Part II. Fading and Diversity Impact of Fading in Detection; Time - - PowerPoint PPT Presentation

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Part II. Fading and Diversity

Impact of Fading in Detection; Time Diversity; Antenna Diversity; Frequency Diversity

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Simplest Model: Single-Tap Rayleigh Fading

Flat fading: single-tap Rayleigh fading V = Hu + Z, H ∼ CN(0, 1), Z ∼ CN(0, N0) Detection:

Detection

ˆ u = aˆ

θ

V = Hu + Z

u = aθ ∈ A {a1, . . . , aM}

Detector (Rx) may or may not know the channel coefficients

Coherent Detection: Rx knows the realization of H Noncoherent Detection: Rx does not know the realization of H

ˆ Θ

Coherent Detection of BPSK

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Detection

ˆ u = aˆ

θ

V = Hu + Z

u ∈ {± p Es} a0 = + p Es, a1 = − p Es H ∼ CN(0, 1), Z ∼ CN(0, N0)

ˆ Θ = φ(V, H) Likelihood function: The detection problem is equivalent to binary detection in ˜ V = u + ˜ Z, ˜ V V /h, ˜ Z Z/h ∼ CN(0, N0/|h|2) Probability of error conditioned on the realization of H = h : fV,H|Θ(v, h|θ) = fV |H,Θ(v|h, θ)fH(h) ∝ fV |H,Θ(v|h, θ) Pe(φ; H = h) = Q

• 2√Es

2√ N0/(2|h|2)

• = Q
• 2|h|2Es

N0

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Probability of error: Pe(φ; H = h) = Q

• 2|h|2Es

N0

• Pe(φ) = EH∼CN (0,1) [Pe(φ; H)]

= EH∼CN (0,1)

• Q
• 2|H|2Es

N0

• ≤ E|H|2∼Exp(0,1)

1

2 exp(−|H|2SNR)

• =

Z ∞ 1 2e−tSNRe−t dt = 1 2(1 + SNR)

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Impact of Fading

• Let us explore the impact of fading by comparing the performance
• f coherent BPSK between AWGN and single-tap Rayleigh fading
• The average received SNRs are the same:
• AWGN: probability of error decays exponentially fast:
• Rayleigh fading: probability of error decays much slower:

EH∼CN (0.1)

• |H|2SNR
• = SNR

Pe(φML) = Q √ 2SNR

• ≤ 1

2 exp(−SNR)

Pe(φML) = EH∼CN (0,1)

• Q
• 2|H|2SNR
• ≤ 1

2 1 1+SNR

e−SNR SNR−1

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10 20 30 40 Non-coherent

• rthogonal

Coherent BPSK BPSK over AWGN

SNR (dB)

10–8 –10 –20 1 10–2 10–4 10–6 10–10 10–12 10–14 10–16

15 dB 3 dB

Pe Availability of channel state information (CSI) at Rx

• nly changes the intercept, but not the slope

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Coherent Detection of General QAM

Probability of error for -ary QAM Pe(φ; H = h) ≤ 4Q

• |h|2d2

min

2N0

• = 4Q
• 3

M−1|h|2SNR

• M = 22ℓ

Pe(φ) ≤ EH∼CN (0,1)

• 4Q
• 3

M−1|H|2SNR

• ≤ E|H|2∼Exp(0,1)
• 2 exp(−|H|2

3 2(M−1)SNR)

• =

2 1 +

3 2(M−1)SNR ≈ 4(M − 1)

3 SNR−1 Using general constellation does not change the order of performance (the “slope” on the log Pe vs. log SNR plot) Different constellation only changes the intercept

Deep Fade: the Typical Error Event

• In Rayleigh fading channel, regardless of constellation size and

detection method (coherent/non-coherent),

• This is in sharp contrast to AWGN:
• Why? Let’s take a deeper look at the BPSK case:
• If channel is good, error probability
• If channel is bad, error probability is
• Deep fade event:

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Pe ∼ SNR−1 Pe ∼ exp(−cSNR) Pe(φ; H = h) = Q

• 2|h|2SNR
• |h|2SNR 1 =

⇒ |h|2SNR < 1 = ⇒ Θ(1) ∼ exp(−cSNR) Pe ≡ P {E} = P

• |H|2 > SNR−1

P

• E | |H|2 > SNR−1

+ P

• |H|2 < SNR−1

P

• E | |H|2 < SNR−1

/ P

• |H|2 < SNR−1

= 1 − eSNR−1 ≈ SNR−1

{|H|2 < SNR−1}

Diversity

• Reception only relies on a single (equivalent) “path” H
• If H is in deep fade ⟹ big trouble (low reliability)
• Increase the number of “paths” ⟺ Increase diversity
• If one path is in deep fade, other paths can compensate!
• If there are L indep. paths, the probability of deep fade becomes
• Find independent paths (diversity) over time, space, and

frequency to increase diversity!

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V = Hu + Z {|H|2 < SNR−1} Deep fade event: 1 −

L

Y

`=1

(1 − P{ i }) ≈ 1 − (1 − SNR−1)L ≈ SNR−L