Overview of Communication Topics Handy Trigonometry Identities - - PowerPoint PPT Presentation

SLIDE 1

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
• f 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

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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

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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-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

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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) =

1 2 cos(a − b) + 1 2 cos(a + b)

sin(a) sin(b) =

1 2 cos(a − b) − 1 2 cos(a + b)

sin(a) cos(b) =

1 2 sin(a − b) + 1 2 sin(a + b)

• J. McNames

Portland State University ECE 223 Communications

• Ver. 1.11

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SLIDE 2

Example 1: Sinusoidal AM of a Random Signal

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −3 −0.2 0.2 x(t) Example of Sinusoidal Amplitude Modulation 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −3 −1 1 c(t) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 −3 −0.2 0.2 Time (s) y(t)

• J. McNames

Portland State University ECE 223 Communications

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Fourier Analysis of Sinusoidal Amplitude Modulation For convenience, we will assume θc = 0. c(t) = cos(ωct) C(jω) = π [δ(ω − ωc) + δ(ω + ωc)] y(t) = x(t) · c(t) Y (jω) =

1 2π [X(jω) ∗ C(jω)]

X(jω) ∗ δ(ω − ωc) = X (j(ω − ωc)) Y (jω) =

1 2X (j(ω − ωc)) + 1 2X (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

• Ver. 1.11

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Example 1: MATLAB Code

function [] = AMTimeDomain(); close all; N = 2000; % No. samples fc = 50e3; % Carrier frequency fs = 1e6; % Sample rate k = 1:N; t = (k-1)/fs; 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; 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 Communications

• Ver. 1.11

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Fourier Analysis of Sinusoidal Amplitude Modulation

1

ω ω ω π

1 2

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

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SLIDE 3

Fourier Analysis of Sinusoidal AM Demodulation

1 2

ω ω ω ω ω π

1 2 1 2

ωc ωc −ωc −ωc 2ωc −2ωc Y (jω) C(jω) W (jω) H(jω) R(jω)

• J. McNames

Portland State University ECE 223 Communications

• Ver. 1.11

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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

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Synchronous AM Demodulation Observations Channel H(s)

Receiver Transmitter

x(t) c(t) c(t) y(t) y(t) w(t) ˆ x(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

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Synchronous Sinusoidal Amplitude Demodulation Channel H(s)

Receiver Transmitter

x(t) c(t) c(t) y(t) y(t) w(t) ˆ x(t) × ×

y(t) = x(t) cos(ωct) w(t) = y(t) cos(ωct) = x(t) cos2(ωct) = x(t) 1

2 + 1 2 cos(2ωct)

• =

1 2x(t) + 1 2x(t) cos(2ωct)

• Synchronous demodulation assumptions

– The carrier c(t) is known exactly – ωc > 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

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SLIDE 4

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

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Synchronous AM Demodulation Carrier Phase Analysis Suppose the transmitter and receiver carrier signals differ by a phase shift: cT (t) = cos(ωct + θ) cR(t) = cos(ωct + φ) w(t) = y(t) cR(t) = x(t) cT (t) cR(t) = x(t) cos(ωct + θ) cos(ωct + φ) = x(t) 1

2 cos(θ − φ) + 1 2 cos(2ωct + θ + φ)

• =

1 2x(t) cos(θ − φ) + 1 2x(t) cos(2ωct + θ + φ)

• J. McNames

Portland State University ECE 223 Communications

• Ver. 1.11

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Example 2: Asynchronous Amplitude Modulation

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10

−3

0.2 0.4 x(t) Example of Asynchronous Sinusoidal AM Modulation 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10

−3

−1 1 c(t) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10

−3

−0.2 0.2 Time (s) y(t)

• J. McNames

Portland State University ECE 223 Communications

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Synchronous AM Demodulation Carrier Phase Comments w(t) = 1

2x(t) cos(θ − φ) + 1 2x(t) cos(2ωct + 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

