1. 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

2. The Concept of Modulation
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)
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3. Amplitude Modulation (AM) of a Complex Exponential Carrier
Book Chapter8: Section1
Amplitude Modulation (AM) of a Complex Exponential Carrier
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4. Demodulation of Complex Exponential AM
Book Chapter8: Section1
Demodulation of Complex Exponential AM
Corresponds to two separate modulation channels (quadratures) with carriers 90˚ out of phase.
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5. Sinusoidal AM Book Chapter8: Section1
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6. Synchronous Demodulation of Sinusoidal AM
Book Chapter8: Section1
Synchronous Demodulation of Sinusoidal AM
Suppose θ= 0 for now, ⇒
Local oscillator is
in phase with
the carrier.
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7. Synchronous Demodulation in the Time Domain
Book Chapter8: Section1
Synchronous Demodulation in the Time Domain
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8. Synchronous Demodulation (with phase error) in the Frequency Domain
Book Chapter8: Section1
Synchronous Demodulation (with phase error) in the Frequency Domain
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9. Alternative: Asynchronous Demodulation
Book Chapter8: Section1
Alternative: Asynchronous Demodulation
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10. Asynchronous Demodulation (continued)Envelope Detector
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)
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11. Double-Sideband (DSB) and Single-Sideband (SSB) AM
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
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12. Single Sideband Modulation
Book Chapter8: Section1
Single Sideband Modulation
Can also get SSB/SC or SSB/WC
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13. Frequency-Division Multiplexing (FDM)
Book Chapter8: Section1
Frequency-Division Multiplexing (FDM)
(Examples: Radio-station signals and analog cell phones)
All the channels can share
the same medium.
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14. FDM in the Frequency-Domain
Book Chapter8: Section1
FDM in the Frequency-Domain
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15. Demultiplexing and Demodulation
Book Chapter8: Section1
Demultiplexing and Demodulation
ωa 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
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16. The Superheterodyne Receiver
Book Chapter8: Section1
The Superheterodyne Receiver
Operation principle:
Down convert from ωc to ωIF, and use a coarse tunable BPF for the front end. (FCC: Federal Communications Commission)
Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.
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17. AM with an Arbitrary Periodic Carrier
Book Chapter8: Section2
AM with an Arbitrary Periodic Carrier
C(t) – periodic with period T, carrier frequency ωc = 2π/T
Remember: periodic in t discrete in ω
𝑎 𝑘 = 1 𝑇  𝑓𝑜𝑟 𝑖𝑚𝑝𝑢𝑙𝑠𝑒 𝑡𝑟𝑎𝑖𝑛 ⇓
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18. Modulating a (Periodic) Rectangular Pulse Train
Book Chapter8: Section2
Modulating a (Periodic) Rectangular Pulse Train
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19. Modulating a Rectangular Pulse Train Carrier, cont’d
Book Chapter8: Section2
Modulating a Rectangular Pulse Train Carrier, cont’d
𝑎𝑛𝑑 𝑎 0 = 𝛥 𝑇 ,   𝑎 𝑘 = sin 𝑘 𝜔 𝑐 𝛥 2 𝜋𝑘
For rectangular pulse
Drawn assuming:
𝜔 𝑐 >2 𝜔 𝑀
Nyquist rate is met
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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 time slots instead of frequency slots to different
channels, e.g. AT&T wireless phones
4) Really only need samples{x(nT)} when ωc> 2ωM⇒Pulse Amplitude
Modulation
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21. Sinusoidal Frequency Modulation (FM)
Book Chapter8: Section2
Sinusoidal Frequency Modulation (FM)
Amplitude fixed
Phase modulation: 𝜃 𝑡 = 𝜔 𝑐 𝑡+ 𝜃 0 + 𝑘 𝑝 𝑥 𝑡
Frequency modulation: 𝑑𝜃 𝑑𝑡 = 𝜔 𝑐 + 𝑘 𝑓 𝑥(𝑡)
X(t) is signal
To be
transmitted
Instantaneous ω
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22. Sinusoidal FM (continued)
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
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)
𝐻 𝑗𝜔 ≅𝑗𝜔 ⇓
…looks like AM envelope detection
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23. DT Sinusoidal AM Multiplication ↔Periodic convolution Example#1:
Book Chapter8: Section2
DT Sinusoidal AM
Multiplication ↔Periodic convolution
Example#1:
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24. Example#2: Sinusoidal AM
Book Chapter8: Section2
Example#2: Sinusoidal AM
Drawn assuming:
i.e.,
No overlap of
shifted spectra
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25. Example #2 (continued):Demodulation
Book Chapter8: Section2
Example #2 (continued):Demodulation
Possible as long as there is no
overlap of shifted replicas of
X(ejω):
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26. Example #3: An arbitrary periodic DT carrier
Book Chapter8: Section2
Example #3: An arbitrary periodic DT carrier
- Periodic convolution
- Regular convolution
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27. Example #3 (continued):
Book Chapter8: Section2
Example #3 (continued):
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28. Book Chapter8: Section2
DT Sampling
Motivation: Reducing the number of data points to be stored or
transmitted, e.g. in CD music recording.
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29. DT Sampling (continued)
Book Chapter8: Section2
DT Sampling (continued)
Note:
𝑥 𝑝 𝑛 = 𝑥 𝑛 , 𝑖𝑓 𝑛 𝑖𝑠 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑁 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ⇒
Pick one out of N
- periodic with period 𝜔 𝑠 = 2𝜋 𝑁
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30. DT Sampling Theorem We can reconstruct x[n] if ωs= 2π/N > 2ωM
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!
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31. Decimation — Downsampling
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
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32. Illustration of Decimation in the Time-Domain (for N= 3)
Book Chapter8: Section2
Illustration of Decimation in the Time-Domain (for N= 3)
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33. Decimation in the Frequency Domain
Book Chapter8: Section2
Decimation in the Frequency Domain
Squeeze in time
Expand in frequency
= 𝑋 𝑝 𝑒 𝑗 𝜔 𝑁
- Still periodic with period 2π
since Xp(ejω) is periodic with 2π/N
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34. Illustration of Decimation in the Frequency Domain
Book Chapter8: Section2
Illustration of Decimation in the Frequency Domain
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35. The Reverse Operation: Upsampling(e.g.CD playback)
Book Chapter8: Section2
The Reverse Operation: Upsampling(e.g.CD playback)
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