Lecture 2 Point-to-Point Communications 1 I-Hsiang Wang - - PowerPoint PPT Presentation

Lecture ¡2 Point-­‑to-­‑Point ¡Communications ¡1

I-Hsiang Wang ihwang@ntu.edu.tw 2/27, 2014

Wire ¡vs. ¡Wireless ¡Communication

2

y[m] = X

l

hlx[m − l] + w[m] y[m] = X

l

hl[m]x[m − l] + w[m]

Wired Channel Wireless Channel

• Deterministic channel gains
• Main issue: combat noise
• Key technique: coding to

exploit degrees of freedom and increase data rate (coding gain)

• Random channel gains
• Main issue: combat fading
• Key technique: coding to

exploit diversity and increase reliability (diversity gain)

• Remark: In wireless channel, there is still additive noise,

and hence the techniques developed in wire communication are still useful.

Plot

• Study detection in flat fading channel to learn
• Communication over flat fading channel has poor performance

due to significant probability that the channel is in deep fade

• How the performance scale with SNR
• Investigate various techniques to provide diversity across
• Time
• Frequency
• Space
• Key: how to exploit additional diversity efficiently

3

Outline

• Detection in Rayleigh fading channel vs. static AWGN

channel

• Code design and degrees of freedom
• Time diversity
• Antenna (space) diversity
• Frequency diversity

4

Detection ¡in ¡Rayleigh ¡ Fading ¡Channel

Baseline: ¡AWGN ¡Channel

6

y = x + w, w ∼ CN

• 0, σ2

BPSK: x = ±a

Transmitted constellation is real, it suffices to consider the real part:

ML rule: Probability of error:

Pr {E} = Pr ⇢ <{w} > a (a) 2

• = Q

a p σ2/2 ! = Q ⇣p 2SNR ⌘ b x = ( a, if |<{y} a| < |<{y} (a)| a,

• therwise

SNR := average received signal energy per (complex) symbol time noise energy per (complex) symbol time a2 σ2 <{y} = x + <{w}, <{w} ⇠ N

• 0, σ2/2

Gaussian ¡Scalar ¡Detection

7

y If y < (uA + uB) / 2 choose uA If y > (uA + uB) / 2 choose uB uA

2

uB (uA+uB)

{y | x = uA} {y | x = uB}

• Sufficient statistic for detection:

projection on to

• Since w is circular symmetric,

Gaussian ¡Vector ¡Detection

8

y ˜ y

uA uB UA UB y2 y1

v := uA − uB ||uA − uB|| e y := v∗ ✓ y − uA + uB 2 ◆ = e x + e w e x := v∗ ✓ x − uA + uB 2 ◆ = ( ||uA−uB||

2

, x = uA − ||uA−uB||

2

, x = uB

y = x + w, w ∼ CN

• 0, σ2I
• =

⇒ e w ∼ CN

• 0, σ2

Binary ¡Detection ¡in ¡Gaussian ¡Noise

9

Binary signaling: It suffices to consider the projection onto Probability of error: y = x + w, w ∼ CN

• 0, σ2I
• x = uA, uB

(uA − uB) Pr ⇢ <{w} > ||uA uB|| 2

• = Q

||uA uB|| 2 p σ2/2 ! e y = x||uA − uB|| + e w, x = ±1 2, e w ∼ CN

• 0, σ2

Rayleigh ¡Fading ¡Channel

• Note: |h| is an exponential random variable with mean 1
• Fair comparison with the AWGN case (same avg. signal power)
• Coherent detection:
• The receiver knows h perfectly (channel estimation through pilots)
• For a given realization of h, the error probability is
• Probability of error:

10

y = hx + w, h ∼ CN (0, 1) , w ∼ CN

• 0, σ2

Pr {E | h} = Q a|h| p σ2/2 ! = Q ⇣p 2|h|2SNR ⌘

Check!

Hint: exchange the order in the double integral

Pr {E} = E h Q ⇣p 2|h|2SNR ⌘i = 1 2 1 − r SNR 1 + SNR !

