ANALOG MODULATION

ANALOG MODULATION

PART II: ANGLE MODULATION

1999 BG Mobasseri 2

What is Angle Modulation?

In angle modulation, information is embedded in the angle of the carrier.

We define the angle of a modulated carrier by the argument of...

s t( )=Ac cosθ t( )( )

1999 BG Mobasseri 3

Phasor Form

In the complex plane we have

t=1

t=0

t=3

Phasor rotates with nonuniform speed

1999 BG Mobasseri 4

Angular Velocity

Since phase changes nonuniformly vs. time, we can define a rate of change

This is what we know as frequency

ωi =dθi(t)

dt

s t( )=Ac cos2πfct+φc

θi t( )1 2 4 3 4

⎛

⎝ ⎜

⎞

⎠ ⎟ ⇒

dθi

dt=2πfc

1999 BG Mobasseri 5

Instantaneous Frequency

We are used to signals with constant carrier frequency. There are cases where carrier frequency itself changes with time.

We can therefor talk about instantaneous frequency defined as

fi t( )=12π

dθi t( )dt

1999 BG Mobasseri 6

Examples of Inst. Freq.

Consider an AM signal

Here, the instantaneous frequency is the frequency itself, which is constant

s t( )= 1+km(t)[ ]cos2πfct+φc

θi t( )1 2 4 3 4

⎛

⎝ ⎜

⎞

⎠ ⎟ ⇒

dθi

dt=2πfc

1999 BG Mobasseri 7

Impressing a message on the angle of carrier

There are two ways to form a an angle modulated signal.– Embed it in the phase of the carrier

Phase Modulation(PM)– Embed it in the frequency of the carrier

Frequency Modulation(FM)

1999 BG Mobasseri 8

Phase Modulation(PM)

In PM, carrier angle changes linearly with the message

Where – 2πfc=angle of unmodulated carrier

– kp=phase sensitivity in radians/volt

s t( )=Ac cosθi t( )( ) =Ac cos2πfct+kpmt( )( )

1999 BG Mobasseri 9

Frequency Modulation

In FM, it is the instantaneous frequency that varies linearly with message amplitude, i.e.

fi(t)=fc+kfm(t)

1999 BG Mobasseri 10

FM Signal

We saw that I.F. is the derivative of the phase

Therefore,

fi t( )=12π

dθi t( )dt

θi t( ) =2πfct+2πkf mt( )0

t

∫

s t( )=Ac cos2πfct+2πkf m(t)dt0

t

∫⎡

⎣ ⎢

⎤

⎦ ⎥


FM for Tone Signals

Consider a sinusoidal message The instantaneous frequency

corresponding to its FM version is

m(t) =Amcos2πfmt( )

fi t( )= fc +kf m(t)

= fcresting frequency

{ +kf Amcos2πfmt( )


Illustrating FM

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

FM message

Inst.frequencyMoves with theMessage amplitude


Frequency Deviation

Inst. frequency has upper and lower bounds given by

fi t( )= fc +Δf cos2πfmt( )

where

Δf = frequency deviation=kf Am

then

fi max= fc +Δf

fi min= fc −Δf


FM Modulation index

The equivalent of AM modulation index is which is also called deviation ratio. It quantifies how much carrier frequency swings relative to message bandwidth

β =ΔfW

baseband{

orΔffm

tone{


Example:carrier swing

A 100 MHz FM carrier is modulated by an audio tone causing 20 KHz frequency deviation. Determine the carrier siwng and highest and lowest carrier frequencies

Δf =20KHz

frequency swing=2Δf =40KHz

frequency range:

fhigh=100MHz+20KHz=100.02MHz

flow =100MHz−20KHz=99.98MHz


Example: deviation ratio

What is the modulation index (or deviation ratio) of an FM signal with carrier swing of 150 KHz when the modulating signal is 15 KHz?

Δf =1502

=75KHz

β =Δffm

=7515

=5


Myth of FM

Deriving FM bandwidth is a lot more involved than AM

FM was initially thought to be a bandwidth efficient communication because it was thought that FM bandwidth is simply 2f

By keeping frequency deviation low, we can use arbitrary small bandwidth


FM bandwidth

Deriving FM bandwidth is a lot more involved than AM and it can barely be derived for sinusoidal message

There is a graphical way to illustrate FM bandwidth


Piece-wise approximation of baseband

Look at the following representation

1/2W

Baseband bandwidth=W


Corresponding FM signal

FM version of the above is an RF pulse for each square pulse.

