ADVANCED SIGNAL PROCESSING TECHNIQUES FOR WIRELESS COMMUNICATIONS Erdal Panayırcı Electronics...
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Transcript of ADVANCED SIGNAL PROCESSING TECHNIQUES FOR WIRELESS COMMUNICATIONS Erdal Panayırcı Electronics...
ADVANCED SIGNAL PROCESSING TECHNIQUES
FOR WIRELESS COMMUNICATIONS
Erdal Panayırcı Electronics Engineering Department
IŞIK University
OUTLINE
IntroductionKnowledge Gaps in GeneralThe essential of EM algorithmThe Sage algorithmSome Application AreasSequential Monte Carlo Method
(SMC)Knowledge Gaps in SMC
INTRODUCTION
Future generation wireless commun. systems are confronted with new challenges mainly due toHostile channel characteristicsLimited bandwidthVery high data rates
Advanced Signal Proc. techniques such asThe Expectation-Maximization
algorithmThe SAGE algorithmThe Baum-Welch algorithmSequential Monte Carlo TechniquesKalman filters and their extensionsHidden Markov modelingStochastic approximation algorithms
In collaboration withInexpensive and Rapid
computational power provide powerful tools to overcome the limitations of current technologies.
Applications of advanced signal processing algorithms, include, but are not limited toJoint/Blind/AdaptiveSequence (data) detectionFrequency, Phase ,timing
synchronizationEqualizationChannel Estimation techniques.
These techniques are employed in advanced wireless communication systems such asOFDM/OFDMACDMAMIMO, Space-time-frequency CodingMulti-User detection
Especially, development of the suitable algorithms for wireless multiple access systems inNon-stationaryInterference-rich
environments presents major challenges to us.
Optimal solutions to these problems mostly can not be implemented in practice mainly due tohigh computational complexity
Advanced signal processing tools, I mentioned before, have provided a promising route for the design of low complexity algorithms with performances approaching the theoretical optimum forFast, and Reliable
communication in highly severe and dynamic wireless environment
Over the past decade, such methods have been successfully applied in several communication problems.
But many technical challenges remain in emerging applications whose solutions will provide the bridge between the theoretical potential of such techniques and their practical utility.
The Key Knowledge GapsTheoretical performance and convergence
analysis of these AlgorithmsSome new efficient algorithms need to be
worked out and developed for some of the problems mentioned above
Computational complexity problems of these algorithms when applied to on-line implementations of some algorithms running in the digital receivers must be handled.
Implementation of these algorithms based on batch processing and sequential (adaptive) processing depending on how the data are processed and the inference is made has not been completely solved for some of the techniques mentioned above.
Some class of algorithms requires efficient generation of random samples from an arbitrary target probability distribution, known up to a normalizing constant. So far two basic types of algorithms, Metropolis algorithm and Gibbs sampler have
been widely used in diverse fields. But it is known that they are substantially complex and difficult to apply for on-line applications like wireless communications.
There are gaps for devising new types of more efficient algorithms that can be effectively employed in wireless applications.
THE EM ALGORITHM
The EM algorithm was popularized in 1977 An iterative “algorithm” for obtaining ML
parameter estimates Not really an algorithm, but a procedure Same problem has different EM
formulations Based on definition of complete and
incomplete data
L. E. Baum, T. Petrie, G. Soules and N. Weiss, A Maximization Technique in Statistical Estimation for Probabilistic Functions of Markov Chains, Annals of Mathematical Statistics, pp. 164-171, 970.
A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum-Likelihood from Incomplete Data Via the EM Algorithm, Journal, Royal Statistical Society, Vol. 39, pp. 1-17, 1977.
C. F. Wu, On the Convergence Properties of the EM Algorithm, Annals of Statistics, Vol. 11, pp. 95-103, 1983.
Main References
The Essential EM Algorithm
Consider estimating parameter vector s from data y (“incomplete” data):
( ) nzsFy += ,
Parameters to be estimated Random parameters
Then, the ML estimate of s is:
( ) ( )[ ]zsypzEsypCsml
s ,maxargˆ =∈
=
Thus, obtaining ML estimates may require:
An Expectation Often analytically intractable
A MaximizationComputationally intensive
The EM Iteration
Define the complete data x
( )xyx → Many-to-one mapping
having conditional density ( )sxf
The EM iteration at the i-th step:
E-step:
M-step:
€
Q s ˆ s i ⎛ ⎝ ⎜
⎞ ⎠ ⎟≡ E log f x s( ) y, ˆ s i
⎡ ⎣ ⎢
⎤ ⎦ ⎥
€
ˆ s i+1 = arg maxs∈C
Q s ˆ s i ⎛ ⎝ ⎜
⎞ ⎠ ⎟
Convergence Properties
At each iteration the likelihood-function is monotonically non-decreasing
If the likelihood-function is bounded, then the algorithm converges
Under some conditions, the limit point coincides with the ML estimate
EM Algorithm Extensions
J. A. Fessler and A. O. Hero, Complete-data spaces and generalized EM algorithms, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP-93), Vol. 4, pp. 1-4, 1993.
J. A. Fessler and A. O. Hero, Space-alternating generalized EM algorithm, IEEE Transactions and Signal Processing, October 1994.
