Quantization Noise: Example
A full-load sinusoidal m(t) of amplitude Am, average
2
power is: PS Am / 2
Full-load
Am=mmax
2
2
Average signal power is: PS Am / 2 mmax / 2
The quantizers output signal-to-noise ratio is:
Average signal power
(SNR ) O
Average
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Quadrature-Carrier Multiplexing
Exploit the quadrature null effect for good use
Quadrature-carrier multiplexing or quadratureamplitude modulation (QAM)
Use two DSB-SC modulated waves to occupy the same
band, yet a
Frequency Modulation
Nonlinear modulation:
s(t) is a nonlinear function of m(t)
Hard for spectral analysis
Analyze the simplest case: single-tone modulation
Information wave:
Instantaneous frequency:
Frequency deviation:
The angle i(t) is:
Modulation inde
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Example 3.1: Single-tone AM
Design factor: modulation factor
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Example 3.1 (contd.)
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Example 3.1
Narrow-Band FM
FM signal with single tone m(t) is
If |x|<1
cos(x
sin(x
For small , we have
x
Similar to a single tone AM signal with the same m(t)
Requires the same transmission bandwidth as AM
See Example 3.1
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Observations
The spectrum of an FM signal
n=0: a carrier component
Amplitude of the carrier is J ( )Ac
n >0 and n < 0: an infinite set of side frequencies located
symmetrically on both sides of the carrier at frequency
separations of fm, 2 fm, 3 fm,
Unli
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Vestigial Sideband Modulation
VSB
For signals do not have an energy gap at the origin
Send: one sideband + vestige of the other sideband
The filter is allowed to have a nonzero transition band
The question: how to
Wide-Band FM
For arbitrary = f/fm, use the complex representation
of band-pass signals
Assuming fc is large enough:
The complex envelope of s(t) is
A periodic signal with period 1/ fm and fundamental
frequency fm:
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Wide-Band
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Spectral Analysis
HW1 Prob. 3
Section 2.6: Fourier Transform of Periodic Signals
g (t nTs )
n
fs
G (nf s ) ( f
nf s )
(2.88)
n
Determine the Fourier transform of g (t) by applying the
duality property and Eqn. (2.88)
g (t )
g (nTs ) (t nTs )
Tells us the
Relationship Between PM and FM
Equivalence of PM and FM
Due to the integral/differentiative relationship between
phase and frequency
An FM signal with m(t) can be regarded as a PM signal with
A PM signal with m(t) can be regarded as an FM signal with
1 d
Reconstruct g(t)
Reconstruct g(t) from the samples
Interchanging the order of sum and integration
Shifted
sinc
function.
We have
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Time-shifting
property
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Reconstruct g(t) (contd.)
Interpolation formula
Reconstruct g(t) from th
Summary
Angle modulation
FM (frequency) and PM (phase)
Equivalence of the two
Properties
Hard for spectral analysis
Study single tone FM for insights
An empirical rule: Carsons rule for approximate evaluation of
the transmission bandwidth of FM BT
Determi
The Sampling
Process
Arbitrary signal
g(t) , but
With finite
energy
Specified for all
time t
Sample inst. at a
uniform rate
c(t)
t
0
Sampling rate:
fs = 1/Ts
Infinite sequence
of samples,
One every Ts
seconds
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The Sampling Pro