Signal Design, Diversity, and Capacity in Multi-access ...

Signal Design, Diversity, and Capacity inMulti-access Communication Systems

Zhifei Fan

Advisor: Prof. Louis L. Scharf

Committee members: Prof. Mahmood R. Azimi-Sadjadi,Prof. Peter J. Brockwell,Prof. Edwin K.P. Chong,

Prof. Donald Estep

Department of Electrical and Computer EngineeringColorado State University

Oct. 19th, 2006

Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 1 / 36

Thesis Work

Thesis Work

Trade-off Between Capacity and Diversity for Block Fading andfor Frequency Selective ChannelsTrade-off between data rate and error probability in time-frequency selectivechannels in wireless communications

Analog Precoder and Equalizer DesignJoint analog precoder and equalizer designs for multi-channel communicationover continuous, time-varying and frequency-selective channels under variouscriteria

Precoder Design in CDMA systemPrecoder design for CDMA systems to significantly reduce the complexity ofMMSE equalizers on the receiver side by exploiting warp convergenceproperty of conjugate gradient recursions for special matrices


Published and Submitted Papers

Published and Submitted PapersJournal

I Z. Fan and L. L. Scharf, “The Approximation of Outage Probability and the Trade-offbetween Capacity and Diversity for the Frequency-Selective Channel,” Submitted toIEEE Transactions on Information Theory. In revision.

I Z. Fan and L. L. Scharf, “Trade-off between Capacity and Diversity for Block FadingSub-Channels,” Submitted to IEEE Transactions on Communications.

I Z. Fan, L. L. Scharf and J. A. Gubner, “Analog Precoder and Equalizer Designs andTheir Geometry for Multichannel Communication,” Submitted to IEEE Transactions onWireless Communications. Under re-review

ConferenceI Z. Fan and L. L. Scharf, “The Approximation of Outage Probability and the Trade-off

between Capacity and Diversity for the Frequency-Selective Channel,” ISIT, Seattle,Washington, July 9-14, 2006.

I Z. Fan and L. L. Scharf, “Trade-off between Capacity and Diversity for Block FadingSub-Channels,” CISS, Princeton, New Jersey, March 22-24, 2006.

I Z. Fan, L. L. Scharf and J. A. Gubner, “Analog Precoder and Equalizer Design forMultichannel Communication,” SPAWC, Lisbon, Portugal, July 11-14, 2004.

I Z. Fan, L. L. Scharf and T. N. Davidson, “Canonical Coordinate Geometry of Precoderand Equalizer Designs for Multichannel Communications,” SPAWC, Lisbon, Portugal,July 11-14, 2004.


Published and Submitted Papers

Outline

1 General Setting

2 Block-fading Channel

3 Frequency-selective Channel

4 Extension to MIMO-OFDM

5 Analog Precoder and Equalizer Design

6 Summary


General Setting

Outline

1 General SettingMotivationThe Work Compared to Related WorkChannel and Communications Models





6 Summary


General Setting Motivation

Motivation

Fundamental goalsIncrease data rate and reduce error probability

ResourcesTime, bandwidth and space

Problem to addressTrade-off between data rate and error probability within time, bandwidth, andspace constraints


General Setting Related Work

The Work Compared to Related Work

Trade-off between capacity andspace diversity [Zheng&Tse 2003]

MIMO and time-invarying,frequency-nonselective channels

Data rate measured bymultiplexing gain r

Error probability measured bydiversity gain d

Trade-off d = (Nt − r)(Nr − r)

x1

x2

y1

y2

w1

w2

xNt

wNr yNr

h11

h22

hNrNt

Problem we solveSISO and time-varying, frequency-selective channels

Trade-off formula like d + r = L

Exploitable degree of freedom in the channel measured by L


General Setting Channel and Communications Models

Channel and Communication ModelsT : communication time

W : communication bandwidth

∆tc : channel coherence time

∆fc : channel coherence bandwidth

� � � � ��

W

T

∆tc

∆fc

� � � ��

T

W

W0hi

Block-Fading Channel

T0

T0 < ∆tc, W0 < ∆fc

� � � ��

� � � �� T

W1W

T1

Frequency-Selective Channel

T1 < ∆tc, W1 = W

� � � � � � � � � � � � � ��

W

Time-Selective Channel

W1

T

T1

T1 = T, W1 < ∆fc


Block-fading Channel

Outline

1 General Setting

2 Block-fading ChannelProblem StatementResults




6 Summary


Block-fading Channel

Block-fading Channels (Ch. 2 in Thesis)

