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Chapter 4

- Angle Modulation

FM Illustration

The frequency of the carrier is varied around wc

in relation with the message signal. wi(t) wc

kf m(t)

Instantaneous Frequency

- The argument of a cosine function represents an

angle. - The angle could be constant cos(300), or

varying with time, cos q(t) - The instantaneous angular frequency (in rad/sec)

is the rate of change of the angle. That iswi

(t) dq (t)/dt . - For cos(wc t f), wi (t) wc as expected.

Representation of Angle Modulation in Time Domain

- For an FM signal wi (t) wc kf m(t)
- For Phase Modulation (PM), the phase of the

carrier is varied in relation to the message

signal f(t) kp m(t)

Relation Between FM and PM

m(t)

gFM(t)

FM Modulator

m(t)

gPM(t)

PM Modulator

Which is Which?

FM and PM Modulation

- kf 2p105 rad/sec/volt 105 Hz/Volt 105

V-1sec-1 - kp 10p rad/Volt 5 v-1
- fc 100 MHz
- FMfi fc kf m(t)
- 108 -105 lt fi lt 108 105 99.9 lt fi lt 100.1

MHZ - PM fi fc kp dm(t)/dt
- 108 -105 lt fi lt 108 -105 99.9 lt fi lt 100.1

MHZ - Power (FM or PM) A2/2

Spectrum of FM/PM

- Unlike Amplitude Modulation, it is not

straightforward to relate the spectrum of the

FM/PM modulated signal to that of the modulating

signal m(t). We can deal with it on a

case-by-case basis. - We are, however, particularly interested in

finding the bandwidth occupied by an FM/PM

signal. - For that purpose, we will make some assumptions

and work on simple modulating messages. - Because of the close relation between FM and PM,

we will do the analysis for FM and extend it to

PM.

What is NOT the bandwidth of FM!

- One may tend to believe that since the modulated

signal instantaneous frequency is varying between

by Df around fc, then the bandwidth of the FM

signal is 2Df. False! - In fact, the motivation behind introducing FM was

to reduce the bandwidth compared to that of

Amplitude Modulation, which turns out to be

wrong. - What was missing from the picture of bandwidth?

FM Visualization

- Think of holding the frequency knob of a signal

generator, and wiggling it back and forth to

modulate the carries in response to some message. - There are two wiggling parameters
- How far you deviate from the center frequency

(Df) - How fast you wiggle (related to Bm)
- The rate of change of the instantaneous frequency

was missing!

Carsons Rule

- BFM 2(DfBm)where Df frequency deviation

kf m(t)max - Bm bandwidth of m(t)
- Define the deviation ratio b Df / Bm.BFM 2(b

1) Bm - The same rule applies to PM bandwidth, BPM

2(DfBm) 2(b 1) Bm where (Df )PM kp

dm(t)/dtmax

Narrow Band and Wide Band FM

- When Df ltlt Bm (or b ltlt1), the scheme is called

Narrow Band (NBFM, NBPM). - BNBFM 2Bm (same for NBPM)
- Therefore, no matter how small we make the

deviation around fc , the bandwidth of the

modulated signal does not get smaller than 2Bm.

Estimate BFM and BPM

- kf 2p105 rad/sec/volt 105 Hz/Volt 105

V-1sec-1 - kp 5p rad/Volt 2.5 v-1
- fc 1000 MHz
- First estimate the Bm.Cn 8/p2n2 for n odd, 0 n

evenThe 5th harmonic onward can be neglected.Bm

15 kHz - For FMDf 100 kHz BFM 230 KHz
- For PMDf 50 kHz BFM 130 KHz

Repeat if m(t) is Doubled

- kf 2p105 rad/sec/volt 105 Hz/Volt 105

V-1sec-1 - kp 5p rad/Volt 2.5 v-1
- fc 1000 MHz
- For FMDf 200 kHz BFM 430 KHz
- For PMDf 100 kHz BFM 230 KHz
- Doubling the signal peak has significant effect

on both FM and PM bandwidth

2

40,000

-40,000

-2

Repeat if the period of m(t) is Doubled

- kf 2p105 rad/sec/volt 105 Hz/Volt 105

V-1sec-1 - kp 5p rad/Volt 2.5 v-1
- fc 1000 MHz
- Bm 7.5 kHz
- For FMDf 100 kHz BFM 215 KHz
- For PMDf 25 kHz BFM 65 KHz
- Expanding the signal varies its spectrum. This

has significant effect on PM.

4x10-4

10,000

-10,000

Spectrum of NBFM (1/2)

where

Spectrum of NBFM (2/2)

- For NBFM, kf a(t)ltlt 1
- Bandwidth of a(t) is equal to the bandwidth of

m(t), Bm. - BNBFM 2 Bm (as expected).
- Similarly for PM (kp m(t)ltlt 1 )
- BNBPM 2 Bm

NBFM Modulator

NBPM Modulator

Immunity of FM to Non-linearities

Frequency Multipliers

Generation of WBFM Indirect Method

- Usually, we are interested in generating an FM

signal of certain bandwidth (or Df or b) and

certain fc. - In the indirect method, we generate a NBFM with

small b then use a frequency multiplier to scale

b to the required value. - This way, fc will also be scaled by the same

factor. We may need a frequency mixer to adjust

fc.

Example From NBFM to WBFM

- A NBFM modulator is modulating a message signal

m(t) with bandwidth 5 kHz and is producing an FM

signal with the following specifications - fc1 300 kHz, ?f1 35 Hz.
- We would like to use this signal to generate a

WBFM signal with the following specifications - fc2 135 MHz, ?f 2 77 kHz.

From NBFM to WBFM System 1

From NBFM to WBFM System 2

Generation of WBFM Direct Method

- Has poor frequency stability. Requires feedback

to stabilize it.

FM Demodulation Signal Differentiation

FM Demodulation Signal Differentiation

Frequency Discriminators

- Any system with a transfer function of the form

H(w) aw b over the band of the FM signal

can be used for FM demodulation - The differentiator is just one example.

Slope Detectors (Demodulators)

Phased-Locked Loop (PLL)

- The multiplier followed by the filter estimates

the error bewteen the angle of gFM(t) and

gVCO(t). - The error is fed to VCO to adjust the angle.
- When the angles are locked, the output of the PLL

would be following m(t) pattern.