DIVERSITY FOR FADING CHANNELS B. Sainath
[email protected]
Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani
April 1, 2019
B. Sainath (BITS, PILANI)
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OUTLINE 1
INTRODUCTION
2
COHERENT DETECTION
3
SEP1 ANALYSIS
4
AWGN VS FADING
5
DIVERSITY
6
COMBINING
7
REFERENCES
1 symbol
error probability
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Introduction
Communication via fading channel poor performance due to high probability of deep fade
Q. How to increase probability of decoding received symbol correctly at Rx ?
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Introduction
Communication via fading channel poor performance due to high probability of deep fade
Q. How to increase probability of decoding received symbol correctly at Rx ? Ans. Diversity Provide independently fading paths across time/frequency/space
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COHERENT DETECTION: BPSK, AWGN Channel Modulation: M-ary phase shift keying (MPSK) Model: BPSK Tx, AWGN channel, ML Rx y [m] = x[m] + w[m]. √ √ x[m] : + Es (bit ‘1’) , x[m] : − Es (bit ‘0’) w[m] ∼ N(0, N20 )
Received SNR: signal energy-to-noise PSD γ=
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COHERENT DETECTION: BPSK, AWGN Channel Modulation: M-ary phase shift keying (MPSK) Model: BPSK Tx, AWGN channel, ML Rx y [m] = x[m] + w[m]. √ √ x[m] : + Es (bit ‘1’) , x[m] : − Es (bit ‘0’) w[m] ∼ N(0, N20 )
Received SNR: signal energy-to-noise PSD γ=
Es N0
Symbol(or bit) error probability: √ Q( 2γ) ≤ exp (−γ) Rate =
log2 M T ,
where T is symbol period
e.g., BPSK: 1 bit/symbol period B. Sainath (BITS, PILANI)
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COHERENT DETECTION: BPSK, Rayleigh Fading plus AWGN Model: BPSK Tx, frequency-flat Rayleigh fading channel, ML Rx Assumption: perfect channel state information (CSI) is available at Rx y [m] = h[m]x[m] + w[m], y , h, w ∈ C √ √ x[m] : + Es (bit ‘1’), x[m] : − Es (bit ‘0’) ⇐ symbols are equally-likely h[m] ∼ CN (0, 1) Q: pdf of |h|2 ?
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COHERENT DETECTION: BPSK, Rayleigh Fading plus AWGN Model: BPSK Tx, frequency-flat Rayleigh fading channel, ML Rx Assumption: perfect channel state information (CSI) is available at Rx y [m] = h[m]x[m] + w[m], y , h, w ∈ C √ √ x[m] : + Es (bit ‘1’), x[m] : − Es (bit ‘0’) ⇐ symbols are equally-likely h[m] ∼ CN (0, 1) Q: pdf of |h|2 ? (Ans. |h|2 ∼ exp(1))
w[m] ∼ CN (0, σ 2 ) ⇐ AWGN For AWGN, σ 2 = N0
Received SNR (instantaneous): Γ=
Es |h|2 N0
Q. Average received SNR? (in class)
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COHERENT DETECTION: ML RULE
Q: Consider BPSK symbols ±a (i.e., x1 = a, x2 = −a). Assuming equiprobable symbols, derive ML rule
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COHERENT DETECTION: ML RULE
Q: Consider BPSK symbols ±a (i.e., x1 = a, x2 = −a). Assuming equiprobable symbols, derive ML rule Conditional pdfs
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COHERENT DETECTION: ML RULE
Q: Consider BPSK symbols ±a (i.e., x1 = a, x2 = −a). Assuming equiprobable symbols, derive ML rule Conditional pdfs
B. Sainath (BITS, PILANI)
|y − ah|2 1 exp − p(y |x1 , h) = 2 πσ σ2
!
|y + ah|2 1 p(y |x2 , h) = exp − 2 πσ σ2
!
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ML RULE Contd., ML rule: p(y|x1 , h) ≷xx12 p(y|x2 , h) ⇒ ln p(y |x1 , h) ≷xx12 ln p(y |x2 , h) After simplification, we get (verify)
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ML RULE Contd., ML rule: p(y|x1 , h) ≷xx12 p(y|x2 , h) ⇒ ln p(y |x1 , h) ≷xx12 ln p(y |x2 , h) After simplification, we get (verify) R{h? y } ≷xx12 0 ?
