Overview of Communication Topics Handy Trigonometry Identities • Sinusoidal amplitude modulation cos( a + b ) = cos( a ) cos( b ) − sin( a ) sin( b ) • Amplitude demodulation (synchronous and asynchronous) sin( a + b ) = sin( a ) cos( b ) + cos( a ) sin( b ) • Double- and single-sideband AM modulation 1 2 cos( a − b ) + 1 cos( a ) cos( b ) = 2 cos( a + b ) • Pulse-amplitude modulation 1 2 cos( a − b ) − 1 sin( a ) sin( b ) = 2 cos( a + b ) • Pulse code modulation 2 sin( a − b ) + 1 1 sin( a ) cos( b ) = 2 sin( a + b ) • Frequency-division multiplexing • Time-division multiplexing • Narrowband frequency modulation J. McNames Portland State University ECE 223 Communications Ver. 1.11 1 J. McNames Portland State University ECE 223 Communications Ver. 1.11 2 Introduction to Communication Systems Introduction to Amplitude Modulation • Communications is a very active and large area of electrical x ( t ) y ( t ) × engineering • Experienced a lot of growth through the nineties with the advent c ( t ) y ( t ) = x ( t ) · c ( t ) c ( t ) = cos( ω c t + θ c ) of wireless cell phones and the internet • Still an active area of research • Modulation : the process of embedding an information-bearing signal into a second signal • Fundamentals of signals and systems are essential to grasping • Demodulation : extracting the information-bearing signal from communications concepts the second signal • The next two lectures will merely introduce some of the fundamental concepts • Sinusoidal Amplitude modulation : a sinusoidal carrier c ( t ) has its amplitude modified by the information-bearing signal x ( t ) • 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 J. McNames Portland State University ECE 223 Communications Ver. 1.11 4
Fourier Analysis of Sinusoidal Amplitude Modulation Fourier Analysis of Sinusoidal Amplitude Modulation For convenience, we will assume θ c = 0 . X ( jω ) 1 c ( t ) = cos( ω c t ) C ( jω ) = π [ δ ( ω − ω c ) + δ ( ω + ω c )] ω ω x y ( t ) = x ( t ) · c ( t ) − ω x 0 1 Y ( jω ) = 2 π [ X ( jω ) ∗ C ( jω )] C ( jω ) π X ( jω ) ∗ δ ( ω − ω c ) = X ( j ( ω − ω c )) 2 X ( j ( ω − ω c )) + 1 1 Y ( jω ) = 2 X ( j ( ω + ω c )) ω ω c - ω c 0 Y ( jω ) • Thus, sinusoidal AM shifts the baseband signal x ( t ) so that it is 1 2 centered at ± ω c ω • Thus, x ( t ) can be recovered only if ω c > ω x so that the replicated - ω c - ω x - ω c - ω c + ω x 0 ω c - ω x ω c ω c + ω x spectra don’t overlap J. McNames Portland State University ECE 223 Communications Ver. 1.11 5 J. McNames Portland State University ECE 223 Communications Ver. 1.11 6 Example 1: Sinusoidal AM of a Random Signal Example 1: MATLAB Code function [] = AMTimeDomain(); close all; Example of Sinusoidal Amplitude Modulation 0.2 N = 2000; % No. samples fc = 50e3; % Carrier frequency x(t) fs = 1e6; % Sample rate 0 k = 1:N; −0.2 t = (k-1)/fs; xh = randn(1,N); % Random high-frequency signal 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 [n,wn] = ellipord(0.01,0.02,0.5,60); [b,a] = ellip(n,0.5,60,wn); 1 −3 x 10 x = filter(b,a,xh); % Lowpass filter to create baseband signal c = cos(2*pi*fc*t); c(t) 0 y = x.*c; figure; −1 FigureSet(1,’LTX’); subplot(3,1,1); 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 h = plot(t,x,’b’); −3 set(h,’LineWidth’,0.2); 0.2 x 10 xlim([0 max(t)]); ylim([-0.3 0.3]); y(t) ylabel(’x(t)’); 0 title(’Example of Sinusoidal Amplitude Modulation’); box off; AxisLines; −0.2 subplot(3,1,2); h = plot(t,c,’b’); 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 set(h,’LineWidth’,0.2); Time (s) xlim([0 max(t)]); −3 x 10 ylim([-1.1 1.1]); J. McNames Portland State University ECE 223 Communications Ver. 1.11 7 J. McNames Portland State University ECE 223 Communications Ver. 1.11 8
