tele3113 wk6tue

TELE3113 Analogue and DigitalCommunications

Transmission of FM Waves

Wei Zhang

[email protected]

School of Electrical Engineering and Telecommunications

The University of New South Wales

Wideband FM Signal

In the last lecture, we studied:

The FM wave of the single-tone message signal is given by

s(t) = Ac

∞∑

n=−∞

Jn(β) cos[2π(fc + nfm)t].

The spectrum of s(t) is given by

S(f) =Ac

2

∞∑

n=−∞

Jn(β)[δ(f − fc − nfm) + δ(f + fc + nfm)].

TELE3113 - Transmission of FM Waves. August 25, 2009. – p.1/9

Transmission BW of FM (1)

In theory, an FM wave contains an infinite number of

side-frequencies, so the BW is infinite.

In practice, the FM wave is effectively limited to a finitenumber of significant side-frequencies.

Specifically, by observing the spectrum,

for large β, the BW approaches 2∆f = 2βfm.

for small β, the BW approaches 2fm.

Carson’s rule:

BT ≈ 2∆f + 2fm = 2∆f

(

1 +1

β

)

.



Carson’s rule is simple, but not an accurate estimate.

An accurate estimate of the BW can be defined as the

separation between the two frequencies beyond whichnone of the side frequencies is greater than 1% of theunmodulated carrier amplitude.

Mathematically, the BW is given by (universal rule)

BT = 2nmaxfm = 2∆fnmax

β

where nmax is the largest value of the integer n that satisfies

|Jn(β)| > 0.01.



β 0.1 0.3 0.5 1 2 5 10 20 30

2nmax 2 4 4 6 8 16 28 50 70

BT

∆f= 2nmax

β

10−1

100

101

100

101

102

Modulation index, β

Nor

mal

ized

ban

dwid

th, B

T/∆

f

BT/∆ f in term of β



Relative error of BW estimate with Carson’s rule compared to

the universal rule is defined by

ε = 1 −BCarson

T

BUniversalT

=

(

1 −1 + β

nmax

)

× 100%.

Usually, Carson’s rule underestimates the BW by 25%.

10−1

100

101

−10

−5

0

5

10

15

20

25

30

35

40

Modulation index, β

Rel

ativ

e er

ror

of B

W e

stim

ate

(\%

)

Relative error of Carson’s rule in term of β


Generation of FM Waves (1)

Armstrong’s method:

The message signal is first used to produce a narrow-band

FM.

Then, a frequency multiplication is used to produce a

wide-band FM.

Integrator

Narrow-band phase Modulator

Message signal m(t) Wide-

band FM wave

Crystal controlled oscillator

Narrow-band phase modulator

Frequency multiplier



Memoryless nonlinear

device

Bandpass filter with mid-band

frequency nfc

FM wave s(t) with carrier frequency fc

and modulation

�

FM wave s’ (t) with

carrier frequency nfc

and modulation

n

�

Frequency multiplier

v(t)

The input-output relation of the nonlinear device is

v(t) = a1s(t) + a2s2(t) + · · · + ansn(t),

where a1, a2, · · · , an are coefficients determined by the device

and n is the highest order of nonlinearity.



Consider the input s(t) is an FM wave defined by

s(t) = Ac cos

[

2πfct + 2πkf

∫ t

0

m(τ)dτ

]

.

Suppose that bandpass filter is designed to have a BW

equal to n times the BW of s(t).

After bandpass filtering of the output v(t), we have

s′(t) = Ac cos

[

2πf′

ct + 2πk′

f

∫ t

0

m(τ)dτ

]

,

where f′

c = nfc and k′

f = nkf .


Demodulation of FM Waves

Recall that the FM signal is given by

s(t) = Ac cos

[

2πfct + 2πkf

∫ t

0

m(τ)dτ

]

.

After taking the derivative of s(t) with respect to t, we get

ds(t)

dt= −2πAc[fc + kfm(t)] sin

(

2πfct + 2πkf

∫ t

0

m(τ)dτ

)

.

The derivative is indeed a band-pass signal with amplitudemodulation and the amplitude is 2πAcfc[1 + kfm(t)/fc].

If fc is large enough such that the carrier is not

overmodulated, then we can recover the message signal

m(t) with an envelop detection. TELE3113 - Transmission of FM Waves. August 25, 2009. – p.9/9

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