BPSK: x = ±a

SNR = E ⇥ |h|2⇤ a2 σ2 = a2 σ2

Non-­‑coherent ¡Detection

• If Rx does not know the realization of h:
• Scalar BPSK (

) completely fails

• Because the phase of h is uniform over [0, 2π]
• Orthogonal modulation:
• Use two time slots m = 0,1
• Modulation:

11

y = hx + w, h ∼ CN (0, 1) , w ∼ CN

• 0, σ2

x = ±a xA = a

• r

xB = 0 a

• m = 1

m = 0 y xB |y[1]| |y[0]| xA

= ⇒ y := y[0] y[1]

• = h

x[0] x[1]

• +

w[0] w[1]

• := hx + w

Non-­‑coherent ¡Detection

• ML rule:
• Given
• LLR:
• Energy detector:

12

Orthogonal modulation: xA =

a

• r

xB = 0 a

• y = hx + w,

h ∼ CN (0, 1) , w ∼ CN

• 0, σ2I2
• x = xA =

⇒ y ∼ CN ✓ 0, a2 + σ2 σ2 ◆ x = xB = ⇒ y ∼ CN ✓ 0, σ2 a2 + σ2 ◆ Λ(y) := ln f(y | xA) f(y | xB) = a2 (a2 + σ2)σ2

• |y[0]|2 − |y[1]|2
• σ2 + a2

|y[0]|2 + σ2|y(1)|2 −

• σ2|y(0)|2 +
• σ2 + a2

|y[0]|2 (a2 + σ2)σ2

b x = xA ⇐ ⇒ |y[0]| > |y[1]| b x = xB ⇐ ⇒ |y[0]| < |y[1]| SNR = a2 2σ2

Non-­‑coherent ¡Detection

• Probability of error:
• Given
• Hence

13

Orthogonal modulation: xA =

a

• r

xB = 0 a

• y = hx + w,

h ∼ CN (0, 1) , w ∼ CN

• 0, σ2I2
• x = xA =

⇒ y ∼ CN ✓ 0, a2 + σ2 σ2 ◆ = ⇒ |y[0]|2 ∼ Exp

• (a2 + σ2)−1

, |y[1]|2 ∼ Exp

• (σ2)−1

|y[0]|2 and |y[1]|2 are independent

Check!

Pr {E} = Pr

• Exp
• (σ2)−1

> Exp

• (a2 + σ2)−1

= (a2 + σ2)−1 (σ2)−1 + (a2 + σ2)−1 = 1 2 + a2/σ2 = 1 2(1 + SNR) SNR = a2 2σ2

Comparison: ¡AWGN ¡vs. ¡Rayleigh

• AWGN: Error probability decays faster than e-SNR
• Rayleigh fading: Error probability decays as SNR-1
• Coherent detection:
• Non-coherent detection:

14

Pr {E} = Q ⇣√ 2SNR ⌘ ≈ 1 √ 2SNR √ 2π e−SNR at high SNR

Q (x) := Pr {N(0, 1) > a} ⇡ 1 x p 2π e−x2/2 when x 1 r x 1 + x = ✓ 1 1 1 + x ◆1/2 ⇡ 1 1 2(1 + x) ⇡ 1 1 2x when x 1

Pr {E} = 1 2 1 − r SNR 1 + SNR ! ≈ (4SNR)−1 at high SNR Pr {E} = 1 2(1 + SNR) ≈ (2SNR)−1 at high SNR

Comparison: ¡AWGN ¡vs. ¡Rayleigh

15

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

Pr {E}

15 dB 3 dB

Coherent ¡Detection ¡under ¡QPSK

• BPSK only makes use of the real dimension (I channel)
• Rate can be increased if an additional bit is sent on the

imaginary dimension (Q channel)

• QPSK:
• (Bit) Probability of error
• Simply a product of two BPSK
• Analysis is the same
• Simply replace SNR by SNR/2

16

b –b b –b QPSK Im Re

Pr {E}AWGN = Q ⇣√ SNR ⌘ Pr {E}Rayleigh = 1 2 1 − r SNR 2 + SNR !

