AM PM Modulation

7/30/2019 AM PM Modulation

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COMS3100/7100n ro uc on o

Lecture 10/11: Angle Modulation

This lecture:

Phase (PM) and Frequency Modulation (FM) Signals

Tone ModulationTransmission BW Estimates and Distortion

Ref: Carlson, Chapter 5,



Phase and Frequency Modulation

2For all linear CW modulation schemes spectrum (× in time ≡ in frequency)

SNR can be improved only by increased transmitted power

Modulated spectrum is not a translated copy of the message

Transmission BW is usually much greater than twice the

SNR can be improved by increased bandwidth.

Phase modulation (PM)

Lecture10/11COMS3100

requency mo u a on



PM and FM Signals

3CW signal with constant envelope ‐)](cos[)( t t At x ccc φ ω +=

Total instantaneous angle defined)(ˆ)( t t t cc φ ω θ +=

Information expressed via angle , )](cos[)(t

ccc t At xθ

θ =

Angle modulation schemesc=

xc(t) is modulated signalAc is carrier amplitude (constant)




Phase Modulation (PM)

4)(ˆ)( t t t cc φ ω θ +=

o180);(ˆ)( ≤= ΔΔ φ φ φ t xt

PM defined : Phase modulation is that form of angle modulation in

which the angle θ c(t ) is varied linearly with the message signal x(t )Instantaneous p ase var es rect y with the modulating signal

ase mo u a on n ex or ase deviation φ Δ is the maximum phase )](cos[)( t xt At x ccc Δ+= φ ω

− 180 o < φ Δ < 180 o (as | x(t) | ≤ 1)




Frequency modulation (FM)

5Total phase angle consists of the

)(ˆ)( t t t cc φ ω θ +=

c

Instantaneous rate of rotation t d t d 11

ˆθ

constant + varying (msg)

is given by dt dt c 22 π π

+==

modulation is that form of angle modulation in which the

cc f f t x f f t f <+= ΔΔ ),(ˆ)(

instantaneous frequency f (t ) is varied linearly with the message signal (t )f

Δis frequency deviation or NB: instantaneous frequency


maximum shift from carrier f c frequency f



Comparison of PM and FM

6If we start from the expression for the derivative of the phase

t d x

dt Δ= π

Integration yields the expression for phase modulation

∫Δ=t

d x f t λ λ π φ )(2)(

The FM waveform is then written as

t

–

Δccc




Power

7As the amplitude of a PM or FM waveform Ac is constant regardless of the modulating signal x (t ), the time average transmitted power is

22 c A== i.e. a sinusoid

2c

systems it is used more frequently

As hence W can be increased inde endentl of SIn optical communications PM is frequently used as an intermediate step leading to intensity modulation. FM is harder to realise using optical hardware.




AM, FM and PM Waveforms

8AM message resides

in

envelope,

PM

&

FM

resides

in

zero

crossings




Narrowband PM and FM

9)](cos[)( t t At x ccc φ ω +=

-- n p aseTaylor series exp.

-- Quadrature

‐Lecture10/11COMS3100



Narrowband PM and FM

10The Fourier transform of the quadrature ‐ carrier equation gives

FM expression was obtained using the integration theorem

, c, c

bandpass signal with bandwidth 2W

Else other terms of Taylor series significant (bandwidth > 2W)




Tone modulation (sinusoidal message waveform)

11

If we allow 90 o phase difference in modulating tones for PM and FM we can jointly carry out the analysis for both.

)()( t xt Δ=φ φ t

Δ= d x f t λ λ π φ )(2)(

, the same result:

with tone modulation.




Narrowband tone modulation

12Narrowband tone

modulation

requires

β <<1,

this

gives

QC

modulation

NB: |A(t)| is constant




Tone modulation with arbitrary β

13

β

Jn(β) are Bessel functions of the first kind, of order n and


.



Bessel functions of the first kind

14

J bJn(β)

0. 8

)(0 β J

0. 61

)(2 β J

0. 2

. 3

2 4 6 8 10 12 14bβ

-0. 4

- .


J0(5.5)=0 , second root of J0(.) J0(2.4)=0 , first root of J0(.)




