Signal and Systems Prof. H. Sameti Chapter 8: 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

The Concept of Modulation  Why?  More efficient to transmit E&M 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) Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 2

Amplitude Modulation (AM) of a Complex Exponential Carrier Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 3

Demodulation of Complex Exponential AM Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 4 Corresponds to two separate modulation channels (quadratures) with carriers 90˚ out of phase.

Sinusoidal AM Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 5

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

Synchronous Demodulation in the Time Domain Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 7

Synchronous Demodulation (with phase error) in the Frequency Domain Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 8

Alternative: Asynchronous Demodulation Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 9

Asynchronous Demodulation (continued)Envelope Detector Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 10 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 [Acosω c t] 2 to make sure A+ x(t) > 0 ⇒Maximum power efficiency = 1/3 (P8.27)

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

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

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

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

Demultiplexing and Demodulation Book Chapter8: Section1 Computer Engineering Department, Signals and Systems 15 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

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

AM with an Arbitrary Periodic Carrier Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 17 C(t) – periodic with period T, carrier frequency ω c = 2π/T Remember: periodic in t discrete in ω

Modulating a (Periodic) Rectangular Pulse Train Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 18

Modulating a Rectangular Pulse Train Carrier, cont’d Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 19 For rectangular pulse

Observations Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 20 1) We get a similar picture with any c(t) that is periodic with period T 2) As long as ω c = 2π/T > 2ω M, there is no overlap in the shifted and scaled replicas of X(jω). 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 > 2ω M ⇒Pulse Amplitude Modulation

Sinusoidal Frequency Modulation (FM) Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 21 Amplitude fixed Instantaneous ω X(t) is signal To be transmitted

Sinusoidal FM (continued) Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 22  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(jω) —Tunable band-limited differentiator, over the bandwidth of y(t) …looks like AM envelope detection

DT Sinusoidal AM Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 23 Multiplication ↔Periodic convolution Example#1:

Example#2: Sinusoidal AM Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 24 i.e., No overlap of shifted spectra Drawn assuming:

Example #2 (continued):Demodulation Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 25 Possible as long as there is no overlap of shifted replicas of X(e jω ):

Example #3: An arbitrary periodic DT carrier Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 26 - Periodic convolution - Regular convolution

Example #3 (continued): Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 27

DT Sampling Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 28 Motivation: Reducing the number of data points to be stored or transmitted, e.g. in CD music recording.

DT Sampling (continued) Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 29 Note: Pick one out of N

DT Sampling Theorem Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 30 We can reconstruct x[n] if ω s = 2π/N > 2ω M Drawn assuming ω s > 2ω M Nyquist rate is met ⇒ ω M < π/N Drawn assuming ω s < 2ω M Aliasing!

Decimation — Downsampling Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 31 x p [n] has (n-1) zero values between nonzero values: Why keep them around? Useful to think of this as sampling followed by discarding the zero values

Illustration of Decimation in the Time-Domain (for N= 3) Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 32

Decimation in the Frequency Domain Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 33 Squeeze in time Expand in frequency - Still periodic with period 2π since X p (e jω ) is periodic with 2π/N

Illustration of Decimation in the Frequency Domain Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 34

The Reverse Operation: Upsampling(e.g.CD playback) Book Chapter8: Section2 Computer Engineering Department, Signal and Systems 35