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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)

x(t)

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Demodulator

Σ

BPF

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 • and is deﬁned for both positive and negative frequency (see Figure 2). % N0 2 ,

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

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receiver in AM and DSB-SC.

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N 2

0

−ω c 4π B

ω 4π B

c

ω

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: Average power of modulated signal s(t) (SN R)I = Average power of noise

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

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• Channel SNR: Average power of modulated signal s(t) (SN R)C = Average power of noise in message bandwidth • Figure of Merit (FoM) of Receiver: (SN R)O F oM = (SN R)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 same power as modulated wave

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Low Pass Filter (B)

Output

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) = CAc cos(ωc t)m(t) (SN R)C P x(t) = =

−2πB

A2 C 2 P c 2BN0

+2πB

SM (ω)dω

= s(t) + n(t) CAc 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

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Band Pass Filter (B)

Product Modulator

v(t)

Low Pass Filter (B)

y(t)

n(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 + [CAc m(t) + nI (t)] cos 2ωc t 2 1 − nQ (t) sin 2ωc t 2

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The Low pass ﬁlter output is: 1 1 Ac m(t) + nI (t) 2 2

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– =⇒ 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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(t) Receiver output is nI2 Average power of nI (t) same as that n(t)

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Average noise power

(SN R)O,DSB−SC

1 = ( )2 2BN0 2 1 BN0 = 2 C 2 A2 P/4 c = BN0 /2 = C 2 A2 P c 2BN0 (SN R)O (SN R)C

F oMDSB−SC • Amplitude Modulation

=

|DSB−SC = 1

– The receiver model is as shown in Figure 5 &

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m(t) message with same power as modulated wave

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Band Pass Filter (B)

x(t)

Envelope Modulator

v(t)

n(t)

Figure 5: Analysis of AM System in Noise

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' s(t) = Ac [1 + ka m(t)] cos ωc t 2 A2 (1 + ka P ) c (SN R)C,AM = 2BN0 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(t) = [Ac + Ac ka m(t) + nI (t)] + 2

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n2 (t) Q

1 2

(SN R)O,AM F oMAM &

≈ Ac + Ac ka m(t) + nI (t) 2 A2 k a P c ≈ 2BN0 2 (SN R)O ka P = |AM = 2 (SN R)C 1 + ka P %

Thus the F oMAM is always inferior to F oMDSB−SC

' – 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

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Σ

Band Pass Filter (B)

x(t)

Limiter

Discriminator

y(t) n(t)

Bandpass low pass filter

Figure 6: Analysis of FM System in Noise

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