AssumptionsHigh SNR

Flat fading: T0 < ∆tc; W0 < ∆fcUnderspread: ∆tc∆fc >> 1

T0W0 >> 1

� � � � ��

T

WT0

W0hi

NotationsNumber of channels N: TW

T0W0

Channel coefficients hi ∼ CN[0, σ2]

Channel coefficient vector h: [h0, · · · , hN−1]′

Covariance matrix Rhh: E[hhH]

Rank{Rhh} = n, n ≤ N.


Block-fading Channel Problem Statement

Problem Statement

DefinitionsFor a fixed coding scheme

Multiplexing gain r := RTWRT0W0

≤ N

I RTW : bits transmitted in TW spaceI RT0W0 : bits transmitted in T0W0 space

Diversity gain d := − limSNR→∞

log Pe

log SNR ≤ N

I Pe.= SNR−d

� � � � ��

T

WT0

W0hi

Problem to addressFor given r, what is the largest d, i.e., smallest error probability, we can get?


Block-fading Channel Results

Result: Bounds on diversity gain

Theorem (Block-fading Channel (in Journal [2], Conf. [2]))Maximum diversity gain d is bounded in terms of the multiplexing gain r:

(n − r)+ ≤ d ≤ n(1 −rN

).

� � � � � � � � � � ��

n

n

rN

d


Block-fading Channel Results

Result: Exact diversity gainf (α) :=

∑n−1l=0 αl

Compute d:d = n − f (α∗)

α∗: solution to the following linear programming problem

Linear programming problem (in Journal [2], Conf. [2])

maxn−1∑

i=0

αi

s.t.N−1∑

i=0

αi ≤ r

0 ≤ αi ≤ 1, i ∈ [0, n − 1]

αk ≥ αkj , ∀k ∈ [n, N − 1];

kj ∈{

i : 0 ≤ i ≤ n − 1, w(k−n),i 6= 0}


Frequency-selective Channel

Outline

1 General Setting


3 Frequency-selective ChannelProblem StatementResultProofGeometric ExplanationOutage Probability Approximation



6 Summary



Frequency-selective Channels (Ch.3 in Thesis)

AssumptionsT > T1, T1 < ∆tc, W1 = W > ∆fcUnderspread: ∆tc∆fc >> 1

T1∆fc >> 1

i.i.d. channels for each block � � � � ��

� � � �� T

W1W

T1


∆fc

..

.. .

t1W

Time domain model for each blockTapped delay line channelh(τ) =

∑L−1l=0 hlδ(τ − l

W )

Channel time spread Td.= 1

∆fc

Number of taps L := WTd = W∆fc

Channel coefficients hl ∼ CN[0, σ2

L ]: i.i.d.



Frequency-selective Channels (Cont’d)

Frequency domain modely = Hx + n

I Spectral channel coefficient Hm: Hm = H( mT1

) =∑L−1

l=0 h( lW )e

−j2πl

WmT1

I Hm ∼ CN[0, σ2]

I Define M := T1WI Spectral channel coefficient matrix H: diag[H0, · · · , HM−1]

I E[HmH̄n] = σ2

L1−e

−j2πL(m−n)M

1−e−j2π(m−n)

M

: correlated

.. .

... . .

