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