2
˜ R , where w ˜ R ∼ N(0, σ2 ) For x1 = a: y˜ = R{ h|h|y } = a |h| + w Symbol error probability (SEP) event ˜ R < −a |h|) SEP|(h, x1 = a) = P(y˜ < 0|h, x1 = a) = P(w q √ 2a2 |h|2 Inst. SEP = Q = Q 2Γ σ2 Γ denotes inst. fading SNR received 2 For a2 = Es , σ 2 = N0 , we get Γ = EsN|h| 0
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ML DETECTION & SEP
Q: Derive fading-averaged SEP & verify it with Craig’s formula (in class) q 2 Inst. Symbol(or bit) error probability: Q 2 |h| γ , where γ = NEs0 Fading-averaged SEP r q γ 1 1 2 2 |h| γ SEP = E Q = 1− ≈ 2 1+γ 4γ √ For AWGN, average SEP = Q 2γ ≤ exp (−γ) Compare with fading plus AWGN channel performance
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AWGN VS FADING
Q. Why poor SEP performance with fading? B. Sainath (BITS, PILANI)
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Typical Error Event
Inst. SEP: SEP|h = Q
q
2
2 |h| γ
2
|h| γ >> 1: very small probability of error (why?) 2
|h| γ < 1: significant error probability (deep fade event) Probability of deep fade event (DFE), pdfe 2
pdfe = P(|h| < γ1 ) ≈
1 γ
(in class) typical error event is due to channel being in deep fade
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Summary So Far
Model: BPSK Tx, AWGN channel, ML Rx (Coherent) Average SEP: ASEP ∝ e−SNR
Wireless fading channel Model: BPSK Tx, frequency-flat Rayleigh fading plus AWGN, ML Rx Fading-Averaged SEP: ASEP ∝ P[deep fade] ∝
1 SNR
Fading Mitigating Techniques: Diversity & combining
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Diversity & Combining Basic idea: transmit same data over independent fading paths Combine independent paths to reduce fading of resultant signal Microdiversity to mitigate effect of multipath fading, eg: Time, Frequency, Space/antenna, Polarization Macrodiversity to mitigate effects of shadow fading Time diversity obtained by interleaving and coding over symbols across different Tc
Figure: consecutive symbols are transmitted sufficiently far apart in time, so that channel taps (h` ) are statistically independent. B. Sainath (BITS, PILANI)
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Remarks on Interleaving
Interleaving: process to rearrange code symbols so as to spread bursts of errors over multiple code-words that can be corrected by error correcting codes Useful technique for improving performance of error correcting codes by combating bursts of errors
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Repetition coding: SEP Performance
Error probability: Q
q
P 2k h k2 γ , where k h k2 = L`=1 h`2 ← Chi-square distribution 2
Let X = k h k , pdf of X pX (x) = For small x, pX (x) =
1 x L−1 exp (−x) , x ≥ 0 (L − 1)!
1 x L−1 , x (L−1)!
≥0
Average SEP as function of received SNR
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Repetition coding: SEP Performance
Error probability: Q
q
P 2k h k2 γ , where k h k2 = L`=1 h`2 ← Chi-square distribution 2
Let X = k h k , pdf of X pX (x) = For small x, pX (x) =
1 x L−1 exp (−x) , x ≥ 0 (L − 1)!
1 x L−1 , x (L−1)!
≥0
Average SEP as function of received SNR Z ∞ √ ASEP = Q 2xSNR pX (x) dx 0
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Repetition coding: SEP curves
Average SEP decreases rapidly as L increases
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Repetition coding: DFE Probability
2
Let X = k h k , pdf of X pX (x) = For small x, pX (x) =
1 x L−1 exp (−x) , x ≥ 0 (L − 1)!
1 x L−1 , x (L−1)!