The frequency of the kth RF pulse at t=tk is given by the height of the pulse. i.e.

fi = fc +kfmtk( )


Range of frequencies?

We have a bunch of RF pulses each at a different frequency.

Inst.freq corresponding to square pulses lie in the following range

fi max= fc +kfmmax

fi min= fc +kf mmin

mmin

mmax


A look at the spectrum

We will have a series of RF pulses each at a different frequency. The collective spectrum is a bunch of sincs

f

highestlowest

4W


So what is the bandwidth?

Measure the width from the first upper zero crossing of the highest term to the first lower zero crossing of the lowest term

f

highestlowest


Closer look

The highest sinc is located at fc+kfmp

Each sinc is 1/2W wide. Therefore, their zero crossing point is always 2W above the center of the sinc.

f2W


Range of frequenices

Above range lies

<fc-kfmp-2W,fc+kfmp+2W>

f

highestlowest


FM bandwidth

The range just defined is one expression for FM bandwidth. There are many more!

BFM=4W+2kfmp

Using =∆f/W with ∆f=kfmp

BFM=2(+2)W


Carson’s Rule

A popular expression for FM bandwidth is Carson’s rule. It is a bit smaller than what we just saw

BFM=2(+1)W


Commercial FM

Commercial FM broadcasting uses the following parameters– Baseband;15KHz– Deviation ratio:5– Peak freq. Deviation=75KHz

BFM=2(+1)W=2x6x15=180KHz


Wideband vs. narrowband FM

NBFM is defined by the condition– ∆f<<W BFM=2W

– This is just like AM. No advantage here

WBFM is defined by the condition– ∆f>>W BFM=2 ∆f

– This is what we have for a true FM signal


Boundary between narrowband and wideband FM

This distinction is controlled by – If >1 --> WBFM– If <1-->NBFM

Needless to say there is no point for going with NBFM because the signal looks and sounds more like AM


Commercial FM spectrum

The FM landscape looks like this

FM station BFM station A FM station C

25KHz guardband

150 KHz

200 KHz

carrier


FM stereo:multiplexing

First, two channels are created; (left+right) and (left-right)

Left+right is useable by monaural receivers

-

Left channel

Right channel

+

+

+

mono


Subcarrier modulation

The mono signal is left alone but the difference channel is amplitude modulated with a 38 KHz carrier

Left channel

Right channel

+

+

+

mono

DSB-SCfsc=38 kHz

+

fsc=38KHz

freqdivider

Composite baseband

-


Stereo signal

Composite baseband signal is then frequency modulated

Left channel

Right channel

+

+

+

mono

DSB-SCfsc=38 kHz

+

fsc=38KHz

freqdivider

Composite baseband

FM transmitter

-


Stereo spectrum

Baseband spectrum holds all the information. It consists of composite baseband, pilot tone and DSB-SC spectrum

38 KHz19 KHz

15 KHz

Left+rightDSB-SC


Stereo receiver

First, FM is stripped, i.e. demodulated Second, composite baseband is lowpass

filtered to recover the left+right and in parallel amplitude demodulated to recover the left-right signal

38 KHz19 KHz

15 KHz

Left+rightDSB-SC


Receiver diagram

FMreceiver

lowpass filter(15K)

bandpassat 38KHz

X lowepass

VCODivide 2

X lowpass

+

+-

+

++

Left+right left

right

PLL

coherent detector

38 KHz19 KHz

15 KHz


Subsidiary communication authorization(SCA)

It is possible to transmit “special programming” ,e.g. commercial-free music for banks, department stores etc. embedded in the regular FM programming

Such programming is frequency multiplexed on the FM signal with a 67 KHz carrier and 7.5 KHz deviation


SCA spectrum

38 KHz19 KHz15 KHz

Left+rightDSB-SC

59.5 67 74.5 f(KHz)