..f1

T1


Frequency-selective Channel Problem Statement

Problem Statement

DefinitionsFor a fixed coding scheme

Multiplexing gain r :=RT1W

RT1∆fc≤ L = W

∆fc

I RT1W : bits transmitted in T1W spaceI RT1∆fc : bits transmitted in T1∆fc space

Diversity gain d := − limSNR→∞

log Pe

log SNR ≤ L

I Pe.= SNR−d

� � � � ��

� � � � �� T

W1W

T1


∆fc

Problem to addressFor given r, what is the largest d, i.e., smallest error probability, we can get?


Frequency-selective Channel Result

Result

Theorem (Frequency-selective Channel (in Journal. [1] and Conf. [1]))The trade-off between the multiplexing gain r and the diversity gain d is

r + d = L.


Frequency-selective Channel Proof

Proof for the Result of Frequency-selective Channel

Theorem (Frequency-selectiveChannel)The trade-off between the multiplexing gain rand the diversity gain d is

r + d = L.

� � � � ��

� � � �� T

W1W

T1


∆fc

Multiplexing GainI RT1∆fc = T1∆fc log SNR + O(1)I RT1W = rT1∆fc log SNR + O(1), for fixed r

Find the maximum d := − limSNR→∞

log Pe

log SNR for fixed r over all codingschemes

Outage probability Pout = infpx(u) P[

I(x; y|H) ≤ rT1∆fc log SNR]



Proof (cont’d)

Pe is exponentially lower bounded by outage probability [Zheng & Tse2003]:

limSNR→∞

log Pout(r, SNR)

log SNR≤ lim

SNR→∞

log Pe(r, SNR)

log SNR

Pe is upper bounded:

Pe(r, SNR) = Pout(r, SNR)P(error|outage) + P(error, no outage)

≤ Pout(r, SNR) + P(error, no outage)

Therefore, the maximum diversity gain d can be obtained through thefollowing “sandwich” inequalities:

Pout(r, SNR).

≤ Pe(r, SNR) ≤ Pout(r, SNR) + P(error, no outage)



Proof (cont’d)

Show

limSNR→∞

log Pout(r, SNR)

log SNR= −d

for all coding schemes

Show

limSNR→∞

log(

Pout(r, SNR) + P(error, no outage))

log SNR= −d

for some coding schemes.



Derivation of Outage ProbabilityCompute outage probability: Pout[r, SNR]

Pout(r, SNR) = infpx(u)

P[

I(x; y|H) ≤ rT1∆fc log SNR]

.

= P[

M−1∑

i=0

log(1 + SNR|Hi|2) ≤ rT1∆fc log SNR

]

Organize channel coefficients vector intop := T1∆fc vectors, L elements in eachvector

H0

Hp...

H(L−1)p

H1

Hp+1...

H(L−1)p+1

· · ·

Hp−1

H2p−1...

HLp−1

� � � � ��

� � � �� T

W1W

T1


∆fc



Derivation of Outage Probability (cont’d)

Decompose mutual informationI

M−1∑

i=0

log(1 + SNR|Hi|2) =

p−1∑

m=0

L−1∑

l=0

log(1 + SNR|Hlp+m|2) =

p−1∑

m=0

Im

I

Im :=

L−1∑

l=0

log(1 + SNR|Hlp+m|2)

Bound outage event

p−1⋂

m=0

{Im ≤ r log SNR} ⊆

{

1p

p−1∑

m=0

Im ≤ r log SNR

}

⊆

p−1⋃

m=0

{Im ≤ r log SNR}


Frequency-selective Channel Geometric Explanation

Geometric Explanation{I0 ≤ r log SNR} =

{

∑L−1l=0 log

(

1 + SNR|Hlp|2)

≤ r log SNR}

{Im ≤ r log SNR} ={

∑L−1l=0 log

(

1 + SNR|Hlp+m|2)

≤ r log SNR}

Set {I0 ≤ r log SNR}

−10 −5 0 5 10

−8

−6

−4

−2

0

2

4

6

8

Set {Im ≤ r log SNR}

−10 −8 −6 −4 −2 0 2 4 6 8 10

−8

−6

−4

−2

0

2

4

6

8



Geometric Explanation{|Hlp|

2 ≤ L(SNRrL −1)

