Adama Science and Technology University

School of Electrical Engineering and Computing

Department of Electronics and Communication Engineering

Introduction to Communication Systems

(ECEg- 3202)

Chapter Six: Noise in Frequency Modulation Sytems

June 4, 2024

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 Outlines

1 Introduction

2 FM Receiver Model

3 Noise in FM Receiver

4 FM Threshold Effect

5 Pre-emphasis and De-emphasis in FM Systems

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 Introduction
Fundamental difference between AM and FM:
Amplitude Modulation: message information contained in the signal
amplitude => Additive noise: corrupts directly the modulated signal.
Frequency Modulation: message information contained in the signal
frequency => The effect of noise on an FM signal is determined by the
extent to which it changes the frequency of the modulated signal.
Consequently, FM signals is less affected by noise than AM signals
A carrier waveform of angle modulated wave
s(t) = Acos(θi (t)), Where θi (t)is the instantaneous phase angle
The relationship between angular frequency ωi and angle θi (t) as
ˆ
dθi dθi
ωi = => 2πfi = => θi (t) = 2π fi (t)dt
dt dt
where fi (t) = fc + kf m(t) is the instantaneous frequency.
ˆ ˆ
θi (t) = 2π (fc + kf m(t))dt, s(t) = Acos(2πfc + 2πkf m(t)dt)

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 Introduction. . . .
Note:
The envelope is constant
Signal s(t) is a non-linear function of the message signal m(t).
Bandwidth of FM
mp ≡ max|m(t)|: Peak message amplitude

fc − kf mp < instantaneous frequency < fc + kf mp

frequency deviation= the deviation of the instantaneous frequency

from the carrier frequency: ∆f = kf mp
∆f kf m p
deviation ratio or modulation index: β = = , Where w is
w w
the message bandwidth
Small β: FM bandwidth 2x message bandwidth (narrow-band FM)
Large β: FM bandwidth ≫ 2x message bandwidth (wide-band FM)
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 Introduction. . . .

Carson’s rule:

BT = 2w (β + 1) = 2(∆f + w )

β ≪ 1 => BT = 2w (as in AM)

β ≫ 1 => BT = ∆f (independent of w)
Angle-modulation systems, particularly FM, provide a high degree of
noise immunity, making them desirable in cases of severe noise and/or
low signal power.
This noise immunity comes at the cost of increased channel
bandwidth requirements, which are significantly higher than those of
amplitude-modulation systems.

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 FM Receiver Model
In FM receivers, the voltage at the output of the audio detector is
directly proportional to the frequency deviation at its input.

Noise in the channel can cause amplitude variations in the FM wave.

An amplitude limiter removes unwanted peaks, maintaining constant
amplitude.
In an FM receiver, a de-emphasis network follows the FM
demodulator. This signal is then amplified by the audio amplifier to
increase power, restoring the original sound signal through the
loudspeaker
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 Noise in FM Receiver

Bandpass filter: Removes any signals outside the bandwidth of

fc ± B2T .
=> The pre-detection noise at the receiver is bandpass with a
bandwidth of BT .
FM signal has a constant envelope.
=> Use a limiter to remove any amplitude variations.
Discriminator: A device with output proportional to the deviation in
the instantaneous frequency.
Recovers the message signal.
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 Noise in FM Receiver. . . .

Final baseband low-pass filter: Has a bandwidth of W .

Passes the message signal and removes out-of-band noise
Linear argument at high SNR:
For high SNR, noise output and message signal are approximately
independent of each other.
Output ≈ Message + Noise.
=> We can compute the signal power for the case without noise and
accept that the result holds for the case with noise too.
Power of signal at the output without noise:
Instantaneous frequency of the input signal: fi = fc + kf m(t)
Output of discriminator: Vout ∝ ∆f = kf m(t)
Output signal power:
Ps = kf2 P
Where:
P : the average power of the message signal.

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 Noise in FM Receiver. . . .
Power of signal at the output with noise:
In the presence of additive noise, the real pre-detection signal is:
 ˆ t 
x(t) = Acos 2πfc + 2πkf m(t)dt + nI (t)cos2πfc t − nQ (t)sin2πfc t
0
For high SNR, noise output is approximately independent of the
message signal.
=> Therefore, we only have the carrier and noise signals present.
=> To calculate the power of output noise, we may use:
x̃(t) = Acos(2πfc t) + nI (t)cos2πfc t − nQ (t)sin2πfc t
Phasor diagram of the FM carrier and noise signals

Where
nc (t) = nI (t)
ns (t) = nQ (t)

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 Noise in FM Receiver. . . .