15

βSignal takes the form (FS expansion)

Eventually, after further manipulation one obtains

∑∞

+= mcncc t n J At x )cos()()( ω ω β −∞=n

. .,





16

βThe FM spectrum consists of a carrier ‐frequency line plus an

infinite number of sideband lines at frequencies f c± nf mAll lines are equally spaced by the modulating frequency and the odd ‐order lower sidebands are reversed in phase.

∞

−∞==

nmcncc n x cos ω ω

The relative amplitude of the carrier line J0(β) varies with

There are spectra where carrier has zero amplitude [for

sideband lines

β that correspond to roots of the J0(β) at β = 2.4,5.5]





17

βNumber of sidebands with appreciable amplitude depends on βLar e im lies a lar e bandwidth to accommodate the extensive sideband structure.

Effect of increasing the β on the FM or PM line spectrumf m is being held fixed





18

βNumber of sidebands with appreciable amplitude depends on βLar e im lies a lar e bandwidth to accommodate the extensive sideband structure.

the FM line spectrum

decreasing f m with Am f Δfixed

Δ= f Am β




FM Phasor diagram (arbitrary β)

19

Carrier plus first pair of sidebands.

Even ‐orders reduce am litude distortionOdd ‐orders reduce phase distortion




Transmission Bandwidth Estimates20In general, the spectrum has infinite extent (pure FM requires infinite

bandwidth, regardless of the message)Practical FM systems having finite BW perform wellSufficiently far form the carrier spectral components become

Omitting any portion of the spectrum will cause distortion in the demodulated signalFM transmission BW: How much of the modulated signal spectrum is significant ?

Jn(β) falls off rapidly for | n/ β| > 1Envelopes of the sideband lines J ( ) is significant only for

m f An Δ=≤ β

All significant lines are contained in the range


Δ±=± f A f f f mcmc β



Transmission Bandwidth Estimates:21All sidebands having relative amplitude | Jm(β)| greater than ε can

be defined as significant, where ε ranges from 0.01 to 0.1 according to t e app icationIf | JM(β)|> ε and | JM+1(β)|< ε, there are M significant sideband pairs

+

1)(,)(2 >= β β M f M B m

The number of significant s e an pa rs as a unct on of β




FM Transmission Bandwidth Estimates:22

B is the

minimum

bandwidth

necessary

for

modulation

by

a tone

of

s ecified am litude and fre uenc . Estimate for the transmission bandwidth BT frequently used is the so called “worst ‐case tone ‐modulation bandwidth”

W f A mm

T

==Δ,1when

,

we define deviation ratio, D, as

maximum deviation

divided

by

f D Δ=

frequencyW

expression is

known

as

Carson’s

rule

11122 <<>>+=+= D DW DW B

More useful approximation is


, >+=+= ΔT



Linear Distortion

23Exponentially

modulated

bandpass signal

x t is a lied to a linear s stem with transfer function H(f) , producing output yc(t)xc(t) constant amplitude, Lowpass equivalent

v u

)(1)( t jclp e At x φ =

where φ (t ) contains the message information. n erms o

lp , e owpass equ va en ou pu spec rum s

)()()()( f X f f u f f H f Y lccl ++=

Lowpass to bandpass transformation finally provides output as

])(Re[2)( t jlpc

cet yt y ω =


. .,



Linear Distortion

24One

of

the

few

cases

that

yield

closed ‐form result is a linear transfer function shown hereThe gain | H ( f )| equals K 0 at f C

y w K 1/ f cThe low ass e uivalent is

)(21

0

10)()( f t f t j

cc

ce f K

K f f u f f H +− +=++ π

Y lp(f) now becomes

Carrier delay t0 and group delay t1




Linear Distortion

25Using the time delay and differential theorems for Fourier transforms

lp

where

Which ives us the out ut si nal

Which has the time varying amplitude




Linear Distortion

26For an FM signal, with the expression for the amplitude becomes

Therefore, passing the signal through this linear network H( f ) pro uces t e ‐ o‐ convers on n a t on to t e t me e ay.

AM with μ = K 1 f Δ / K 0 f c .

However, the delay distortion (phase nonlinearity) is a more serious problem than the amplitude distortion (FM‐to ‐AM conversion)This problem must be addressed by equalisation


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