Post on 30-Jun-2020
transcript
Lecture
Noise Analysis in Angle Modulation
Agenda
Noise Analysis in FM
Introduction
Noise in FM Receivers
Angle Modulation
Introduction
Frequency Modulation
The analysis for FM is rather complex
The receiver model is as shown in Figure
Angle Modulation
Introduction
The noise n(t) is modeled as white Gaussian noise of zero mean and power spectral density No/2
The received FM signal s(t) has a carrier frequency fc and transmission bandwidth B, such that only a negligible amount of power lies outside the frequency band fc ± B /2 for positive frequencies
The bandpass filter has a mid-band frequency fc and bandwidth B and therefore passes the FM signal essentially without distortion
Angle Modulation
Introduction
Ordinary, B is small compared with the mid-band frequency fc so that we may use a narrowband representation for n(t) in terms of its in-phase and quadrature components
In an FM system, the message information is transmitted by variations of the instantaneous frequency of a sinusoidal carrier wave, and its amplitude is maintained constant
Any variations of the carrier amplitude at the receiver input must result from noise or interference
Angle Modulation
Introduction The limiter is used to remove amplitude variations by
clipping the modulated wave at the filter output almost to the zero axis
The resulting rectangular wave is rounded off by another bandpass filter that is an integral part of the limiter, thereby suppressing harmonics of the carrier frequency
The filter output is again sinusoidal, with an amplitude that is practically independent of the carrier amplitude at the receiver input
Angle Modulation
Introduction The discriminator consists of two components:
A slope network or differentiator with a purely imaginary transfer function that varies linearly with frequency. It produces a hybrid-modulated wave in which both amplitude and frequency vary in accordance with the message signal
An envelope detector that recovers the amplitude variation and thus reproduces the message signal
The slope network and envelope detector are usually implemented as integral parts of a single physical unit
Angle Modulation
Introduction
The post-detection filter, labeled “low-pass filter,” has a bandwidth that is just large enough to accommodate the highest frequency component of the message signal
This filter removes the out-of-band components of the noise at the discriminator output and thereby keeps the effect of the output noise to a minimum
Angle Modulation
Noise in FM Receivers The filtered noise at the band-pass filter
output is defined as:
The incoming FM signal s(t) is given by
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Noise in FM Receivers
The noisy signal at the band-pass filter output is
The envelope of x(t) is of no interest to us, because any envelope variations at the band-pass output are removed by the limiter (Ac >> r(t))
Noise in FM Receivers With the discriminator assumed ideal, its
output is proportional to θ’(t)/2 π where θ’(t) is the derivative of θ(t) with respect to time
We need to make certain simplifying approximations so that the analysis may yield useful results
Angle Modulation
Noise in FM Receivers
This means that the additive noise nd(t)
appearing at the discriminator output is
determined effectively by the carrier
amplitude Ac and the quadrature
component nQ(t) of the narrowband noise
n(t)
The output of the discriminator v(t)
)(
2
1)(
dt
tdtv
)()( tntmk df
message additive noise
Noise in FM Receivers
Assume (t)-(t) is uniformly distributed over (0, 2), then the noise component nd(t) is independent on the message signal
From the definition of filtered noise we have:
Angle Modulation
)(sin)(2
1)( ttr
dt
d
Atn
c
d
)(sin)()( ttrtnQ
dt
tdn
Atn
Q
c
d
)(
2
1)(
Here the quadrature
component appears
Noise in FM Receivers Recall that:
Thus, we have:
Angle Modulation
fjdt
d TF
2.
nQ(t) nd(t)
dt
d
Ac2
1
)( fSQN )( fS
dN
)()(2
2
fSA
ffS
Qd N
c
N 2
,2
2
0 T
c
Bf
A
fN
Noise in FM Receivers
Angle Modulation
At the receiver output:
Noise output power:
Signal output power:
WfA
fNfS
c
N , )(2
2
0
0
W
Wc
dffA
Ntn
2
2
02
0 )(
)(22tmk f
3
22
3
0
cA
WN effect quieting noise
12
cA
Noise in FM Receivers
The output signal-to-noise ratio is defined as the ratio of the average output signal power to the average output noise power
In FM system, increasing the carrier power has a noise-quieting effect
Average power of output noise
The channel signal to noise ratio
Angle Modulation
Noise in FM Receivers
The average power in the modulated signal s(t) is
The average noise power in the message bandwidth is WNo
The channel signal to noise ratio
Figure of merit for frequency modulation
Angle Modulation
Noise in FM Receivers
Example: Single-Tone Modulation
A sinusoidal wave of frequency fm as the modulating signal, and assume a peak frequency deviation Δf . The FM signal is define by
The average power of message signal m(t)
Angle Modulation
)2sin()(20
t ff
fdm k m
m
t
f
)2cos()( sideboth t fk
ftm
dt
dm
f
Noise in FM Receivers
Example: Single-Tone Modulation
output signal-to-noise ratio
Where β = Δf /W is the modulation index and we get the figure of merit
It is important to note that the modulation index β = Δf /W is
determined by the bandwidth W of the postdetection low-pass filter and is related to the sinusoidal message frequency fm
Angle Modulation
Noise in FM Receivers
For an AM system operating with a sinusoidal modulating signal and 100 percent modulation, we have
It is of particular interest to compare the noise performance of AM and FM systems
Define β = 0.5 as the transition between narrowband FM and wide-band FM Angle Modulation
Noise in FM Receivers
Angle Modulation
Power spectral density of noise
at FM receiver output. Power spectral density of
a typical message signal.
Lecture Summary
Covered material Noise Analysis in FM
Angle Modulation