Signal and Systems Chapter 8: Modulation Complex Exponential - PowerPoint PPT Presentation

Signal and Systems Chapter 8: Modulation Complex Exponential Amplitude Modulation • Sinusoidal AM • Demodulation of Sinusoidal AM • Single-Sideband (SSB) AM • Frequency-Division Multiplexing • Superheterodyne Receivers • AM with an Arbitrary Periodic Carrier • Pulse Train Carrier and Time-Division Multiplexing • Sinusoidal Frequency Modulation • DT Sinusoidal AM • DT Sampling, Decimation, and Interpolation •

Book Chapter8: Section1 The Concept of Modulation  Why?  More efficient to transmit signals at higher frequencies  Transmitting multiple signals through the same medium using different carriers  Transmitting through “ channels ” with limited pass-bands  Others …  How?  Many methods  Focus here for the most part on Amplitude Modulation (AM) Computer Engineering Department, Signals and Systems 2

Book Chapter8: Section1 A mplitude M odulation (AM) of a Complex Exponential Carrier     j t c t ( ) e , c carrier frequency c   j t y t ( ) x t e ( ) c 1      Y j ( ) X j ( ) C j ( )  2 1        X j ( ) 2 ( )  c 2     X j ( ( )) c Computer Engineering Department, Signals and Systems 3

Book Chapter8: Section1 Demodulation of Complex Exponential AM      j t e cos t j sin t c c c  Corresponds to two separate modulation channels (quadratures) with carriers 90 ˚ out of phase. Computer Engineering Department, Signals and Systems 4

Book Chapter8: Section1 Sinusoidal AM 1               Y j ( ) X j ( ) { ( ) ( )}  c c 2 1 1         X j ( ( )) X j ( ( ))  c c 2 2 Drawn assuming    c M Computer Engineering Department, Signals and Systems 5

Book Chapter8: Section1 Synchronous Demodulation of Sinusoidal AM  Suppose θ= 0 for now, ⇒ Local oscillator is in phase with the carrier . Computer Engineering Department, Signals and Systems 6

Book Chapter8: Section1 Synchronous Demodulation in the Time Domain 1       2 w t ( ) y t ( )cos t x t ( )cos t x t ( )cos2 t c c c 2 High-Frequency Signals filterd out by the LRF  Then ( ) r t x t ( )   Now suppose there is phase difference, i.e. 0, then          w t ( ) y t ( )cos( t ) x t ( )cos t cos( t ) c c c 1 1       x t ( )cos x t ( )(cos(2 t )) c 2 2 HF signal   Now ( ) r t x t ( )cos Two special cases:      0 1) 2, the local oscillator is 90 out of phase with the carrier, ( ) r t 0,signal unrecoverable.         2) ( ) t slowly var ying with time, ( ) r t cos[ ( )] t x t ( ), time-varying "gain". Computer Engineering Department, Signals and Systems 7

Book Chapter8: Section1 Synchronous Demodulation (with phase error) in the Frequency Domain   Demodulating signal has phase difference w.r.t. the modulating si gnal 1 1            j j t j j t t e e e e cos( ) c c c 2 2  F               j j e ( ) e ( ) c c       Again, the low-frequency signal ( )when 2. M Computer Engineering Department, Signals and Systems 8

Book Chapter8: Section1 Alternative: Asynchronous Demodulation    Assume ,so signal envelope looks like ( ) x t c M   A 0 DSB/SC (Double Side Band, Suppressed Carrier)   DSB/WC (Double Side Band, With Carrier ) A 0 Computer Engineering Department, Signals and Systems 9

Book Chapter8: Section1 Asynchronous Demodulation (continued)Envelope Detector In order for it to function properly, the envelope function must be positive for all time, i.e. A + x ( t ) > 0 for all t. Demo: Envelope detection for asynchronous demodulation. Advantages of asynchronous demodulation: — Simpler in design and implementation. Disadvantages of asynchronous demodulation: — Requires extra transmitting power [ A cosω c t ] 2 to make sure A + x ( t ) > 0 ⇒ Maximum power efficiency = 1/3 (P8.27) Computer Engineering Department, Signals and Systems 10

Book Chapter8: Section1 Double-Sideband (DSB) and Single- Sideband (SSB) AM Since x ( t ) and y ( t ) are real , from Conjugate symmetry both LSB and USB signals carry exactly the same information. DSB, occupies 2 ω M bandwidth in ω> 0 Each sideband approach only occupies ω M bandwidth in ω> 0 Computer Engineering Department, Signals and Systems 11

Book Chapter8: Section1 Single Sideband Modulation Can also get SSB/SC or SSB/WC Computer Engineering Department, Signals and Systems 12

