ECE 4710: Lecture #22 1 Frequency Modulation FM spectrum is very difficult to calculate in general Useful to develop simple approximations when the

ECE 4710: Lecture #22 1

Frequency Modulation

FM spectrum is very difficult to calculate in general Useful to develop simple approximations when the

modulation index, , is small or large

Recall that

“small” so F B / 5 Narrowband FM “large” so F > 5 B Wideband FM

FM complex envelope

where

BF

)()( tjceAtg

dmDtt

f )()(

rad/s are units )(

abledummy vari

mD f


Narrowband FM

Narrowband FM = NBFM Restrict phase angle to small value For small values the exponential function in complex

envelope can be approximated using first two terms of Taylor series expansion

So NBFM signal at Tx output

2.0|)(| t

1for 1 xxex

1 |)(|for )](1[)( )( tjtjAeAtg ctj

c

]2sin[]2cos[)(Re

]2sin[]2cos[)](1[Re

)(Re)( 2

tfjtftjAA

tfjtftjA

etgts

cccc

ccc

tfj c


Narrowband FM

NBFM signal at Tx output (continued)

Carrier Term does NOT depend on (t) modulation» Just like large carrier (LC) in AM

Sideband Term depends on (t) modulation and is 90° out of phase with respect to carrier» Note that (t) is modulating amplitude of sine !!

» Different than AM where sidebands are in phase with LC

]2sin[)(]2cos[

]2sin[]2cos[)(Re)(

tftAtfA

tfjtftjAAts

cccc

cccc

Carrier Term Sideband Term


NBFM Spectrum

2.0BF

f

)2sin()( tftm m

Carrier

Sidebands

]2sin[)(]2cos[)( tftAtfAts cccc

)()([2

)]()([2

)( ccf

ccc ffMffM

f

Dffff

AfS

LC DSB


NBFM Tx

LC

DSB

]2sin[)(]2cos[)( :NBFM tftAtfAts cccc


Wideband FM

FM instantaneous frequency of s(t) varies in direct proportion to modulating signal voltage m(t) If m(t) stays at one particular voltage more than another then s(t) will

spend more time at one particular frequency

PSD of s(t) should have more power at frequencies associated with m(t) voltages that are more likely

Probability Density Function (PDF) of m(t) can be used to approximate PSD for s(t) for wideband FM (WBFM)

)(

2)(

2

2)(

2

c

f

mc

f

m

f

c ffD

fffD

fD

Af

P

)( of PDF where tmfm


Wideband FM

Equal probability for any voltage between Vp

and +Vp

Uniform PDF for m(t) and rectangular spectrum for

WBFM

Triangle Modulating m(t) PDF of m(t)

Approximate WBFM spectrum for triangle modulation


Wideband FM Tx

VCO = Voltage Controlled Oscillator

Phase Lock Loop (PLL) configuration produces DC output at LPF to

ensure stable carrier when center frequency of VCO drifts away from fc


Digital FM Signals

What happens if a digital waveform is used for m(t) in an FM system?? Instantaneous frequency of s(t) varies in time about

assigned carrier frequency in direct proportion to modulating signal voltage m(t)

Two voltage states (± Ac) two discrete frequency values

(fH, fL) @ output of FM Tx

)(21

)( tmDftf fci

)(tm+Ac

VCOL

H

f

f

Ac


Digital FM Signals

)(ts

VCOL

H

f

f

Hf Lf Hf Lf Hf Lf

Binary Frequency Shift Keying (BFSK)

)(tm+Ac

Ac

1 0 1 0 1 0

1 0 1 0 1 0

special case of WBFM with digital m(t)


BFSK Signal Spectrum

Binary Digital m(t) PDF of m(t)

Approximate WBFM spectrum for BFSK


Digital AM & FM

BPSK Signal

PolarModulation

BFSK Signal

Special Case of DSB-SC

Special Case of

FM

1 0 1 0 1 0

1 0 1 0 1 0

1 0 1 0 1 0


FDM

Frequency Division Multiplexing = FDM Technique to transmit multiple messages simultaneously

over a wideband channel Modulate multiple messages onto subcarriers

» Composite baseband signal formed from multiple signals

Composite signal spectrum most not have overlapping spectrum message “crosstalk” (interference)

Composite signal modulated onto main RF carrier Any type of modulation can be used for carrier and sub-

carriers (AM, FM, DSB, etc.)» Different modulation types can be used within same FDM system» Example: DSB-SC for subcarriers and FM for carrier FM Stereo


FDM Tx

Transmitter

Composite Baseband Signal Spectrum


FDM Rx

Receiver

1. Received FDM signal demodulated to recover composite baseband signal

2. Passed thru multiple bandpass filters to separate individual subcarriers

3. Subcarriers demodulated to recover multiple message signals


FM Stereo Broadcast

FM Stereo Broadcast in U.S. is FDM system FM carrier modulation DSB-SC sub-carrier modulation Left & Right (L/R) audio channels for stereo signal L+R signal used to allow mono Rx (outdated)

» 15 kHz baseband audio signal BW

L-R signal modulated on 38 kHz DSB-SC sub-carrier» 38 kHz sub-carrier signal BW (2 x 15 kHz)» 38 / 2 = 19 kHz pilot tone for coherent demodulation of sub-carrier

Rx uses sum & difference of L+R and L-R to recover L and R channels for stereo speakers


FM Stereo Tx

Transmitter


23kHz

53kHz

fL + R


23kHz

53kHz

L + R

DSB-SC SpectrumLR


FM Stereo Rx

L + R

L R

fp = 19 kHz

fVCO = 38 kHz


FM Stereo Broadcast

FCC can authorize FM station for subsidiary communication authorization (SCA) Up to 4 SCA channels authorized First SCA subcarrier is usually 67 kHz

» Second analog audio program» Background “elevator” music Musac service

Secondary service of FM station for revenue generation

Other subcarriers used for data or analog audio» Radio Broadcast Data System (RBDS)

57 kHz subcarrier with 1200 bps dataAuxillary text information (music titles, artist, traffic reports, etc.)


FM Stereo Spectrum

fComposite Baseband Signal Spectrum with SCA Channels

23kHz

53kHz

L + R

DSB-SC SpectrumLR

67kHz

57kHz

2nd Audio“Musac”Data

75kHz

59kHz

1. Non-Commercial Stations 88.1 to 91.9 MHz

2. Commercial Stations 92.1 to 107.9 MHz

3. RF Channel BW = 200 kHz

4. F = 75 kHz from RF channel carrier frequency

5. 2F = 150 kHz < 200 kHz channel BW guard BW to minimize interference between stations

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ECE 4710: Lecture #22 1 Frequency Modulation FM spectrum is very difficult to calculate in general Useful to develop simple approximations when the