Signal and Systems Prof. H. Sameti

1. Signal and SystemsProf. 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

2. Book Chapter8: Section1 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)

3. Book Chapter8: Section1 Amplitude Modulation (AM) of a Complex Exponential Carrier

4. Book Chapter8: Section1 Demodulation of Complex Exponential AM Corresponds to two separate modulation channels (quadratures) with carriers 90˚ out of phase.

5. Book Chapter8: Section1 Sinusoidal AM

6. Book Chapter8: Section1 Synchronous Demodulation of Sinusoidal AM • Suppose θ= 0 for now, ⇒ Local oscillator is in phase with the carrier.

7. Book Chapter8: Section1 Synchronous Demodulation in the Time Domain

8. Book Chapter8: Section1 Synchronous Demodulation (with phase error) in the Frequency Domain

9. Book Chapter8: Section1 Alternative: Asynchronous Demodulation

10. 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 [Acosωct]2to make sure A+ x(t) > 0 ⇒Maximum power efficiency = 1/3 (P8.27)

11. 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 2ωMbandwidth in ω> 0 Each sideband approach only occupies ωM bandwidth in ω> 0

12. Book Chapter8: Section1 Single Sideband Modulation Can also get SSB/SC or SSB/WC

13. Book Chapter8: Section1 Frequency-Division Multiplexing (FDM) • (Examples: Radio-station signals and analog cell phones) All the channels can share the same medium.

14. Book Chapter8: Section1 FDM in the Frequency-Domain

15. Book Chapter8: Section1 Demultiplexing and Demodulation ωaneeds 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

16. Book Chapter8: Section1 The Superheterodyne Receiver • Operation principle: • Down convert from ωc to ωIF, and use a coarse tunable BPF for the front end. • Use a sharp-cutofffixed BPF at ωIF to get rid of other signals.

17. Book Chapter8: Section2 AM with an Arbitrary PeriodicCarrier C(t) – periodic with period T, carrier frequency ωc = 2π/T Remember: periodic in t discrete in ω

18. Book Chapter8: Section2 Modulating a (Periodic) Rectangular Pulse Train

19. Book Chapter8: Section2 Modulating a Rectangular Pulse Train Carrier, cont’d For rectangular pulse Drawn assuming: Nyquist rate is met

20. Book Chapter8: Section2 Observations 1) We get a similar picture with any c(t) that is periodic with period T 2) As long as ωc= 2π/T > 2ωM, there is no overlap in the shifted and scaled replicas of X(jω). Consequently, assuming a0≠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 timeslots instead of frequencyslots to different channels, e.g. AT&T wireless phones 4) Really only need samples{x(nT)} when ωc> 2ωM⇒Pulse Amplitude Modulation

21. Book Chapter8: Section2 Sinusoidal Frequency Modulation (FM) Amplitude fixed Phase modulation: Frequency modulation: X(t) is signal To be transmitted Instantaneous ω

22. Book Chapter8: Section2 Sinusoidal FM (continued) • Transmitted power does not depend on x(t): average power = A2/2 • Bandwidth of y(t) can depend on amplitude of x(t) • Demodulation • 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

23. Book Chapter8: Section2 DT Sinusoidal AM Multiplication ↔Periodic convolution Example#1:

24. Book Chapter8: Section2 Example#2: Sinusoidal AM Drawn assuming: i.e., No overlap of shifted spectra

25. Book Chapter8: Section2 Example #2 (continued):Demodulation Possible as long as there is no overlap of shifted replicas of X(ejω):

26. Book Chapter8: Section2 Example #3: An arbitrary periodic DT carrier - Periodic convolution - Regular convolution

27. Book Chapter8: Section2 Example #3 (continued):

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

29. Book Chapter8: Section2 DT Sampling (continued) Note: Pick one out of N - periodic with period

30. Book Chapter8: Section2 DT Sampling Theorem We can reconstruct x[n] if ωs= 2π/N > 2ωM Drawn assuming ωs > 2ωM Nyquist rate is met ⇒ ωM< π/N Drawn assuming ωs < 2ωM Aliasing!

31. Book Chapter8: Section2 Decimation — Downsampling xp[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

32. Book Chapter8: Section2 Illustration of Decimation in the Time-Domain (for N= 3)

33. Book Chapter8: Section2 Decimation in the Frequency Domain Squeeze in time Expand in frequency - Still periodic with period 2π since Xp(ejω) is periodic with 2π/N

34. Book Chapter8: Section2 Illustration of Decimation in the Frequency Domain

35. Book Chapter8: Section2 The Reverse Operation: Upsampling(e.g.CD playback)