Book Chapter8: Section1 Frequency-Division Multiplexing (FDM)  (Examples: Radio-station signals and analog cell phones) All the channels can share the same medium. Computer Engineering Department, Signals and Systems 13

Book Chapter8: Section1 FDM in the Frequency-Domain Computer Engineering Department, Signals and Systems 14

Book Chapter8: Section1 Demultiplexing and Demodulation ω a needs to be tunable  Channels must not overlap ⇒ Bandwidth Allocation  It is difficult (and expensive) to design a highly selective band-pass filter with a tunable center frequency  Solution – Superheterodyne Receivers Computer Engineering Department, Signals and Systems 15

Book Chapter8: Section1 The Superheterodyne Receiver  Operation principle:  Down convert from ω c to ω IF , and use a coarse tunable BPF for the front end. (FCC: Federal Communications Commission)  Use a sharp-cutoff fixed BPF at ω IF to get rid of other signals. Computer Engineering Department, Signals and Systems 16

Book Chapter8: Section2 AM with an Arbitrary Periodic Carrier C ( t ) – periodic with period T , carrier frequency ω c = 2 π/T Remember: periodic in t discrete in ω  𝑏 𝑙 = 1         C ( j ) 2 a ( k ) 𝑈 𝑔𝑝𝑠 𝑗𝑛𝑞𝑣𝑚𝑡𝑓 𝑢𝑠𝑏𝑗𝑜 k c  k ⇓  1            Y ( j ) X ( j )* C j ( ) X ( j )* a ( k )  k c 2  k       a X ( ( j k )) k c  k Computer Engineering Department, Signal and Systems 17

Book Chapter8: Section2 Modulating a (Periodic) Rectangular Pulse Train  y ( t ) x ( t ). c ( t ) Computer Engineering Department, Signal and Systems 18

Book Chapter8: Section2 Modulating a Rectangular Pulse Train Carrier, cont ’ d          C j ( ) 2 a ( k ) k c  k 𝑏𝑜𝑒 𝛦 2 sin 𝑙𝜕 𝑑 𝑏 0 = 𝛦 𝑈 , 𝑏 𝑙 = 𝜌𝑙 For rectangular pulse 1     Y ( j ) X ( j )* C ( j )  2 Drawn assuming: 𝜕 𝑑 > 2𝜕 𝑁 Nyquist rate is met Computer Engineering Department, Signal and Systems 19

Book Chapter8: Section2 Observations 1) We get a similar picture with any c ( t ) that is periodic with period T 2 ) As long as ω c = 2 π/ T > 2 ω M , there is no overlap in the shifted and scaled replicas of X ( j ω). Consequently, assuming a 0 ≠ 0: x ( t ) can be recovered by passing y ( t ) through a LPF 3) Pulse Train Modulation is the basis for Time-Division Multiplexing Assign time slots instead of frequency slots to different channels, e.g. AT&T wireless phones 4) Really only need samples { x ( nT )} when ω c > 2 ω M ⇒ Pulse Amplitude Modulation Computer Engineering Department, Signal and Systems 20

Book Chapter8: Section2 Sinusoidal F requency M odulation (FM)   y ( t ) A cos( ( t )) Amplitude fixed Phase modulation: 𝜄 𝑢 = 𝜕 𝑑 𝑢 + 𝜄 0 + 𝑙 𝑞 𝑦 𝑢 𝑒𝜄 X(t) is signal Frequency modulation: 𝑒𝑢 = 𝜕 𝑑 + 𝑙 𝑔 𝑦(𝑢) To be Instantaneous ω transmitted Computer Engineering Department, Signal and Systems 21

Book Chapter8: Section2 Sinusoidal FM (continued)  Transmitted power does not depend on x ( t ): average power = A 2 /2  Bandwidth of y ( t ) can depend on amplitude of x ( t )  Demodulation a) Direct tracking of the phase θ( t ) (by using phase-locked loop) b) Use of an LTI system that acts like a differentiator H ( j ω) — Tunable band-limited differentiator, over the bandwidth of y ( t ) 𝐼 𝑘𝜕 ≅ 𝑘𝜕 ⇓ … looks like AM dy t ( )      u t ( ) ( w k x t ( )) A sin ( ) t envelope c f dt  d / dt detection Computer Engineering Department, Signal and Systems 22

Book Chapter8: Section2 DT Sinusoidal AM Multiplication ↔ Periodic convolution Example#1: e   j n c n [ ] c             j c e ( ) 2 ( 2 k ) c  k 1      j j j Y e ( ) X e ( ) C e ( )  2 Computer Engineering Department, Signal and Systems 23