The SAGE Algorithm
The SAGE algorithm is an extension of EM algorithm
It provides much faster convergence than EM
Algorithm alternates several hidden data spaces rather than just using one complete data space, and
Updates only a subset of elements of the parameters in each itteration
Some Application Areas
Positron-Emission-Tomography (PET) Genetics Neural Networks Radar Imaging Image / Speech processing Communications
Channel Estimation / Equalization Multiuser detection Squence estimation Interference rejection
SEQUENTIAL MONTE CARLO TECHNIQUE (SMC)
Emerged in the field of statistics, J. S. Liu and R. Chen, “Sequential Monte
Carlo Methods for Dynamics Systems”, J. American Stat. Assoc., Vol. 93, pp. 1032-1044, 1998.
Recently, SMC has been successfully applied to several problems in wireless communications, such as,Blind equalizationDetection/decoding in fading channels
It is basically based on approximating the expectation operation by means of sequentially generated Monte Carlo samples from either unknow state variables or system parameters.
Main Advantages
SMC is self adaptive and no training/pilot symbols or decision feedback are needed
Tracking of fading channels and the estimation of the data sequence are naturally in integrated
Channel noise can be either Gaussian on Non-Gaussian
It is suitable for MAP receiver design
If the system employs channel coding, the coded signal structure can be easily exploited to improve the accuracy of both channel and data estimation
SMC is suitable for high-speed parallel implementation using VLSI
Does not require iterations like in EM algorithm
Updating with new data can be done more efficiently
SMC Method
Let denote the parameter vector of interest Let denote the complete data
so that is assumed to be simple is partially observed It can be partitioned as where
denotes the observed part
( )Tt21t xxxX ,...,,=( )tX p θ
tX( )ttt SYX ,=
( )tt yyyY ,...,, 21=
( )tt sssS ,...,, 21= denotes in the incomplete data or unobservable or missing data.
θ
Example
1. Fading channel
,...2,1,1
=+=∑=
− tnshyL
ititit
Problem: Joinly estimate the data signal and the unknown channel
parameters
{ } Lihi ,...,2,1, =
,...2,1 , =tst
2. Joint Phase Offset and SNR Estimation
,...2,1 , tnesy t
j
tt
θ
θ is unknown phase offset
2 is unknown noise variance
t21t s s sS ,...,, is the data to be transmitted
Problem: Estimate 21 p θ , based on
complete data ttt SYX , where
tt yyyY ,...,, 21 observed part
tt sssS ,...,, 21 incomplete data
MAP SOLUTION USING SMC METHOD
MAP solution of the unknown parameter vector θ is
θθθθθθ
d Y p Y ttMAP ˆ
tttttS
t dS YS p SY pY pt
,θθ
Where p(θ |Yt) can be computed by
means of incomplete data sequence as
To implement SMS, we need to draw m independent samples (Monte Carlo samples)
m
j
j
ts1
from the conditional distribution of
yyysss pYS p t21t21tt ,...,,,...,,
Usually, directly drawing samples from this distribution is difficult.
But, drawing samples from some trial-distribution tt YSq is easy.
In this case, we can use the idea of importance sampling as follows:
Suppose a set of samples mjs j
t ,...,2,1, is drawn
from the trial distribution . YS q tt By associating
weight
t
j
t
t
j
tj
t Ys q
Ys pw
to the sample ,j
ts
jt
m
1j
j
t
t
ttMAP w S W
1Y S
θ̂
where,
m
1j
j
jt w W
The pair mjwS jt
jt ,...,2,1,, is called
a properly weighted sample. w. r. t. distribution . YS p tt
We can now estimate MAPθ̂ as follows;
By properly choosing the trial distribution q(.), the weighted samples
wS j
t
j
t ,
can be generated sequentially. That is, suppose a set of properly weighted samples
m
1j
j
1t
j
1t w S ,
Then SMC algorithm generates from this set, a new one
m
1j
j
t
j
t w S , at time t.
at time t-1 is given.
1. Draw samples j
t
j
1t
j
t s SS ,
jts from the trial distribution
q(.) and let
2. Compute the important weight jtw from
j1tw sequentially.
3. Compute the MAP estimate
jt
m
1j
j
tMAP w S m1
θ̂
As a summary SMC algorithm is given as follows for j = 1, 2,..., m
KNOWLEDGE GAPS IN SMC Coosing the effective sample size m (empirically
usually ,20 < m < 100). The sampling weights measures the
“quality” of the corresponding drawn data sequence
Small weights implies that these samples do not really represent the distribution from which they are drawn and have small contribution in the final estimation
Resampling procedure was developed for it. It needs to be improved for differential applications
50m j
tw
jtS
Delay Estimation Problem:
Since the fading process is highly correlated, the future received signals contain information about current data and channel state.
A delay estimate seems to be more efficient and promising than the present estimate summarized above.
In delay estimation:
Instead of making inference on (St, θ) with posterior density p(θ, St|Yt), we
delay this inference to a later time (t+) with the distribution p(θ, St|Yt+)
Note: Such a delay estimation method does not increase computational cost but it requires some extra memory.
Knowledge Gap: Develop computationally efficient delayed-sample estimation techniques which will find applications in channel with strong memory (ISI channel).
Turbo Coding Applications
Because, SMC is soft-input and soft-output in nature, the resulting algorithms is capable of exchanging extrinsic information with the MAP outher channel decoder and sucessively improving the overall receiver performance. Therefore blind MAP decoder in turbo receivers can be worked out.