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SLIDE 5

Diodes

0.7 0.7

• +

Ideal Model Real Model

I I I V V 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

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Example 2: MATLAB Code

function [] = AAMTimeDomain(); close all; N = 2000; % No. samples fc = 50e3; % Carrier frequency fs = 1e6; % Sample rate k = 1:N; t = (k-1)/fs; 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 signal x = 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 Communications

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Example 3: Full Wave Rectifier

vo

• +

vs R

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

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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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SLIDE 6

Example 4: Envelope Tracking

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10 −3 −0.2 0.2 y(t) Example of Envelop Tracking of an Asynchronous AM Signal 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10 −3 0.1 0.2 0.3 Half−wave Rectifier 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x 10 −3 0.1 0.2 0.3 Full−wave Rectifier Time (s)

• J. McNames

Portland State University ECE 223 Communications

• Ver. 1.11

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Example 3: Workspace

• J. McNames

Portland State University ECE 223 Communications

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Example 4: MATLAB Code

function [] = EnvelopeTracking(); close all; N = 2000; % No. samples fc = 50e3; % Carrier frequency fs = 1e6; % Sample rate al = 0.95; % First-order filter parameter k = 1:N; t = (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 Communications

• Ver. 1.11

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Envelope Detectors

R C R

• +
• +
• +
• +

i(t) i(t) y(t) y(t) e(t) 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

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SLIDE 7

Spectrum of Asynchronous Amplitude Modulation

1

ω ω ω π

1 2

Aπ ωx −ωx ωc ωc ωc-ωx ωc+ωx

• ωc
• ωc
• ωc-ωx
• ωc+ωx

X(jω) C(jω) Y (jω)

• J. McNames

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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

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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 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

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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 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

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SLIDE 8

Frequency-Division Multiplexing

1 1 1

ω ω ω ω

1 2

X1(jω) X2(jω) X3(jω) Y (jω)

• 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 Communications

• Ver. 1.11

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Single Sideband AM

ω

1 2

ωc ωc-ωx ωc+ωx

• ωc
• ωc-ωx
• ωc+ωx

Y (jω)

• 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

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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

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Single Sideband AM Continued

ω

1 2

ωc ωc-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 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 Communications

• Ver. 1.11

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SLIDE 9

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

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Example 5: Time-Division Multiplexing

1 2 3 4 5 6 7 8 9 10 11 0.5 1 x1[n] Example of Time−Division Multiplexing 1 2 3 4 5 6 7 8 9 10 11 0.5 1 x2[n] 1 2 3 4 5 6 7 8 9 10 11 0.5 1 x3[n] 5 10 15 20 25 30 0.5 1 y[n]

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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

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Example 5: MATLAB Code

function [] = TDMultiplexing(); close all; N = 10; % No. samples k = 1:N; x1 = rand(N,1); x2 = rand(N,1); x3 = rand(N,1); y = zeros(3*N,1); k1 = 1:3:3*N; k2 = 2:3:3*N; k3 = 3:3:3*N; y(k1) = x1; y(k2) = x2; y(k3) = x3; wc = pi; T = 1; % Sample rate t = 0:0.01:3*N+1; n = 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);

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Portland State University ECE 223 Communications

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SLIDE 10

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

• verlapping 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 π sinc tW π

• ⇔ PW (jω)
• J. McNames

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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) 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

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Example 6: Workspace

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Pulse-Code Modulation

p(t)

Sign

Modulation/ Demodulation

Sign

x[n] x(t) r(t) r[n] × T 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

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SLIDE 11

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 = 2*pi*50; % Limit pulse bandwidth to 100 Hz (-50 to 50) T = pi/wc; % Sample period Ts = 0.0005; t = 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

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Example 6: Plot of x[n] and x(t)

1 2 3 4 5 6 7 8 9 10 11 0.2 0.4 0.6 0.8 1 x1[n] Example of Pulse−Code Modulation 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −2 −1 1 2 x(t)

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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;

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Example 6: Eye Diagram

−0.01 −0.008 −0.006 −0.004 −0.002 0.002 0.004 0.006 0.008 0.01 −2 −1.5 −1 −0.5 0.5 1 1.5 2 x(t) Time (sec) Eye Diagram

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SLIDE 12

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?

