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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 2 / 36
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.
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 3 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 4 / 36
General Setting
Outline
1 General SettingMotivationThe Work Compared to Related WorkChannel and Communications Models
2 Block-fading Channel
3 Frequency-selective Channel
4 Extension to MIMO-OFDM
5 Analog Precoder and Equalizer Design
6 Summary
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 5 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 6 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 7 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 8 / 36
Block-fading Channel
Outline
1 General Setting
2 Block-fading ChannelProblem StatementResults
3 Frequency-selective Channel
4 Extension to MIMO-OFDM
5 Analog Precoder and Equalizer Design
6 Summary
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 9 / 36
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.
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 10 / 36
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?
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 11 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 12 / 36
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}
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 13 / 36
Frequency-selective Channel
Outline
1 General Setting
2 Block-fading Channel
3 Frequency-selective ChannelProblem StatementResultProofGeometric ExplanationOutage Probability Approximation
4 Extension to MIMO-OFDM
5 Analog Precoder and Equalizer Design
6 Summary
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 14 / 36
Frequency-selective Channel
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
Frequency-Selective Channel
∆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.
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 15 / 36
Frequency-selective Channel
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 16 / 36
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
Frequency-Selective Channel
∆fc
Problem to addressFor given r, what is the largest d, i.e., smallest error probability, we can get?
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 17 / 36
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.
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 18 / 36
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
Frequency-Selective Channel
∆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]
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 19 / 36
Frequency-selective Channel Proof
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)
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 20 / 36
Frequency-selective Channel Proof
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.
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 21 / 36
Frequency-selective Channel Proof
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
Frequency-Selective Channel
∆fc
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 22 / 36
Frequency-selective Channel Proof
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}
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 23 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 24 / 36
Frequency-selective Channel Geometric Explanation
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 25 / 36
Frequency-selective Channel Geometric Explanation
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 26 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 27 / 36
Frequency-selective Channel Outage Probability Approximation
Outage Probability Approximation at Low SNR
−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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 27 / 36
Frequency-selective Channel Outage Probability Approximation
Outage Probability Approximation at Low SNR
−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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 27 / 36
Frequency-selective Channel Outage Probability Approximation
Outage Probability Approximation at High SNR
−10 −5 0 5 10
−8
−6
−4
−2
0
2
4
6
8
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 28 / 36
Frequency-selective Channel Outage Probability Approximation
Outage Probability Approximation at High SNR
−60 −40 −20 0 20 40 60
−40
−30
−20
−10
0
10
20
30
40
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 28 / 36
Frequency-selective Channel Outage Probability Approximation
Outage Probability Approximation at High SNR
−100 −50 0 50 100
−80
−60
−40
−20
0
20
40
60
80
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 28 / 36
Extension to MIMO-OFDM
Outline
1 General Setting
2 Block-fading Channel
3 Frequency-selective Channel
4 Extension to MIMO-OFDMMIMO-OFDM Communication Systems
5 Analog Precoder and Equalizer Design
6 Summary
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 29 / 36
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]
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 30 / 36
Analog Precoder and Equalizer Design
Outline
1 General Setting
2 Block-fading Channel
3 Frequency-selective Channel
4 Extension to MIMO-OFDM
5 Analog Precoder and Equalizer DesignProblem StatementResults
6 Summary
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 31 / 36
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)
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 32 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 33 / 36
Summary
Outline
1 General Setting
2 Block-fading Channel
3 Frequency-selective Channel
4 Extension to MIMO-OFDM
5 Analog Precoder and Equalizer Design
6 SummarySummary for Capacity Vs. DiversitySummary for Analog Precoders
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 34 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 35 / 36
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
Zhifei Fan Signal Design, Diversity and Capacity Thesis Defense, Oct. 19th, 2006 36 / 36
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