≥0
Q. What is probability of DFE? (In class) ‘L’ is diversity gain of the system More sophisticated codes can achieve coding gain (e.g., Rotation code) Important: tradeoff between throughput vs. diversity
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Time Diversity: Rotation Code
Rotate square constellation by angle θ Consider L = 2 scenario ASEP upper bound: corresponds to optimum θ = 0.5 tan−1 (2) =
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Time Diversity: Rotation Code
Rotate square constellation by angle θ Consider L = 2 scenario ASEP upper bound: corresponds to optimum θ = 0.5 tan−1 (2) = 31.7 degrees maximize the minimum Euclidean distance
ASEP ≤
15
SNR2 Coding gain over repetition code = 3.5 dB Read Ch. 3. Section 3.2.2. Tse & Viswanath B. Sainath (BITS, PILANI)
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GSM Uses Time Diversity
Coded bits are interleaved across 8 consecutive time slots Maximum possible time-diversity: 8 Actual gain depends on mobile speed
Exercise: Tc = 5 ms, fc = 900 MHz, Ds = ? & mobile speed = ? Express Tc in terms of Ds
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Frequency Diversity
Wideband channels: transmission BW > coherence BW (of channel) Sending symbols more frequently ⇒ ISI issue How to mitigate ISI while exploiting channel’s diversity ⇐ challenge Modulate the transmitting signal through ‘L’ different carriers Approaches: single-carrier systems with equalization, DSSS, Multi-carrier systems System Single-carrier system Direct-sequence spread-spectrum Multi-carrier system
Example GSM IEEE 802.11b, IS-95 IEEE 802.11a2
2 https://en.wikipedia.org/wiki/IEEE_802.11 B. Sainath (BITS, PILANI)
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Direct-Sequence Spread-Spectrum Principle: spread data via pseudo-random noise(PN) sequence across bandwidth much larger than data rate The signal received along ‘L’ nearly orthogonal is maximal-ratio combined using RAKE Rx
Average probability of error when anti-podal modulation (XA = −XB = u) used h √ i E Q 2Γ Γ=
kuk2
PL
2 l=0 |hl |
N0
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DSSS: Asymptotic Average SEP
Channel gains are i.i.d. & hl ∼ CN (0, 1L ) 2
Eb , k u k is average total energy received per bit of information SNR per branch ?
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DSSS: Asymptotic Average SEP
Channel gains are i.i.d. & hl ∼ CN (0, 1L ) 2
Eb , k u k is average total energy received per bit of information SNR per branch ? 1 L
Eb N0
, factor
1 L
accounts for splitting of energy due to spreading
As L → ∞, average SEP → Q
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q
2Eb N0
⇐ performance of AWGN
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Space Diversity
Antenna diversity: receive diversity, transmit diversity, & transmit and receive diversity
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Combining
Combining techniques to achieve array gain Eg. Linear combining at coherent Rx (assuming AWGN channel and BPSK transmitter model) Combining ’M’ branches each with SNR of γ = NEs 0 Total SNR = M × γ in each branch (details & proof discussed in class) Array gain is M What is the symbol/bit error probability?
Combining techniques for fading channels (to be discussed in class) Maximal ratio combining Selection combining
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MRC: AWGN Scenario (No fading)
AWGN w ∼ CN(0, N0 ) L− branch Linear combining: Received SNR (derivation in class) ΓE =
LEs N0
L− fold increase in SNR due to coherent combining SNR increase in the absence of fading ⇒ array gain
Q:Write down expressions for outage, average SEP, spectral efficiency
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MRC: Fading Scenario Consider L− branch MRC, frequency-flat Rayleigh fading Principle Maximally combine signal by choosing optimal weights ⇒ maximize SNR Received signal:
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MRC: Fading Scenario Consider L− branch MRC, frequency-flat Rayleigh fading Principle Maximally combine signal by choosing optimal weights ⇒ maximize SNR Received signal: yML = w1 y1 + . . . + wL yL Received instantaneous fading SNR 2
ΓE =
Es |h1 | Es |hL | + ... + N0 N0
2
Q: What is the average SNR? Diversity order indicates how the slope of the average symbol error probability (SEP) as a function of average SNR changes with diversity
Exercise: Derive expressions for Outage probability Fading-averaged SEP (Hint: Use MGF approach) & diversity order Fading-averaged spectral efficiency B. Sainath (BITS, PILANI)
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Selection Combining
Combiner outputs the signal on the branch with the highest SNR Average SNR of the combiner output in i.i.d. Rayleigh fading is ΓE = Γ
L X 1 n=1
n
Exercise: Derive expressions for Outage probability Fading-averaged SEP (Hint: Use MGF approach) & diversity order Fading-averaged spectral efficiency
Compare MRC vs SC performance
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References
Fundamentals of wireless communication by Tse & Viswanath Wireless communication by Andrea Goldsmith www.wirelesscommunication.nl/reference/about.htm
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