Double of BPSK! SNR = 2b2 σ2 x ∈ {b(1 + j), b(1 − j), b(−1 + j), b(−1 − j)}

≈ (2SNR)−1 at high SNR

Degrees ¡of ¡Freedom

• A complex scalar channel has 2 degrees of freedom
• BPSK only uses 1 but QPSK uses 2 ⟹ QPSK rate is doubled
• QPSK is 2.5× more energy efficient than 4-PAM
• QPSK avg. Tx energy = 2b2
• 4-PAM avg. Tx energy = 5b2

17

Re b –b b –b QPSK Im Re –3b –b b 3b 4-PAM Im

symbol error probability = 3 2Q r 2b2 σ2 ! 4-PAM symbol error probability ≈ 2Q r 2b2 σ2 ! QPSK

Typical ¡Error ¡Event: ¡Deep ¡Fade

• In Rayleigh fading channel, regardless of constellation

size and detection method (coherent/non-coherent),

• For BPSK,
• If
• If
• Hence,

18

Pr {E} ∼ 1 SNR Pr {E | h} = Q ⇣p 2|h|2SNR ⌘

probability of deep fade |h|2 SNR−1 = ) the conditional probability is very small |h|2 < SNR−1 = ⇒ the conditional probability is very large / Pr

• |h|2 < SNR−1

= 1 − eSNR−1 ≈ SNR−1 Pr {E} = Pr

• |h|2 > SNR−1

Pr

• E | |h|2 > SNR−1

+ Pr

• |h|2 < SNR−1

Pr

• E | |h|2 < SNR−1

Diversity

• Reception only relies on a single “path” h
• If h is in deep fade ⟹ trouble (low reliability)
• Increase the number of “paths” ⟺ Increase diversity
• If one path is in deep fade, other paths can compensate!
• Diversity over time, space, and frequency

19

y = hx + w, h ∼ CN (0, 1) , w ∼ CN

• 0, σ2

Deep Fade Event:

• |h|2 < SNR−1

Outlook

• Time diversity
• Coding + Interleaving: obtain diversity over time
• Repetition coding
• Rotation coding: utilize degrees of freedom better
• Space (Antenna) diversity
• Receive diversity: multiple Rx antennas
• Transmit diversity: multiple Tx antennas
• Space-time codes
• Frequency diversity
• ISI mitigation
• Time-domain equalization
• Direct-sequence spread spectrum
• OFDM

20

Time ¡Diversity

Repetition ¡Coding ¡+ ¡Interleaving

• A simple idea: Repetition Coding
• Repeat the symbol over L time slots (note: L is NOT the # of taps)
• As long as the channels {hl | l = 1, 2, ... L} are not ALL in deep

fade, there is a good probability that we can decode the symbol

• Interleaving:
• Channels within coherence time are highly correlated
• Diversity is obtained if we interleave the codeword across multiple

coherence time periods

22

• Info. Symbol b → ENC → Codeword x :=

⇥b b · · · b⇤ yl = hlxl + wl, l = 1, 2, . . . , L

Coding ¡+ ¡Interleaving ¡Increases ¡Diverirsit

23

Interleaving x2 Codeword x3 Codeword x0 Codeword x1 Codeword | hl | L = 4 l No interleaving

h1 h2 h3 h4 h1 h2 h4 h3

All are bad Only one is bad

Analysis ¡of ¡Repetition ¡Coding

• Equivalent vector channel
• Original channel:
• Sufficient interleaving
• Repetition coding
• Vector channel:
• Analysis of error probability: under BPSK x = ±a,
• After projection we get a scalar equivalent channel
• Probability of error:

24

= ⇒ {hl | l = 1, 2, . . . , L} : i.i.d. CN(0, 1) = ⇒ xl = x, l = 1, 2, . . . , L y = hx + w

y := ⇥y1 y2 · · · yL ⇤T h := ⇥h1 h2 · · · hL ⇤T w := ⇥w1 w2 · · · wL ⇤T

yl = hlxl + wl, l = 1, 2, . . . , L, wl ∼ CN

• 0, σ2

e y = ||h||x + e w, x = ±a, e w ∼ CN

• 0, σ2

E h Q ⇣p 2||h||2SNR ⌘i ≈ ✓2L − 1 L ◆ 1 (4SNR)L

Probability ¡of ¡Deep ¡Fade

• Deep fade event:
• Chi-squared distribution with 2L degrees of freedom
• Probability of deep fade:
• Approximation:

25

• ||h||2 < SNR−1

||h||2 =

L

X

l=1

|hl|2 : sum of i.i.d. Exp(1) RV’s = ⇒ density of ||h||2 : f(x) = 1 (L − 1)!xL−1e−x, x ≥ 0 f(x) = 1 (L 1)!xL−1e−x ⇡ 1 (L 1)!xL−1, 0  x ⌧ 1 = ⇒ Pr

• ||h||2 < SNR−1

≈ Z SNR−1 1 (L − 1)!xL−1 dx = 1 L! 1 SNRL

Deep ¡Fades ¡Become ¡Rarer

26

0.7 0.8 0.9 1.0 5 7.5 10 0.5 0.4 0.3 0.2 0.1 0.6

2 2L

2.5

χ

L = 1 L = 2 L = 3 L = 4 L = 5

pdf

Pr

• ||h||2 < SNR−1

≈ 1 L! 1 SNRL

Diversity ¡Gain: ¡1 ¡→ ¡L

• Comparison of probabilities of deep fade
• Without coding and interleaving:
• With coding and interleaving:
• Diversity: increase from 1 to L

27

∼ SNR−1 ∼ SNR−L

–10

L = 1 L = 2 L = 3 L = 4 L = 5

–5 5 10 15 25 35 30 40 20 1 10–5 10–10 10–15 10–20 10–25 SNR (dB)

Error Prob.

Beyond ¡Repetition ¡Coding

• Repetition coding:
• Achieves full diversity gain L
• Only one symbol per L symbol times
• Does not fully exploit the degrees of freedom
• How to do better?

28

Rotation ¡Code ¡(L=2)

• 2 BPSK symbols (x1, x2 = ±a) over two time slots (L = 2)
• No diversity, as each BPSL symbol experiences only one “path”
• Rotation:
• 4 codewords:

29

x1 x2 (a, a) (a, −a) (−a, −a) (−a, a) x1 x2 xA xB xC xD xA = R a a

• ,

xB = R −a a

• ,

xC = R −a −a

• ,

xD = R  a −a

• x = R

 x1 x2

• ,

R :=  cos θ − sin θ sin θ cos θ

• unitary ¡matrix

Rotation ¡vs ¡Repetition ¡Coding

• Again, like QPSK vs 4-PAM, rotation code uses the

available DoF better.

• Coding Gain: saving power by 3.5 dB (
• )

30

xC xD xB = (b, b) xA = (3b, 3b) xC = (–b, –b) x2 x1 (–a, a) (–a, –a) (a, –a) (a, a) xA x2 xB x1 xD = (–3b, –3b)

Rotation Code Repetition Code √ 5

Vector ¡Channel

31

y = Hx + w = u + w

y = y1 y2

• ,

H = h1 h2

• ,

x = x1 x2

• ,

w = w1 w2

• y

˜ y

uA uB UA UB y2 y1

Probability of error (given channel):

Pr {xA ! xB | H, xA is sent} = Pr ⇢ <{w} > ||uA uB|| 2

• = Q

||uA uB|| 2 p σ2/2 !

Pairwise ¡Error ¡Probability

• Hard to compute the exact error probability
• Union bound: WLOG assume xA is sent.
• Pairwise error probability:

32

Pr {E} ≤ Pr {xA → xB} + Pr {xA → xC} + Pr {xA → xD}

x1 x2 xA xB xC xD = Q r |h1|2(ad1)2 + |h2|2(ad2)2 2σ2 ! ad1 ad2 = Q r SNR (|h1|2|d1|2 + |h2|2|d2|2) 2 ! ≤ exp −SNR

• |h1|2|d1|2 + |h2|2|d2|2

4 !