SNR } ⊆{

∑L−1l=0 log

(

1 + SNR|Hlp|2)

≤ r log SNR}

{|Hlp+m|2 ≤ L(SNR

rL −1)

SNR } ⊆{

∑L−1l=0 log

(

1 + SNR|Hlp+m|2)

≤ r log SNR}

−10 −5 0 5 10

−8

−6

−4

−2

0

2

4

6

8

−10 −8 −6 −4 −2 0 2 4 6 8 10

−8

−6

−4

−2

0

2

4

6

8



Geometric Explanation{|Hlp|

2 ≤ L(SNRrL −1)

SNR } ⊆⋂p−1

m=0 {Im ≤ r log SNR}

−10 −5 0 5 10

−8

−6

−4

−2

0

2

4

6

8


Frequency-selective Channel Outage Probability Approximation

Outage Probability Approximation at Low SNR

−10 −5 0 5 10

−8

−6

−4

−2

0

2

4

6

8




−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5




−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05



Outage Probability Approximation at High SNR

−10 −5 0 5 10

−8

−6

−4

−2

0

2

4

6

8




−60 −40 −20 0 20 40 60

−40

−30

−20

−10

0

10

20

30

40




−100 −50 0 50 100

−80

−60

−40

−20

0

20

40

60

80


Extension to MIMO-OFDM

Outline

1 General Setting



4 Extension to MIMO-OFDMMIMO-OFDM Communication Systems


6 Summary


Extension to MIMO-OFDM MIMO-OFDM

MIMO-OFDM Extension

AssumptionsNt transmit antennas, Nr receiveantennas

L multipaths for each channel

Fix r = sL + q

x1

x2

y1

y2

w1

w2

xNt

wNr yNr

h11

h22

hNrNt

Finished WorkOutage probability bound

Future WorkError probability without outage

Conjectured = q[(Nr − s − 1)(Nt − s − 1)] + (L − q)[(Nr − s)(Nt − s)]

Special case: d = (Nr − r)(Nt − r), when L = 1 [Zheng & Tse]


Analog Precoder and Equalizer Design

Outline

1 General Setting




5 Analog Precoder and Equalizer DesignProblem StatementResults

6 Summary


Analog Precoder and Equalizer Design Problem Statement

Problem Statement (Ch.4 in thesis)

AssumptionsNon-parametric channel h(t, τ)

Criteria: maximizing mutual information, minimizing mean-squared errorand minimizing error probability under power constraint

Purpose: finding optimal f and g

g1(t)u1 f1(t) v1

g2(t)u2

gm(t)um

f2(t) v2

fm(t) vm

n(t)

h(t, τ)


Analog Precoder and Equalizer Design Results

Results

EqualizersMatched filter for maximizing mutual information

Wiener filter for minimizing mean-squared error

Zero forcing equalizer for minimizing error probability

Precoder Design Steps (in Journal [3] and Conf. [3] and [4])Subspace design is determined by the second order characterization ofthe channel

(h∗h)(τ ′, τ) =

∫

h̄(s, s − τ ′)h(s, s − τ)ds

Dimension of the subspace decided by the power supply, correspondingeigenvalues of the eigenfunctions spanning this subspace, and noisepower

Coefficient design depends on these criteria


Summary

Outline

1 General Setting





6 SummarySummary for Capacity Vs. DiversitySummary for Analog Precoders


Summary Summary for Capacity Vs. Diversity

Summary for Capacity Vs. Diversity

Problem addressedIs it possible to have fast data rate and small error probability at the sametime?

For fixed multiplexing gain r, what is the maximum diversity gain d we canachieve?

ResultsBlock-fading channel: d = n − f (α∗)

Frequency-selective channel: d = L − r

MIMO-OFDM: Partially completed


Summary Summary for Analog Precoders

Summary for Analog Precoders

Problem addressedHow to precode and equalize for analog channel h(t, τ)?

ResultsOptimal subspace design

Coefficients design according to criteria


Signal Design, Diversity, and Capacity in Multi-access ...

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