Instantaneous phase

nQ (t)
θi (t) = tan−1
A + nI (t)

nQ (t) nQ (t)
For large carrier power (Large A):θi (t) = tan−1 ≈
A A
Discriminator = instantaneous frequency

1 dθi (t) 1 dnQ (t)

fi (t) = =
2π dt 2πA dt
The Discriminator Output in the presence of signal and noise

1 dnQ (t)
kf m(t) +
2πA dt

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 Noise in FM Receiver. . . .
PSD of fi (t)
 2
1
PSD of fi (t) = |j2πf |2 × No = SD (f )
2πA
,  
BT
where PSD of nQ (t) = No within band ±
2

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 Noise in FM Receiver. . . .

Average nose power at the receiver output:

ˆ w
PN = SD (f )df
−w
ˆ w 2
2No w 3

1
= |j2πf |2 No df =
−w 2πA 3A2

Average noise power at the output of a FM receiver

1
∝
carrier power A2
A ↑=> Noise ↓ called quieting effect

3A2 kf2 P
SNRo = = SNRFM
2No W 3

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 Noise in FM Receiver. . . .

A2
Transmitted power of an FM waveform: PT =
2
PT
SNRbaseband =
No W

3kf2 P P
SNRFM = 2
SNRbaseband = 3β 2 2 SNRbaseband
W mP
Figure of merit(FoM)
P
FFM = 3β 2
mP2
Valid when the carrier power is large compared with the noise power

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 FM Threshold Effect
If carrier to noise ratio at the discriminator input ≫1
FM receiver breaks. From breaks to sputtering sounds.
3A2 kf2 P
SNRo = formula does not hold.
2No W 3
The SNRo of FM receiver is valid only if the carrier to noise ratio
measured at the discriminator input is high compared to unity.
As the input noise is increased so that the SNRo decreased, the FM
receiver breaks.
Near the break point SNRo equation begins to fail predicting values
of output SNR larger than the actual ones. This phenomenon is also
known as the threshold effect.
Threshold effect is the minimum carrier to noise ratio that gives the
output SNR not less than the value predicted by the usual signal to
noise formula assuming a small noise power.
The onset of FM threshold effects can be delayed by using techniques
such as FM feedback, phase locked loops and frequency locked loops.
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 Pre-emphasis and De-emphasis in FM Systems
Major encountered in FM receiver are capture and threshold effect
Used in FM transmitter and FM receiver to improve capture and
threshold effect
FM improvement not deteriorate from actual value

a. PSD of the noise at the detector output ∝ square of frequency.

b. PSD of a typical message typically rolls off at around 6dB per decade.
To increase SNRFM : Use a LPF to fc at the output
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 Pre-emphasis and De-emphasis in FM Systems. . . .

Hpe (f ) : used to artificially emphasize the high frequency components

of the message prior to modulation, and hence, before noise is
introduced.
hde (f ): used to de-emphasize the high frequency components at the
receiver, and restore the original PSD of the message signal.
In theory,
Hpe (f ) ∝ f , Hde (f ) ∝ 1/f
Pre-emphasis at the transmitter:
A filter that artificially emphasize the high-frequency components of
the message signal prior to the modulation.
f
Hpe = 1 + j
fo
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 Pre-emphasis and De-emphasis in FM Systems. . . .
Dolby noise reduction uses an analogous pre-emphasis technique to
reduce the effects of noise (hissing noise in audiotape recording is also
concentrated on high frequency)
De-emphasis at the receiver:
An inverse operation performed by a filter placed after the
demodulation.
The de-emphasis filter restores the original signal by de-emphasizing
the high-frequency components.
1
Hde =
f
1+j
fo
Effects of pre-emphasis and de-emphasis filters cancel each other:
 
 
f   1 =1

Hpe Hde = 1 + j
fo  f 
1+j
fo
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 Pre-emphasis and De-emphasis in FM Systems. . . .

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 Example

1. A frequency modulator is having frequency sensitivity

kf = 1KHz/volt. If the modulator is used modulate a single tone
signal of m(t) = 10cos(4000πt) and the modulated signal is
demodulated signal is demodulated at receiver by using frequency
discriminator. Determine the figure of merit of a receiver

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 Thank You

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