SCA signal


FM receiver

FM receiver is similar to the superhet layout

RF

mixer

LO

limiterDiscrimi-

natordeemphasis

AF poweramp

IF


Frequency demodulation

Remember that message in an FM signal is in the instantaneous frequency or equivalently derivative of carrier angle

s t( )=Ac cos2πfct+2πkf m(t)dt0

t

∫⎡

⎣ ⎢

⎤

⎦ ⎥

′ s t( )=Ac 2πfc +2πkf mt( )[ ]sin 2πfct+2πkf m(t)dt−∞

t

∫⎛

⎝ ⎜

⎞

⎠ ⎟

Do envelope detection on s’(t)


Receiver components:RF amplifier

AM may skip RF amp but FM requires it FM receivers are called upon to work with

weak signals (~1V or less as compared to 30 V for AM)

An RF section is needed to bring up the signal to at least 10 to 20 V before mixing


Limiter

A limiter is a circuit whose output is constant for all input amplitudes above a threshold

Limiter’s function in an FM receiver is to remove unwanted amplitude variations of the FM signal

Limiter


Limiting and sensitivity

A limiter needs about 1V of signal, called quieting or threshold voltage, to begin limiting

When enough signal arrives at the receiver to start limiting action, the set quiets, i.e. background noise disappears

Sensitivity is the min. RF signal to produce a specified level of quieting, normally


Sensitivity example

An FM receiver provides a voltage gain of 200,000(106dB) prior to its limiter. The limiter’s quieting voltage is 200 mV. What is the receiver’s sensitivity?

What we are really asking is the required signal at RF’s input to produce 200 mV at the output

200 mV/200,000= 1V->sensitivity


Discriminator

The heart of FM is this relationship

What we need is a device that linearly follows inst. frequency

fi(t)=fc+kfm(t)

Disc.output

f

Deviation limits

+75 KHz-75 KHz

fcarrier

fcarrier is at the IF frequencyOf 10.7 MHz


Examples of discriminators

Slope detector - simple LC tank circuit operated at its most linear response curve

This setup turns an FM signalinto an AM

fc fo

output

f


Phase-Locked Loop

PLL’s are increasingly used as FM demodulators and appear at IF output

Phase

comparator

Lowpass

filter

VCO

fin Error signal

fvcoVCO input

Control signal:constantWhen fin=fvco

Output proportional toDifference between fin and fvco


PLL states

Free-running– If the input and VCO frequency are too far apart,

PLL free-runs

Capture– Once VCO closes in on the input frequency, PLL

is said to be in the tracking or capture mode

Locked or tracking– Can stay locked over a wider range than was

necessary for capture


PLL example

VCO free-runs at 10 MHZ. VCO does not change frequency until the input is within 50 KHZ.

In the tracking mode, VCO follows the input to ±200 KHz of 10 MHz before losing lock. What is the lock and capture range?– Capture range= 2x50KHz=100 KHz– Lock range=2x200 KHz=400 KHz


Advantages of PLL

If there is a carrier center frequency or LO frequency drift, conventional detectors will be untuned

PLL, on the other hand, can correct itself. PLL’s need no tuned circuits

fc fo

output

f

If fc drifts detector has no way of correcting itselfSlope detector


Zero crossing detector

Hard limiter

Zero Crossingdetector

Multi-vibrator

Averagingcircuot

FM Output

FM input

Hard limiter

ZC detector

multiV

more frequentZC’s meanshigher inst freqin turn meansLarger messageamplitudes

Averaging circuit

NOISE IN ANALOG MODULATION

AMPLITUDE MODULATION


Receiver Model

The objective here is to establish a relationship between input and and output SNR of an AM receiver

BPF detector

Noise n(t)

Modulated signal s(t)l

output

filter

fc-fc

BT=2W


Establishing a reference SNR

Define “channel” SNR measured at receiver input

(SNR)c=avg. power of modulated signal/

avg. noise power in the message bandwidth


Noise in DSB-SC Receiver

Tuner plus coherent detection

BPF LPFDSB-SC

n(t) Cos(2πfct)

x(t) v(t)

s(t)

s t( )=Acm(t)cos2πfct( )