Synchronous Sinusoidal Amplitude Demodulation ylabel(’c(t)’); box off; AxisLines; subplot(3,1,3); Transmitter Receiver h = plot(t,y,’b’,t,x,’g’,t,-x,’r’); set(h,’LineWidth’,0.2); y ( t ) y ( t ) w ( t ) xlim([0 max(t)]); ylim([-0.3 0.3]); Channel H ( s ) x ( t ) ˆ x ( t ) × × xlabel(’Time (s)’); ylabel(’y(t)’); box off; AxisLines; c ( t ) c ( t ) AxisSet(6); y ( t ) = x ( t ) cos( ω c t ) print -depsc AMTimeDomain; w ( t ) = y ( t ) cos( ω c t ) x ( t ) cos 2 ( ω c t ) = � 1 2 + 1 � = x ( t ) 2 cos(2 ω c t ) 2 x ( t ) + 1 1 = 2 x ( t ) cos(2 ω c t ) • Synchronous demodulation assumptions – The carrier c ( t ) is known exactly – ω c > ω x • 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 9 J. McNames Portland State University ECE 223 Communications Ver. 1.11 10 Fourier Analysis of Sinusoidal AM Demodulation Synchronous AM Demodulation Observations Transmitter Receiver Y ( jω ) 1 2 y ( t ) y ( t ) w ( t ) ω Channel H ( s ) x ( t ) x ( t ) ˆ × × ω c − ω c 0 C ( jω ) π c ( t ) c ( t ) ω − ω c ω c 0 • The lowpass filter H ( s ) should have a passband gain of 2 W ( jω ) 1 2 • The transition band is very wide so the filter does not need to be ω close to ideal (e.g. it can be low order) − 2 ω c 2 ω c 0 H ( jω ) 2 • We learned how to design this type of filter in ECE 222 ω • We assumed the signal spectrum X ( jω ) was real 0 R ( jω ) • The same ideas hold if X ( jω ) is complex 1 ω • 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 11 J. McNames Portland State University ECE 223 Communications Ver. 1.11 12
Synchronous AM Demodulation Carrier Phase Analysis Synchronous AM Demodulation Carrier Phase Comments Suppose the transmitter and receiver carrier signals differ by a phase w ( t ) = 1 2 x ( t ) cos( θ − φ ) + 1 2 x ( t ) cos(2 ω c t + θ + φ ) shift: c T ( t ) = cos( ω c t + θ ) • If θ = φ then we have the same case as before and we recover c R ( t ) = cos( ω c t + φ ) x ( t ) exactly after a lowpass filter with a passband gain of 2 w ( t ) = y ( t ) c R ( t ) • If | θ − φ | = π 2 , we lose the signal completely = x ( t ) c T ( t ) c R ( t ) • Otherwise, the received signal is attenuated = x ( t ) cos( ω c t + θ ) cos( ω c t + φ ) • The phase relationship of the oscillators must be maintained over � 1 2 cos( θ − φ ) + 1 � = x ( t ) 2 cos(2 ω c t + θ + φ ) time 2 x ( t ) cos( θ − φ ) + 1 1 = 2 x ( t ) cos(2 ω c t + θ + φ ) • 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 ω c 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 13 J. McNames Portland State University ECE 223 Communications Ver. 1.11 14 Introduction to Asynchronous AM Demodulation Example 2: Asynchronous Amplitude Modulation • Asynchronous modulation does not require the carrier signal Example of Asynchronous Sinusoidal AM Modulation 0.4 c ( t ) be available in the receiver 0.2 • Thus, there is no need for synchronization x(t) • Asynchronous Modulation Assumptions: 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ω c ≫ ω x 1 −3 x 10 x ( t ) > 0 for all t c(t) 0 • However, it does require that the baseband signal x ( t ) be positive −1 • In this case, the envelope of the modulated signal y ( t ) is 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −3 approximately the same as the baseband signal x ( t ) x 10 0.2 y(t) • Thus, we can recover a good approximation of x ( t ) with an 0 −0.2 envelope detector 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (s) −3 x 10 J. McNames Portland State University ECE 223 Communications Ver. 1.11 15 J. McNames Portland State University ECE 223 Communications Ver. 1.11 16
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