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Noise Analysis - AM, FM

The following assumptions are made:

• Channel model

– distortionless

– Additive White Gaussian Noise (AWGN)

• Receiver Model (see Figure 1)

– ideal bandpass filter

– ideal demodulator

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Modulated signals(t) Σ

w(t)

x(t)

DemodulatorBPF

Figure 1: The Receiver Model

• BPF (Bandpass filter) - bandwidth is equal to the message

bandwidth B

• midband frequency is ωc.

Power Spectral Density of Noise

• N0

2 , and is defined for both positive and negative frequency (see

Figure 2).

• N0 is the average power/(unit BW) at the front-end of the

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receiver in AM and DSB-SC.

ω cc−ω

B B

2N0

ω

4π 4π

Figure 2: Bandlimited noise spectrum

The filtered signal available for demodulation is given by:

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x(t) = s(t) + n(t)

n(t) = nI(t) cosωct

−nQ(t) sinωct

nI(t) cosωct is the in-phase component and

nQ(t) sinωct is the quadrature component.

n(t) is the representation for narrowband noise.

There are different measures that are used to define the Figure of

Merit of different modulators:

• Input SNR:

(SNR)I =Average power of modulated signal s(t)

Average power of noise

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• Output SNR:

(SNR)O =Average power of demodulated signal s(t)

Average power of noise

The Output SNR is measured at the receiver.

• Channel SNR:

(SNR)C =Average power of modulated signal s(t)

Average power of noise in message bandwidth

• Figure of Merit (FoM) of Receiver:

FoM =(SNR)O

(SNR)C

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To compare across different modulators, we assume that (see

Figure 3):

• The modulated signal s(t) of each system has the same average

power

• Channel noise w(t) has the same average power in the message

bandwidth B.

m(t)

message with samepower as modulated wave

n(t)

Low Pass Filter(B)

OutputΣ

Figure 3: Basic Channel Model

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Figure of Merit (FoM) Analysis

• DSB-SC (see Figure 4)

s(t) = CAc cos(ωct)m(t)

(SNR)C =A2

cC2P

2BN0

P =

∫ +2πB

−2πB

SM (ω)dω

x(t) = s(t) + n(t)

CAc cos(ωct)m(t)

+nI(t) cosωct + nQ(t) sinωct

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m(t)

message with same

n(t)

Σpower as modulated wave

(B)Band Pass Filter

ModulatorProduct

Local Oscillator

(B)

y(t)v(t) Low Pass Filter

Figure 4: Analysis of DSB-SC System in Noise

The output of the product modulator is

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v(t) = x(t) cos(ωct)

=1

2Acm(t) +

1

2nI(t)

+1

2[CAcm(t) + nI(t)] cos 2ωct

−1

2nQ(t) sin 2ωct

The Low pass filter output is:

=1

2Acm(t) +

1

2nI(t)

– =⇒ ONLY inphase component of noise nI(t) at the output

– =⇒ Quadrature component of noise nQ(t) is filtered at the

output

– Band pass filter width = 2B

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Receiver output is nI(t)2

Average power of nI(t) same as that n(t)

Average noise power = (1

2)22BN0

=1

2BN0

(SNR)O,DSB−SC =C2A2

cP/4

BN0/2

=C2A2

cP

2BN0

FoMDSB−SC =

(

(SNR)O

(SNR)C

)

|DSB−SC = 1

• Amplitude Modulation

– The receiver model is as shown in Figure 5

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m(t)

message with same

n(t)

Σpower as modulated wave

(B)Band Pass Filter

ModulatorEnvelope v(t)x(t)

Figure 5: Analysis of AM System in Noise

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s(t) = Ac[1 + kam(t)] cosωct

(SNR)C,AM =A2

c(1 + k2aP )

2BN0

x(t) = s(t) + n(t)

= [Ac + Ackam(t) + nI(t)] cosωct

−nQ(t) sinωct

y(t) = envelope of x(t)

=[

[Ac + Ackam(t) + nI(t)]2 + n2

Q(t)]

1

2

≈ Ac + Ackam(t) + nI(t)

(SNR)O,AM ≈A2

ck2aP

2BN0

FoMAM =

(

(SNR)O

(SNR)C

)

|AM =k2

aP

1 + k2aP

Thus the FoMAM is always inferior to FoMDSB−SC

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– Frequency Modulation

∗ The analysis for FM is rather complex

∗ The receiver model is as shown in Figure 6

m(t)

message with same

n(t)

Σpower as modulated wave

(B)Band Pass Filter x(t)

Limiter Discriminator

Bandpasslow pass filter

y(t)

Figure 6: Analysis of FM System in Noise

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(SNR)O,FM =3A2

ck2fP

2N0B3

(SNR)C,FM =A2

c

2BN0

FoMFM =

(

(SNR)O

(SNR)C

)

|FM =3k2

fP

B2

The significance of this is that when the carrier SNR is

high, an increase in transmission bandwidth BT provides a

corresponding quadratic increase in output SNR or FoMFM