Noise in AM FM

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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 signal s(t)

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x(t)

Σ

BPF

Demodulator

w(t)

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 •

N 0

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

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• N 0 is the average power/(unit BW) at the front-end of the

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receiver in AM and DSB-SC.

N

0

2

−ω c

4π B

ω

c

ω

4π B

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)cos ωc t −nQ (t)sin ωc t

nI (t)cos ωc t is the in-phase component and nQ (t)sin ωc t 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:

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Average power of modulated signal s( s(t) (S N R)I  = Average power of noise

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• Output SNR: Average power of demodulated signal s( s(t) (S N R)O = Average power of noise The Output SNR is measured at the receiver. • Channel SNR:

(S N R)C  =

Average power of modulated signal s( s(t) Average power of noise in message bandwidth

• Figure of Merit (FoM) of Receiver:

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(S N R)O F oM  = (S N R)C 

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To compare across different modulators, we assume that (see Figure 3):

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• 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 . Output

m(t) message with same power as modulated wave

Σ

Low Pass Filter (B)

n(t)

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Figure 3: Basic Channel Model

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   $ Figure of Merit (FoM) Analysis

• DSB-SC (see Figure 4)

s(t) = C Ac cos(ω cos(ωc t)m(t) (S N R)C  =

Ac2 C 2 P  2B N 0 +2πB

P  =

 

S M  M (ω )dω

−2πB

x(t)

= s(t) + n(t) C Ac cos(ω cos(ωc t)m(t) +nI (t)cos ωc t + nQ (t)sin ωc t

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   $ m(t)

message with same power as modulated wave

Σ

n(t)

Band Pass Filter (B)

Product Modulator

v(t)

Low Pass Filter (B)

y(t)

Local Oscillator

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(ωc t) 1 1 = Ac m(t) + nI (t) 2 2 1 + [C Ac m(t) + nI (t)]cos2ω )]cos2ωc t 2 1 − nQ (t)sin2ω )sin2ωc t 2

The Low pass filter output is:

=

1 1 Ac m(t) + nI (t) 2 2

– =⇒ 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 2B

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Receiver output is n 2(t) Average power of  nI (t) same as that n(t) I

Average noise power

(S N R)O,DSB−S C 

1 2 = ( ) 2B N 0 2 1 = B N 0 2 C 2 Ac2 P /4 = B N 0 /2 =

F oM DS DS B −S C  =

C 2 Ac2 P  2B N 0 (S N R)O |DS B−S C  = 1 (S N R)C 

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• Amplitude Modulation – The receiver model is as shown in Figure 5

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   $ m(t)

message with same power as modulated wave

Σ

Band Pass Filter (B)

x(t)

Envelope

v(t)

Modulator

n(t)

Figure 5: Analysis of AM System in Noise

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   $ s(t) = Ac [1 + ka m(t)]cos ωc t Ac2 (1 + ka2 P ) P ) (S N R)C,AM  = 2B N 0 x(t) = s(t) + n(t) = [Ac + Ac ka m(t) + nI (t)]cos ωc t −nQ (t)sin ωc t y(t)

= envelope of x( x(t) =

(S N R)O,AM  F oM AM  AM 

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2

[A + A k m(t) + n (t)] + n (t) c

c a

I 

1 2

Q

≈ Ac + Ac ka m(t) + nI (t) Ac2 ka2 P  ≈ 2B N 0 (S N R)O ka2 P  = |AM  = (S N R)C  1 + ka2 P 





Thus the F oM AM  AM  is always inferior to F oM DS DS B −S C 

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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 power as modulated wave

Σ

Band Pass Filter

x(t) Limiter

(B)

y(t) n(t)

Discriminator

Bandpass low pass filter

Figure 6: Analysis of FM System in Noise

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   $ (S N R)O,FM  = (S N R)C,FM  = F oM F M  =

3Ac2 kf 2 P  2N 0 B 3 Ac2 2B N 0

 (SN R)  O

(S N R)C 

3kf 2 P  |F M  = 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  F oM F M 

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