<s2 t( )>=avg.power=Ac2 <m2(t)>/2=Ac

2P /2

P =avg. message power


Receiver input SNR

Also defined as channel SNR:

(SNR)c =Ac

2P /2WNo

noise power in the message bandwidth{

=Ac

2P2WNo

W-W

No/2Flat noise spectrum:white noise

Noise power=hatched area


Output SNR

Carrying signal and noise through the rest of the receiver, it can be shown that output SNR comes out to be equal to the input. Hence

Therefore, any reduction in input SNR is linearly reflected in the output

SNR( )oSNR( )c

=1


(SNR)o for DSB-AM

Following a similar approach,

Best case is achieved for 100% modulation index which, for tone modulation, is only 1/3

SNR( )oSNR( )c

=k2P

1+k2P<1

k: AM modulation index

P :avg. message power


DSB-AM and DSB-SC noise performance

An AM system using envelope detection needs 3 times as much power to achieve the same output SNR as a suppressed carrier AM with coherent detection

This is a result similar to power efficiency of the two schemes


Threshold effect-AM

In DSB-AM (not DSB-SC) there is a phenomenon called threshold effect

This means that there is a massive drop in output SNR if input SNR drops below a threshold

For DSB-AM with envelope detection, this threshold is about 6.6 dB

NOISE IN ANALOG MODULATION

FREQUENCY MODULATION


Receiver model

Noisy FM signal at BPF’s output is

BFP LimiterFM

detectorLPF(W)

n(t)

FMs(t)

x t( )=s t( )+n(t) =

Ac cos2πfct+φ t( )( )+r(t)cos2πfct+ψ t( )( )noise

1 2 4 4 4 3 4 4 4

where

φ t( )= m(t)dt∫


Phasor model

We can see the effect of noise graphically

(t)-(t)

reference

(t) (t)

FM signal

nois

e

Received signal

The angle FM detector will extract

(t)


Small noise

For small noise, it can be approximated that the noise inflicted phase error is

=[r⁄Ac]Sin( So the angle available to the FM detector

is + FM Detector computes the derivative of

this angle. It will then follow that...


FM SNR for tone modulation

Skipping further detail, we can show that for tone modulation, we have the following ratio

SNR rises as power of 2 of bandwidth; e.g. doubling deviation ratio quadruples the SNR

SNR( )oSNR( )c

=32

β2

Bandwidth-SNR exchange


Comparison with AM

In DSB-SC the ratio was 1 regardless. For commercial FM, =5. Therefore,

(SNR)o/(SNR)c=(1.5)x25=37.5

Compare this with just 1 for AM


Capture effect in FM

An FM receiver locks on to the stronger of two received signals of the same frequency and suppresses the weaker one

Capture ratio is the necessary difference(in dB) between the two signals for capture effect to go into action

Typical number for capture ratio is 1 dB


Normalized transmission bandwidth

With all these bandwidths numbers, it is good to have a normalized quantity.

Define

normalized bandwidth=Bn=BT/W

Where W is the baseband bandwidth


Examples of Bn

For AM:

Bn=BT/W=2W/W=2

For FM

Bn=BT/W~2 to 3 For =5 in commercial FM, this is a very

large expenditure in bandwidth which is rewarded in increased SNR


Noise/bandwidth summary

AM-envelope detection

SNR( )o =μ2

2+μ2 SNR( )c

Bn =2



DSB-SC/coherent detection

(SNR)o=(SNR)c

Bn=2

SSB

(SNR)o=(SNR)c

Bn=1



FM-tone modulation and =5

(SNR)o=1.5 2(SNR)c=37.5 (SNR)c

Bn~16 for =5


Preemphasis and deemphasis

High pitched sounds are generally of lower amplitude than bass. In FM lower amplitudes means lower frequency deviation hence lower SNR.

Preemphasis is a technique where high frequency components are amplified before modulation

Deemphasis network returns the baseband to its original form


Pre/deemphasis response

Flat up to ~500Hz, rises from 500-15000 Hz

500 Hz 2120 Hz 15KHz

-17dB

17dB

+3dB

-3dB

preemphasis

deemphasis

Deemphasis circuitIs between the detectorAnd the audio amplifier


Suggested homework

3.41 5.3 5.7

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ANALOG MODULATION