximum amplitude is 1
c = cos(wc*t); % Carrier signalyp = cos(wc*t + kp*x);theta = cumsum(wc + kf*x)/fs; % Approximate integral of angleyf = 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 Communications Ver. 1.11 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
θ(t) = ωct + θ0 + kpx(t)dθ(t)dt
= ωc + kfx(t)
• These two forms are easily related.
• PM with x(t) is equivalent to FM with dx(t)dt
• FM with x(t) is equivalent to PM with∫ t
0x(τ) dτ
• For c(t), instantaneous frequency is defined as
ωi(t) ≡ dθ(t)dt
• Frequency modulation: ωi(t) = ωc + kfx(t)
• Phase modulation: ωi(t) = ωc + kpdx(t)
dt
J. McNames Portland State University ECE 223 Communications Ver. 1.11 61
Frequency Modulation
• Consider a sinusoidal baseband signal x(t) = A cos(ωxt)
• This models a bandlimited signal limited to ±ωx
• Thenωi(t) = ωc + kfA cos(ωxt)
• The instantaneous frequency varies between ωc + kfA andωc − kfA
• The modulated signal is then of the form
y(t) = cos(
ωct + kf
∫ t
−∞x(τ) dτ
)= cos
(ωct +
�ω
ωxsin(ωxt) + θ0
)
• Define the following variables
�ω ≡ kfA m ≡ �ω
ωx
• m is called the modulation index
J. McNames Portland State University ECE 223 Communications Ver. 1.11 62
When m is small (m � π2 ), this is called narrowband FM modulation
and we may use the following approximations.
cos(m sin(ωmt)) ≈ 1 sin(m sin(ωmt)) ≈ m sin(ωmt)
Thus
y(t) = cos(ωct) − m sin(ωct) sin(ωmt)= cos(ωct) − m
2 cos(ωct − ωmt) + m2 cos(ωct + ωmt)
J. McNames Portland State University ECE 223 Communications Ver. 1.11 63
Narrowband Frequency Modulation Continued
0
ω
ω
π
mπ/2
−mπ/2
A2
ωc
ωc
ωc-ωx
ωc-ωx
ωc+ωx
ωc+ωx
-ωc
-ωc
-ωc-ωx
-ωc-ωx
-ωc+ωx
-ωc+ωx
Y (jω)
Y (jω)
• 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 of1 Hz and a carrier sinusoidal signal at 12 Hz. Plot these signals andthe modulated signals after applying amplitude modulation andfrequency modulation. Use a scaling factor kf = 1 and a modulationindex 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
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−5
0
5
x(t)
Example of Sinusoidal AM and FM Modulation
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−1
0
1
c(t)
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−5
0
5
Time (s)
AM
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−1
0
1
Time (s)
FM
J. McNames Portland State University ECE 223 Communications Ver. 1.11 66
Example 10: Relevant MATLAB Code%function [] = AMFM();close all;
fx = 1; % Signal frequency (Hz)fc = 15; % Carrier frequency (Hz)fs = 100; % Sampling frequencykf = 1; % FM scaling coefficientm = 0.5; % Modulation index
wx = 2*pi*fx; % Signal frequency (rad/s)wc = 2*pi*fc; % Carrier frequency (rad/s)A = m*wx/kf; % Modulating amplitude
J. McNames Portland State University ECE 223 Communications Ver. 1.11 67
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 Ver. 1.11 68
Summary
• Modulation is the process of embedding one signal in another withdesirable 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 isalso more expensive
• FM is more tolerant of noise than FM, but requires morebandwidth and cost
• Filters and frequency analysis using the Fourier transform have acrucial role in communication systems
• Frequency- (FDM) and time-division (TDM) multiplexing can beused to merge multiple bandlimited signals into a single compositesignal with a larger bandwidth
J. McNames Portland State University ECE 223 Communications Ver. 1.11 69