Q(x) ≤ e−x2/2

= ⇒ Pr {xA → xB} ≤ 1 1 + SNR|d1|2

4

! 1 1 + SNR|d2|2

4

! ≈ 16 |d1|2|d2|2 1 SNR2 Pr {xA → xB | h1, h2} = Q ||uA − uB|| 2 p σ2/2 !

uA = h1xA,1 h2xA,2

• uB =

h1xB,1 h2xB,2

• conditional probability

Product ¡Distance

• Diversity Order = 2
• Intuition:
• Squared product distance:

33

Pr {xA → xB | h1, h2} = Q r SNR (|h1|2|d1|2 + |h2|2|d2|2) 2 ! = ⇒ Pr {Deep Fade} ≈ Pr ⇢ |h1|2 < 1 |d1|2SNR, |h2|2 < 1 |d2|2SNR

• ≈

1 |d1|2|d2|2 1 SNR2

δAB := |d1d2|2 = ⇒ Pr {xA → xB} / 16 δAB 1 SNR2

dAB := xA − xB a = 2 cos θ 2 sin θ

• =

d1 d2

• x1

x2 xA xB xC xD ad1 ad2

Rotation ¡Code ¡Achieves ¡Full ¡Diversity

• Total probability of error: upper bounded by
• Diversity Order = 2
• Coding Gain: maximize the minimum product distance
• max-min is achieved when

34

Pr {E} ≤ Pr {xA → xB} + Pr {xA → xC} + Pr {xA → xD} / 16 ✓ 1 δAB + 1 δAC + 1 δAD ◆ SNR−2 ≤ 48 min {δ}SNR−2

x1 x2 xA xB xC xD

δAB = δAD = 4 sin2 2θ, δAC = 16 cos2 2θ 4 sin2 2θ = 16 cos2 2θ = ⇒ θ = 1 2 tan−1 2

Summary: ¡Time-­‑Diversity ¡Code

• Code:
• Union bound on error probability:
• Pairwise error probability:
• Diversity order:
• Squared product distance

35

x ∈ {x1, x2, · · · , xM}, xi ∈ CL Pr {E} ≤ 1 M X

i6=j

Pr {xi → xj} Pr {xi → xj} ≤

L

Y

l=1

1 1 + SNR|xi,l − xj,l|2/4 min

i6=j {Lij} ,

Lij =

L

X

l=1

I {xi,l 6= xj,l} δij =

L

Y

l=1

|xi,l − xj,l|2 If full diversity L is obtained / 4L M X

i6=j

✓ 1 δij ◆ SNRL

Antenna ¡Diversity

Multiple ¡Antennas

37

Receive Diversity Transmit Diversity Both SIMO MISO MIMO

Typical antenna separation for space diversity ~ λc

Receive ¡Diversity

• Same as repetition coding in time diversity
• Except that there is a further power gain
• Receive SNR in repetition coding =
• Receive SNR in SIMO =
• Probability of Error:

38

h x y a2 σ2 E " Q s 2 ✓ 1 L||h||2 ◆La2 σ2 !#

tends to 1 as L tends to ∞: Diversity Gain

L a2 σ2

L-fold Power Gain

Diversity Order = L L-fold Power Gain y = hx + w ∈ CL, x = ±a e y = ||h||x + e w, e w ∼ CN

• 0, σ2

↓ MRC, h∗ ||h|| ↓

Transmit ¡Diversity

• SIMO: Rx beamforming
• MISO: if Tx knows the channel, it can send
• Same as SIMO: diversity order = L; L-fold power gain
• What if Tx does not know the channel?

39

h x y y = h∗x + w ∈ C, x, h ∈ CL Tx Beamforming x = x h ||h|| ↓ Tx Beamform, x = xh/||h|| ↓ h∗ = ⇥h1 h2 ⇤ y = x||h|| + w

Space-­‑Time ¡Codes

• Transmit the same symbol at all antennas

simultaneously won’t work: (diversity order = 1)

• Time-diversity code can be used to get full Tx diversity:
• Idea: use just one antenna at one time (let x = [x1 x2]T be the time-

diversity codeword)

• Space-time codes

40

x = x1 = ⇒ y = x

L

X

l=1

hl + w,

L

X

l=1

hl ∼ CN(0, L)

x1 x2 ⇥y1 y2 ⇤ = h∗ x1 x2

• +

⇥w1 w2 ⇤

X, space-time codeword

Space-­‑Time ¡Codes: ¡Simple ¡Examples

• Convert a time-diversity code x to a space-time code X:
• Spatial coding: turning one antenna on per time
• Achieves full diversity; waste available DoF
• Better design is out there!

