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Signal and System Lecture 16

This document summarizes key concepts in signals and systems including AM modulation using periodic carriers, pulse train carriers for time-division multiplexing, sinusoidal frequency modulation, discrete-time sinusoidal AM, sampling theory, and decimation/interpolation. Specifically, it discusses how AM, FM, and pulse train modulation work in both continuous and discrete time. It also covers the sampling theorem, decimation which reduces the sample rate by discarding zeros, and interpolation which increases the sample rate.

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189 views20 pages

Signal and System Lecture 16

This document summarizes key concepts in signals and systems including AM modulation using periodic carriers, pulse train carriers for time-division multiplexing, sinusoidal frequency modulation, discrete-time sinusoidal AM, sampling theory, and decimation/interpolation. Specifically, it discusses how AM, FM, and pulse train modulation work in both continuous and discrete time. It also covers the sampling theorem, decimation which reduces the sample rate by discarding zeros, and interpolation which increases the sample rate.

Uploaded by

ali_rehman87
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Signals and Systems

Fall 2003
Lecture #16
30 October 2003

1. AM with an Arbitrary Periodic Carrier


2. Pulse Train Carrier and Time-Division Multiplexing
3. Sinusoidal Frequency Modulation
4. DT Sinusoidal AM
5. DT Sampling, Decimation,
and Interpolation
AM with an Arbitrary Periodic Carrier
Modulating a (Periodic) Rectangular Pulse Train
Modulating a Rectangular Pulse Train Carrier, cont’d

for rectangular pulse


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 ao ≠ 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 > 2 ωM
⇒ Pulse Amplitude Modulation
Sinusoidal Frequency Modulation (FM)

x(t) is signal
to be
transmitted

FM
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)

… looks like AM
envelope detection
DT Sinusoidal AM
Multiplication ↔ Periodic convolution

Example #1:
Example #2: Sinusoidal AM

No overlap of
shifted spectra
Example #2 (continued): Demodulation

Possible as long as there is


no overlap of shifted replicas
of X(ejω):

Misleading drawing – shown for a


very special case of ωc = π/2
Example #3: An arbitrary periodic DT carrier
Example #3 (continued):

2πa3 = 2πa0

No overlap when: ωc > 2ωM (Nyquist rate) ⇒ ωM < π/N


DT Sampling
Motivation: Reducing the number of data points to be stored or
transmitted, e.g. in CD music recording.
DT Sampling (continued)
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!
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

compressed in
time by N
Illustration of Decimation in the Time-Domain (for N = 3)
Decimation in the Frequency Domain

Squeeze in time
Expand in frequency
Illustration of Decimation in the Frequency Domain

After sampling

After discarding zeros


The Reverse Operation: Upsampling (e.g. CD playback)
N
x[n]

s s

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