Lecture Slides On Diversity.pdf
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Lecture Slides On Diversity.pdf as PDF for free.
More details
- Words: 2,352
- Pages: 37
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)
MOBILE TELECOM: Diversity
April 1, 2019
1 / 27
OUTLINE 1
INTRODUCTION
2
COHERENT DETECTION
3
SEP1 ANALYSIS
4
AWGN VS FADING
5
DIVERSITY
6
COMBINING
7
REFERENCES
1 symbol
error probability
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
2 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
3 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
3 / 27
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=
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
4 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
4 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
5 / 27
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)
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
5 / 27
COHERENT DETECTION: ML RULE
Q: Consider BPSK symbols ±a (i.e., x1 = a, x2 = −a). Assuming equiprobable symbols, derive ML rule
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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
!
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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)
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
7 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
7 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
8 / 27
AWGN VS FADING
Q. Why poor SEP performance with fading? B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
9 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
10 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
11 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
12 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
13 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
14 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
14 / 27
Repetition coding: SEP curves
Average SEP decreases rapidly as L increases
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
15 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
16 / 27
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) =
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
17 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
17 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
18 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
19 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
20 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
21 / 27
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
B. Sainath (BITS, PILANI)
q
2Eb N0
⇐ performance of AWGN
MOBILE TELECOM: Diversity
April 1, 2019
21 / 27
Space Diversity
Antenna diversity: receive diversity, transmit diversity, & transmit and receive diversity
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
22 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
23 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
24 / 27
MRC: Fading Scenario Consider L− branch MRC, frequency-flat Rayleigh fading Principle Maximally combine signal by choosing optimal weights ⇒ maximize SNR Received signal:
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
25 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
25 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
26 / 27
References
Fundamentals of wireless communication by Tse & Viswanath Wireless communication by Andrea Goldsmith www.wirelesscommunication.nl/reference/about.htm
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
27 / 27
Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani
April 1, 2019
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
1 / 27
OUTLINE 1
INTRODUCTION
2
COHERENT DETECTION
3
SEP1 ANALYSIS
4
AWGN VS FADING
5
DIVERSITY
6
COMBINING
7
REFERENCES
1 symbol
error probability
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
2 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
3 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
3 / 27
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=
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
4 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
4 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
5 / 27
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)
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
5 / 27
COHERENT DETECTION: ML RULE
Q: Consider BPSK symbols ±a (i.e., x1 = a, x2 = −a). Assuming equiprobable symbols, derive ML rule
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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
!
MOBILE TELECOM: Diversity
April 1, 2019
6 / 27
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)
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
7 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
7 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
8 / 27
AWGN VS FADING
Q. Why poor SEP performance with fading? B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
9 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
10 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
11 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
12 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
13 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
14 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
14 / 27
Repetition coding: SEP curves
Average SEP decreases rapidly as L increases
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
15 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
16 / 27
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) =
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
17 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
17 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
18 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
19 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
20 / 27
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 ?
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
21 / 27
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
B. Sainath (BITS, PILANI)
q
2Eb N0
⇐ performance of AWGN
MOBILE TELECOM: Diversity
April 1, 2019
21 / 27
Space Diversity
Antenna diversity: receive diversity, transmit diversity, & transmit and receive diversity
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
22 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
23 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
24 / 27
MRC: Fading Scenario Consider L− branch MRC, frequency-flat Rayleigh fading Principle Maximally combine signal by choosing optimal weights ⇒ maximize SNR Received signal:
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
25 / 27
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)
MOBILE TELECOM: Diversity
April 1, 2019
25 / 27
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
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
26 / 27
References
Fundamentals of wireless communication by Tse & Viswanath Wireless communication by Andrea Goldsmith www.wirelesscommunication.nl/reference/about.htm
B. Sainath (BITS, PILANI)
MOBILE TELECOM: Diversity
April 1, 2019
27 / 27
Related Documents
Lecture Slides On Cpfsk.pdf
November 2019 33
Lecture Slides On Networking
May 2020 17
Lecture Slides On Diversity.pdf
November 2019 23
Lecture Slides On Small Scale Fading.pdf
November 2019 21
Lecture Slides 1
June 2020 20
Additional Cancer Lecture Slides
May 2020 22More Documents from ""
Lecture Slides On Small Scale Fading.pdf
November 2019 21
Readme.txt
November 2019 20
Assignment 1a And 1b
August 2019 18
Lecture Slides On Diversity.pdf
November 2019 23
Report_vlsi.docx
November 2019 21