Angle Modulation: Phase Modulation or Frequency Modulation

cosc cV t V t t Basic Form of FM signal:•Constant Amplitude•Information is contained in (t)

Define Phase Modulation Index, mp, to restrict (t) such that

or

p

p MAX

m t t

m

Instantaneous Phase: (t)

(t) = mpa(t)

ct t t

Total Instantaneous

Phase

Reference Phase

Phase Deviation

Normalized Information

Signal

Visualizing the FM Phasor . . .…down at the “Complex Plane”

Reference Phasor

c c

FM Phasor

(t)

mp = |MAX|

Reference Phase

Constant Amplitude

c

(t) = mpa(t)

Instantaneous Frequency Deviation

cosc

j tc

V t V t

V V e

Time Waveform:Phasor Notation:

c c c

d d dt t t t t t

dt dt dt

p p p

d d dt t m a t m a t m a t

dt dt dt

“Carrier Frequency”

“Frequency Deviation”

Can be very large and still have |a(t)| < 1High rates of change implies Wide Bandwidth

Instantaneous “Frequency Deviation”

Voltage Controlled Oscillator (VCO)

fc (hz or R/s )k0 (hz/v or R/s/v)

VCO

( ) ( )inv t V s

fc

k0

vin

fout

0 0( ) cos 2 cos 2 2out c c in c c inv t V f k v t t V f t k v t t

“Free running” frequency

Frequency deviation: f(t) “Reference” phase

Phase Deviation:

(t)

0

0

0

( )

( )

( )

inf t k v t

F s k V s

k V ss

s

Frequency of VCO Output

Deviation Sensitivity

Simple Case For Analysis

cos

cos sin

m

p m p m m

MAX p m

MAX MAXp MAX f

m m

a t t

dt m t m t

dtm

fm m

f

FM Spectrum

Spectral Analysis: High Math. Spectrum is characterized by spectral components spaced at + nm from c, with amplitudes determined by Bessel Functions Jn(mp).

cos cosc c c c pV t V t t V t m a t

cos cosc c p mV t m t Now all we have to do is take the Fourier Transform of this thing . . .

c

m

J0

J3

J2

J1

J4

J1(m)

J0(m)

J2(m)J3(m) J4(m) J5(m) J6(m) J7(m) J8(m) J9(m) J10(m)

Bessel Function Tabulationm f J 0 J 1 J 2 J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 10 J 11 J 12 J 13 J 140.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.25 0.98 0.12 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.50 0.94 0.24 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.001.00 0.77 0.44 0.11 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.001.50 0.51 0.56 0.23 0.06 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.002.00 0.22 0.58 0.35 0.13 0.03 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.002.50 -0.05 0.50 0.45 0.22 0.07 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.003.00 -0.26 0.34 0.49 0.31 0.13 0.04 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.004.00 -0.40 -0.07 0.36 0.43 0.28 0.13 0.05 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.005.00 -0.18 -0.33 0.05 0.36 0.39 0.26 0.13 0.05 0.02 0.01 0.00 0.00 0.00 0.00 0.006.00 0.15 -0.28 -0.24 0.11 0.36 0.36 0.25 0.13 0.06 0.02 0.01 0.00 0.00 0.00 0.007.00 0.30 0.00 -0.30 -0.17 0.16 0.35 0.34 0.23 0.13 0.06 0.02 0.01 0.00 0.00 0.008.00 0.17 0.23 -0.11 -0.29 -0.11 0.19 0.34 0.32 0.22 0.13 0.06 0.03 0.01 0.00 0.009.00 -0.09 0.25 0.14 -0.18 -0.27 -0.06 0.20 0.33 0.31 0.21 0.12 0.06 0.03 0.01 0.0010.00 -0.25 0.04 0.25 0.06 -0.22 -0.23 -0.01 0.22 0.32 0.29 0.21 0.12 0.06 0.03 0.01

The Bessel Function values determine the relative voltage amplitudes of their respective sidebands.The squares of the Bessel Function values determine the relative power amplitudes of their respective sidebands.

For any value of m:

If our Bandwidth includes N sidebands, then :

2 2 2 2 20 1 2 3 42 1J m J m J m J m J m

1

22 2 2 2 2 20 1 2 3

1

1 2 N nn N

THD J m J m J m J m J m J m

2 2 2 2 2 210 0 1 2 3 10

1

10log 1 2 10logN nn N

THD dB J m J m J m J m J m J m