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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 = 2*pi*50; % Limit pulse bandwidth to 100 Hz (-50 to 50) T = pi/wc; % Sample period Ts = 0.0005; t = 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’);

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Example 7: Plot of x[n] and x(t)

1 2 3 4 5 6 7 8 9 10 11 0.5 1 x1[n] Example of Pulse−Code Modulation 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −2 2 x(t) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −2 2 4 r(t)

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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;

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SLIDE 13

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;

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Example 7: Eye Diagram

−0.01 −0.008 −0.006 −0.004 −0.002 0.002 0.004 0.006 0.008 0.01 −3 −2 −1 1 2 3 x(t) Time (sec) Eye Diagram

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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.

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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 = 2*pi*50; % Limit pulse bandwidth to 100 Hz (-50 to 50) T = pi/wc; % Sample period Ts = 0.0002; t = 0:Ts:(N+1)*T; nt = 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;

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SLIDE 14

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 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) = ωct + θ0 + kpx(t) – Frequency modulation (FM): dθ(t)

dt

= ωc + kfx(t)

• Note that for FM, θ(t) = (ωc + kfx(t)) t
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Example 8: Workspace 1

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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
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Example 8: Workspace 2

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SLIDE 15

Example 9: MATLAB Code

%function [] = AngleModulation(); close all; N = 500; % No. samples fc = 3; % Carrier frequency (Hz) fs = 50; % Sample rate (Hz) fx = 1; % Bandlimit of baseband signal kp = 3; % PM scaling coefficient kf = 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 signal x = 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’);

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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.

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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;

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Example 9: Signal Plot

1 2 3 4 5 6 7 8 9 −1 1 x(t) Example of Sinusoidal AM Modulation 1 2 3 4 5 6 7 8 9 −1 1 c(t) 1 2 3 4 5 6 7 8 9 −1 1 Time (s) PM 1 2 3 4 5 6 7 8 9 −1 1 Time (s) FM

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SLIDE 16

Narrowband Frequency Modulation y(t) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt)) − sin(ωct) sin(m sin(ωmt)) When m is small (m ≪ π

2 ), 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) + m 2 cos(ωct + ωmt)

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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

0 x(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 + kp

dx(t) dt

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Narrowband Frequency Modulation Continued

ω ω π mπ/2 −mπ/2

A 2

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
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Frequency Modulation

• Consider a sinusoidal baseband signal x(t) = A cos(ωxt)
• This models a bandlimited signal limited to ±ωx
• Then

ωi(t) = ωc + kfA cos(ωxt)

• The instantaneous frequency varies between ωc + kfA and

ωc − kfA

• The modulated signal is then of the form

y(t) = cos

• ωct + kf

t

−∞

x(τ) dτ

• = cos
• ωct + △ω

ωx sin(ωxt) + θ0

• Define the following variables

△ω ≡ kfA m ≡ △ω ωx

• m is called the modulation index
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SLIDE 17

Example 10: Relevant MATLAB Code

%function [] = AMFM(); close all; fx = 1; % Signal frequency (Hz) fc = 15; % Carrier frequency (Hz) fs = 100; % Sampling frequency kf = 1; % FM scaling coefficient m = 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 t = -0.75:0.001:0.75; x = A*cos(wx*t); % Baseband signal c = cos(wc*t); % Carrier signal ya = x.*c; % Amplitude modulated signal yf = 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)]);

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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.

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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;

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Example 10: Signal Plot

−0.6 −0.4 −0.2 0.2 0.4 0.6 −5 5 x(t) Example of Sinusoidal AM and FM Modulation −0.6 −0.4 −0.2 0.2 0.4 0.6 −1 1 c(t) −0.6 −0.4 −0.2 0.2 0.4 0.6 −5 5 Time (s) AM −0.6 −0.4 −0.2 0.2 0.4 0.6 −1 1 Time (s) FM

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SLIDE 18

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

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