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
4) Really only need samples {x(nT)} when ω
c
> 2 ω
M
Pulse Amplitude Modulation
x(t) can be recovered by passing y(t) through a LPF
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
o
≠ 0:
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
Sinusoidal Frequency Modulation (FM)
FM
x(t) is signal
to be
transmitted
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(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(e
jω
):
Misleading drawing – shown for a
very special case of ω
c
= π/2
Example #3,An arbitrary periodic DT carrier
Example #3 (continued):
No overlap when,ω
c
> 2ω
M
(Nyquist rate)?ω
M
< π/N
2πa
3
= 2πa
0
DT Sampling
Motivation,Reducing the number of data points to be stored or
transmitted,e.g,in CD music recording.
DT Sampling (continued)
Drawn assuming
ω
s
< 2ω
M
Aliasing!
DT Sampling Theorem
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
Decimation — Downsampling
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
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
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
4) Really only need samples {x(nT)} when ω
c
> 2 ω
M
Pulse Amplitude Modulation
x(t) can be recovered by passing y(t) through a LPF
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
o
≠ 0:
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
Sinusoidal Frequency Modulation (FM)
FM
x(t) is signal
to be
transmitted
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(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(e
jω
):
Misleading drawing – shown for a
very special case of ω
c
= π/2
Example #3,An arbitrary periodic DT carrier
Example #3 (continued):
No overlap when,ω
c
> 2ω
M
(Nyquist rate)?ω
M
< π/N
2πa
3
= 2πa
0
DT Sampling
Motivation,Reducing the number of data points to be stored or
transmitted,e.g,in CD music recording.
DT Sampling (continued)
Drawn assuming
ω
s
< 2ω
M
Aliasing!
DT Sampling Theorem
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
Decimation — Downsampling
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
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