7 - demodulation of fm

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Demodulation of FM SignalsKEEE343 Communication Theory

Lecture #18, May 17, 2011Prof. Young-Chai [email protected]


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Summary

BandwidthofWidebandFrequencyModulationCarlsonsrule

DemodulationofFMSignals

BasebandrepresentationofPassbandsignal

DemodulationofFMsignals


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Recall the single-tone frequency modulated wave given as

and its FT is given as

where for the message signal

To see the bandwidth let us consider two different cases

1. Case 1: Fix and vary (phase deviation is varied but the BW

of message signal is fixed.)

2. Case 2: Fix and vary (phase deviation is fixed but the BWof message signal is varied.)

Transmission Bandwidth of FM Waves

s(t) = Ac cos[2fct + sin(2fmt)]

S(f) = Ac

2

1X

n=1

Jn()[(f fc nfm) + (f+ fc + nfm)]

m(t) = Am cos(2fmt)

fm Am

Am fm

=kfAm

fm=

f

fm


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Case 1

[Ref: Haykin & Moher, Textbook]


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Case 2



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In theory, an FM wave contains an infinite number of side-frequencies.

However, we find that the FM wave is effectively limited to a finite numberof significant side-frequencies compatible with a specified amount of

distortion.

Observations of two limiting cases

1. For large values of the modulation index , the bandwidth

approaches, and is only slightly greater than the total frequencyexcursion .

2. For small values of the modulation index, the spectrum of the FM

wave is effectively limited to one pair of side-frequencies atso that the bandwidth approaches .

Transmission Bandwidth of the FM Wave

2f

fc fm2fm


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Carsons rule is the approximate rule for the transmission bandwidth of anFM wave

Single-tone case

Arbitrary modulating wave

where is deviation ratio.

Carsons Rule

BT 2f+ 2fm = 2f

1 +

1

BT 2(f+ W) = 2f

1 +

1

D

D =f

W


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Carsons rule is simple but unfortunately it does not always provide a goodestimate of the transmission bandwidth, in particular, for the widebandfrequency modulation.

Universal Curve for FM TransmissionBandwidth



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Baseband Representation of Modulated Waves andBandpass Filters

Band-pass representation (Modulated waves)

Let

and

Then

s(t) = sI(t)cos(2fct) sQ(t) sin(2fct)

s t = sI t + jsQ t

c(t = c(t + jc(t

= cos(2fct) + j sin(2fct)

= exp(j2fct)

s(t) = < [s(t)c(t)]

= < [s(t) exp(j2fct)]


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We can derive the baseband representation of the modulated wave infrequency domain as follows

Define a signal, called analytical signal that contains only the positivefrequencies in s(t)

Then its inverse Fourier transform is

|S(f |

ffcfc 0

S+(f = 2u(f S(f

1

2S+(f)

s+(t) =

Z1

1

S+(f)ej2ft df

= 2F1 [2u(f)] F1[S(f)]

=

(t) +

j

t

s(t)

= s(t) +j

t s(t)


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Using the Hilbert transform given as

the analytical signal can be written as

Now define the equivalent lowpass representation by performing a frequency

translation of such as

The equivalent time-domain relation is

or equivalently

s

(t) =

1

t s

(t)

s+(t = s(t + js(t

S+(f

S(f) = S+(f+ fc)

s

(t) =

s+(

t)ej2fct

= [s

(t) +

js

(t)]ej2fct

s(t) + js(t) = s(t)ej2fct


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In general,

Then

Complex envelope representation

s(t) = sI(t) + jsQ(t)

s(t) = sI(t) cos(2fct) sQ(t) sin(2fct)

s(t) = sI(t) sin(2fct) + sQ(t) cos(2fct)

s(t) = 00, f < 0

H(f fc)

Then H(f fc) = 0, f > 0H

(f

), f

7 - demodulation of fm

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