41

⇥y1 y2 ⇤ = h∗ x1 x2

• +

⇥w1 w2 ⇤

Space Time

x = ⇥x1 x2 ⇤ : time-diversity codeword l X =  x1 x2

• : space-time codeword

yT = h∗X + wT

Alamouti ¡Scheme

42

Time 1 Time 2

u1 u2 −u∗

2

u∗

1

X = u1 −u∗

2

u2 u∗

1

• u1, u2 ∈ C

space-time codeword Equivalent Channel: y1 y∗

2

• =

h1 h2 h∗

2

−h∗

1

u1 u2

• +

w1 w2

• = u1

h1 h∗

2

• + u2

 h2 −h∗

1

• +

w1 w2

• h1

h2 h1 h2 Projection onto the two column vectors respectively, we can get two clean channels for u1 and u2! e h1 e h2 e h1 ⊥ e h2

Performance ¡of ¡Alamouti ¡Scheme

• Projection onto two orthogonal directions
• Double the rate of repetition coding
• Diversity order = 2

Full diversity

• 3dB loss in Tx power compared to Tx beamforming

43

y1 y∗

2

• = u1

h1 h∗

2

• + u2

 h2 −h∗

1

• +

w1 w2

• e

h1 ⊥ e h2 e y = u1e h1 + u2e h2 + e w e h∗

1

||e h1|| e y = u1||e h1|| + e w1 = u1||h|| + e w1 e h∗

2

||e h2|| e y = u2||e h2|| + e w2 = u2||h|| + e w2 Two parallel channels, each for one symbol!

Space-­‑Time ¡Code ¡Design

• In general we can extract L Tx diversity by using an L×L

space-time code in an L×1 MISO channel

• Similar to time-diversity code!
• Channel:
• Pairwise error probability:

44

X ∈ {XA, XB, · · · }, X ∈ CL×L yT = h∗X + wT

= Q s SNR 2 X

l=1L

|e hl|2λ2

l

! (XA − XB) (XA − XB)∗ = UΛU∗ e h := U∗h Λ = diag

• λ2

1, · · · , λ2 L

• Pr {XA → XB | h} = Q

||h∗ (XA − XB) || 2 p σ2/2 ! = Q r SNR 2 h∗ (XA − XB) (XA − XB)∗ h ! Pr {XA → XB} ≤

L

Y

l=1

1 1 + SNR|λl|2/4 / 4L QL

l=1 |λl|2 SNR−L

det

• (XA − XB) (XA − XB)∗

Design ¡Criteria

• Time-diversity code:
• Maximize the min squared product distance
• Space-time code
• Maximize the min determinant
• Full diversity ⟺ (Xi - Xj) is full rank for all i, j

45

min

i,j δij,

δij :=

L

Y

l=1

|xi,l − xj,l|2 min

i,j det

• (Xi − Xj) (Xi − Xj)∗

Frequency ¡Diversity

Diversity ¡in ¡Frequency-­‑Selective ¡Channel

• Resolution of multipaths provides diversity.
• Full diversity is achieved by sending one symbol every L

symbol times.

• But this is inefficient (like repetition coding).
• Sending symbols more frequently may result in

intersymbol interference.

• Challenge is how to mitigate the ISI while extracting the

inherent diversity in the frequency-selective channel.

47

y[m] = X

l

hlx[m − l] + w[m]

Approaches

• Time-domain equalization (eg. GSM)
• Direct-sequence spread spectrum (eg. CDMA)
• Orthogonal frequency-division multiplexing OFDM (eg.

802.11a, Flash-OFDM, LTE)

48

ISI ¡Equalization

• Suppose a sequence of uncoded symbols are

transmitted.

• Maximum likelihood sequence detection is performed

using the Viterbi algorithm.

• full diversity can be achieved.
• Complexity grows exponentially with number of taps L.

