Wireless Multiple Access Schemes in a Class of Frequency Selective Channels
with Uncertain Channel State Information
Christopher Steger
February 2, 2004
Outline
Introduction and MotivationProblem FormulationAnalysisSimulationsConclusions
Outline
Introduction and MotivationProblem FormulationAnalysisSimulationsConclusions
Introduction: What is 4G?
We know that we desire data rates far exceeding existing systems.
We know that our PHY layer design depends heavily on our choice of multiple access scheme.
We still don’t know what 4G will be.
How do we choose?
We determine that there are four basic candidates: OFDM TDMA MC-CDMA DS-CDMA
We contend that spectral efficiency is an essential metric.
f
t
t
t
OFDM
TDMA
CDMA
How do we measure?
Observe that spectral efficiency is proportional to mutual information.
Mutual information can be determined without fixing a particular bandwidth.
Fewer assumptions are better
*
*
(b/sym) ninformatio mutual),(
(s/sym) period symbol
(b/s/Hz)
efficiency spectral
CT
B
RTYSIT
B
RC
S
S
S
PREAMBLE DATA
How can we add realism?
Two important aspects are missing from previous analyses: Frequency selective
channels Estimation-based
channel state information.
Outline
Introduction and MotivationProblem Formulation
Channel Model System Models
AnalysisSimulationsConclusions
Problem Formulation
This is a first step in several directions.
We need to stay simple. The simplest frequency
selective channels. The simplest versions
of our multiple access schemes
Channel Model
The simplest multipath channel: 2 paths.
The simplest frequency domain channel: 2 subcarriers.
Discrete in time and frequency.
Block fading. f1 f2
h1 h2
DFT
Channel Model
We define channels in both time and frequency to assure fairness.
Fading is complex Rayleigh or Ricean.
We vary correlation in one domain by varying variances in the other.
22
22
,
21
22
22
,
21
21
21
21
21
21
21
tly...independen fade ),( ssubcarrier twoWhen the
tly...independen fade ),(multipath twoWhen the
ff
ffhh
hh
hhff
ff
hh
System Models
Simplest broadcast scenario: 2 users. Avoid giving any system an unfair advantage. All systems form channel estimates from a
preamble signal.
Frame of Length L Symbols
Preamble of Length L (1-)L Data Symbols
Common Assumptions
Gaussian signaling No feedback.
No power control. Fixed resource allocation.
Size and number of subcarriers is constant. Time slot and spreading code allocations are constant.
Nonlinear interference cancellation. “Genie-aided”
LMMSE equalization.
OFDM System
Two subcarrier OFDM is indistinguishable from FDMA.
Each user gets one subcarrier Flat fading. Frequency allocation is
independent of power allocation.
0 B/2 B
User 1 User 2
0 t 2t
OFDM Block Diagram
Estimate Channel
Equalize
User 1 Data
User 2 Data
Channel FFTDetect and
Decode
Output
IFFT
TDMA System
Each user receives half of the frame and the full bandwidth. Users can resolve both
multipath Time allocation is
independent of power allocation.
Nonlinear ISI cancellation. Cancel edge effects as
well.
s0 h1 s0 h2
s1 h1 s1 h2
s2 h1 s2 h2
Interval of Interest
TDMA Block Diagram
Estimate Channel
Equalize
User 1 Data
User 2 Data
ChannelISI
CancellationDetect and
Decode
Output
MC-CDMA System
Complex orthogonal spreading codes. Length 2 Spread over two
subcarriers. Both users use full
bandwidth and full frame. Each subcarrier is flat
fading Code allocation and
spreading length is independent of power allocation.
s1c11f1 s1c12f2
s2c21f1 s2c22f2
Full Bandwidth
Half Bandwidth
User 1
User 2
First Subcarrier
Second Subcarrier
MC-CDMA Block Diagram
Estimate Channel
FFT
Despread
Spread
Spread IFFT
IFFT
ChannelInterference Cancellation
Equalize
Detect and Decode
User 1 Data
User 2 Data
Output
DS-CDMA System
Complex, orthogonal spreading codes. Length 2
Synchronous transmission Users can resolve both
multipath components. Nonlinear interference
cancellation ISI Other user
Code assignment and spreading length are independent of power allocation.
s1c11h s1c12h
s2c21h s2c22h
Symbol Interval
Chip Interval
User 1
User 2
DS-CDMA ChannelInterval of Interest
s10 c11 h1 s10 c12 h1
User 2
Signal
User 1
Signal
s10 c11 h2 s10 c12 h2
s11 c11 h1 s11 c12 h1
s11 c11 h2 s11 c12 h2
s12 c11 h1 s12 c12 h1
s12 c11 h2 s12 c12 h2
s20 c21 h1 s20 c22 h1
s20 c21 h2 s20 c22 h2
s21 c21 h1 s21 c22 h1
s21 c21 h2
s22 c21 h1
s21 c22 h2
s22 c22 h1
s22 c21 h2 s22 c22 h2
DS-CDMA Block Diagram
Estimate Channel
Despread
Detect and Decode
User 1 Data
User 2 Data
Spread
Spread
ChannelInterference Cancellation
Equalize
Output
Outline
Introduction and MotivationProblem FormulationAnalysis
Sketch of Derivation Calculating Achievable Rate Regions Results
SimulationsConclusions
Analysis
We are deriving a lower bound on mutual information using a method developed by Medard in 2000.
It is a lower bound because it assumes that uncertain CSI yields an additional AWGN term.
The bound depends on the variance of our LMMSE equalizer.
HHHH
HHHNHSY
H
~cov
error estimation~
estimate channelchannel actual
~
~
Sketch of Derivation
First, we find the LMMSE equalizer [Anderson and Moore, Optimal Filtering].
Then we find the variance of the equalizer.
We lower bound our mutual information by finding the difference between the entropy of the signal and the entropy of Gaussian noise with variance equal to that of the equalizer.
121
1
log
,0,0
|,
cov
,,
S
S
TT
MAI
NhNh
YShShYSI
YS
YYEYSE
INHSY
Calculating Achievable Regions
To find average mutual information, take expectation over all channel states.
To define the region, find the average mutual information for all divisions of transmit power between the two users.
Results: Equations
IHHYSI
IFFYSI
hhhhYSI
ffYSI
TS
SCHCSCHC
NSCHCSCHCCDMADS
TSCNSCFSCFCDMAMC
nsh
IRIRsTDMA
nsf
IRsFDMA
ISIISIISIISIISIISI1
221111
221111
112211
1
~~
~~
1~~
222~
22
22
21
21
2
222~
21
21
2
log2
1,
log2
1,
14
log,
12
log,
Results: Achievable Rate Regions
Results: Achievable Rate Regions
Results: Achievable Rate Regions
Recall that in order to achieve
correlation in time we have
made one subcarrier much
stronger than the other.
Therefore, one FDMA user is
favored.
Outline
Introduction and MotivationProblem FormulationAnalysisSimulations
Methods Results
Conclusions
Simulations
Try an alternative evaluation method for our multiple access schemes.
Verify our analytical results. Verify that we have calculated lower
bounds. Assess the tightness of the bounds. Verify convergence to analytical results with
perfect CSI.
Methods
Perform actual channel estimation, interference cancellation and equalization.
Determine the SNR of the output.
Use that SNR to determine mutual information.
Average over many (10000) channel states. noise
signal
effectivenoise
effective
effectivesignal
effective
P
PSNR
NP
SSN
SP
S
SSS
2
2
ˆ
ˆ,
Simulation Block Diagram
Estimate Channel
Equalize
Generate Signal 1
Generate Signal 2
ChannelCancel
InterferenceProcess
Calculate SNR
Process
Process
Multiplex
Calculate Mutual Info
Generate Fading
Generate Noise
Simulation Results
Simulation Results
Simulation Results
Recall that in order to achieve
correlation in time we have
made one subcarrier much
stronger than the other.
Therefore, one FDMA user is
favored.
Outline
Introduction and MotivationProblem FormulationAnalysisSimulationsConclusions
Evaluating Schemes Our Tools
Conclusions: Evaluating Schemes
In several cases, we did not achieve a clear differentiation between schemes.
In the cases where we were able to see strong trends: TDMA and MC-CDMA often had nearly identical
performance. When FDMA performs well, DS-CDMA tends to do
badly. It makes a large difference whether fading is defined in
time or frequency. The difference between Ricean and Rayleigh fading is
also a strong indicator of performance. All 4 schemes were best and worst at least
once.
Conclusions: Tools
Our analytical solutions don’t scale well for future work.
The agreement between our analytical and simulation results is mutually validating.
Simulations scale much more easily to more challenging channels.
Lower bound analysis is not always accurate.
Simulations are the best method.
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