49

Reduction ¡to ¡Transmit ¡Diversity

50

h0 h0 h1 h0 h1 h2 h0 h1 h2 x [1] y[1] y[2] y[3]

y[4]

x [3]

x [3] x [4]

x [2]

x [2]

x [2] Increasing time x [1] x [3]

MLSD ¡Achieves ¡Full ¡Diversity

51

Space-time code matrix for input sequence Difference matrix for two sequences first differing at is full rank.

Direct ¡Sequence ¡Spread ¡Spectrum

• Information symbol rate is much lower than chip rate

(large processing gain).

• Signal-to-noise ratio per chip is low.
• ISI is not significant compared to interference from other

users and match filtering (Rake) is near-optimal.

52

Channel decoder Modulator Channel encoder Pseudorandom pattern generator Pseudorandom pattern generator Information sequence Output data Demodulator Channel

Frequency ¡Diversity ¡via ¡Rake ¡Receiver

• Considered a simplified situation (uncoded).
• Each information bit is spread into two pseudorandom

sequences xA and xB (xB = -xA).

• Each tap of the match filter is a finger of the Rake.

53

XA XB h w[m] ˜ XA ˜ h ˜ XB ˜ h Decision Estimate h +

ISI ¡vs ¡Frequency ¡Diversity

• In narrowband systems, ISI is mitigated using a complex

receiver.

• In asynchronous CDMA uplink, ISI is there but small

compared to interference from other users.

• But ISI is not intrinsic to achieve frequency diversity.
• The transmitter needs to do some work too!

54

OFDM: ¡Basic ¡Concept

• Most wireless channels are underspread
• Delay spread ≪ Coherence time.
• Can be approximated by a linear time invariant channel
• ver a long time scale.
• Complex sinusoids are the only eigenfunctions of linear

time-invariant channels.

• Should signal in the frequency domain and then

transform to the time domain.

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OFDM

56 d[N–1] ˜ y0 x [N + L – 1] = d[N – 1] Cyclic prefix y [N + L – 1] ˜ dN–1 IDFT DFT Remove prefix ˜ yN–1 y[L] y[N + L – 1] y[1] y[L – 1] y[L] x [L – 1] = d[N – 1] x [L] = d[0] x [1] = d[N – L + 1] ˜ d0 d[0] Channel

OFDM

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OFDM transforms the communication problem into the frequency domain: a bunch of non-interfering sub-channels, one for each sub-carrier. Can apply time-diversity techniques.

Cyclic ¡Predix

• The Nc data symbols constitute one OFDM symbol:
• Cyclic prefix prevents inter-OFDM-symbol interference.
• It also converts linear convolution into circular

convolution.

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e d0, e d1, . . . , e dNc−1

Cyclic ¡Predix ¡Overhead

• OFDM overhead
• = length of cyclic prefix / OFDM symbol time
• Cyclic prefix dictated by delay spread.
• OFDM symbol time limited by channel coherence time.
• Equivalently, the inter-carrier spacing should be much

larger than the Doppler spread.

• Since most channels are underspread, the overhead can

be made small.

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Example ¡1: ¡Flash ¡OFDM

• Bandwidth = 1.25 Mz
• OFDM symbol = 128 samples = 100 μ s
• Cyclic prefix = 16 samples = 11 μ s delay spread
• 11 % overhead.

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Example ¡2: ¡Long-­‑term ¡Evolution ¡(LTE) ¡

• Bandwidth = 1.25 - 20MHz
• OFDM symbol = 128 – 2048 samples (100 μ s)
• Inter-carrier spacing = 15 kHz
• Cyclic prefix = 9 – 144 samples = 5 μ s delay spread
• 5 % overhead.

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Channel ¡Uncertainty

• In fast varying channels, tap gain measurement errors

may have an impact on diversity combining performance.

• The impact is particularly significant in channel with

many taps each containing a small fraction of the total received energy. (eg. Ultra-wideband channels)

• The impact depends on the modulation scheme.

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Summary

• Fading makes wireless channels unreliable.
• Diversity increases reliability and makes the channel

more consistent.

• Smart codes yields a coding gain in addition to the

diversity gain.

• This viewpoint of the adversity of fading will be

challenged and enriched in later parts of the course.

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