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July 20 05 Behro uz Fa rhang Slide 1 doc.: IEEE 802.11-05/0790r1 Submission A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers Notice: This document has been prepared to assist IEEE 802.11. It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein. Release: The contributor grants a free, irrevocable license to the IEEE to incorporate material contained in this contribution, and any modifications thereof, in the creation of an IEEE Standards publication; to copyright in the IEEE’s name any IEEE Standards publication even though it may include portions of this contribution; and at the IEEE’s sole discretion to permit others to reproduce in whole or in part the resulting IEEE Standards publication. The contributor also acknowledges and accepts that this contribution may be made public by IEEE 802.11. Patent Policy and Procedures: The contributor is familiar with the IEEE 802 Patent Policy and Procedures < http:// ieee802.org/guides/bylaws/sb-bylaws.pdf >, including the statement "IEEE standards may include the known use of patent(s), including patent applications, provided the IEEE receives assurance from the patent holder or applicant with respect to patents essential for compliance with both mandatory and optional portions of the standard." Early disclosure to the Working Group of patent information that might be relevant to the standard is essential to reduce the possibility for delays in the development process and increase the likelihood that the draft publication will be approved for publication. Please notify the Chair <[email protected] > as early as possible, in written or electronic form, if patented technology (or technology under patent application) might be incorporated into a draft standard being developed within the IEEE 802.11 Working Group. If Date: 2005-07-21 Author Name Company Address Phone Email Behrouz Farhang- Boroujeny Univ of Utah Sal Lake City, UT 84112 +801-587-7959 [email protected]

Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

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Page 1: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 1

doc.: IEEE 802.11-05/0790r1

Submission

A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

Notice: This document has been prepared to assist IEEE 802.11. It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein.

Release: The contributor grants a free, irrevocable license to the IEEE to incorporate material contained in this contribution, and any modifications thereof, in the creation of an IEEE Standards publication; to copyright in the IEEE’s name any IEEE Standards publication even though it may include portions of this contribution; and at the IEEE’s sole discretion to permit others to reproduce in whole or in part the resulting IEEE Standards publication. The contributor also acknowledges and accepts that this contribution may be made public by IEEE 802.11.

Patent Policy and Procedures: The contributor is familiar with the IEEE 802 Patent Policy and Procedures <http:// ieee802.org/guides/bylaws/sb-bylaws.pdf>, including the statement "IEEE standards may include the known use of patent(s), including patent applications, provided the IEEE receives assurance from the patent holder or applicant with respect to patents essential for compliance with both mandatory and optional portions of the standard." Early disclosure to the Working Group of patent information that might be relevant to the standard is essential to reduce the possibility for delays in the development process and increase the likelihood that the draft publication will be approved for publication. Please notify the Chair <[email protected]> as early as possible, in written or electronic form, if patented technology (or technology under patent application) might be incorporated into a draft standard being developed within the IEEE 802.11 Working Group. If you have questions, contact the IEEE Patent Committee Administrator at <[email protected]>.

Date: 2005-07-21Author

Name Company Address Phone Email

Behrouz Farhang-Boroujeny Univ of Utah Sal Lake City, UT 84112 +801-587-7959 [email protected]

Page 2: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 2

doc.: IEEE 802.11-05/0790r1

Submission

Outline Introduction Channel model Soft Information: Log-likelihood ratio, LLR values

■ what is the problem?

Zero-forcing / MMSE / VBLAST detectors■ computation of LLR values

Our solution to LLR computation Simulation results

Conclusions

Page 3: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 3

doc.: IEEE 802.11-05/0790r1

Submission

IntroductionWe answer the following question:

■ In a MIMO set-up how one can efficiently obtain soft information, e.g., log-likelihood ratio (LLR) values, of the data bits?

Page 4: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 4

doc.: IEEE 802.11-05/0790r1

Submission

Introduction

We answer the following question:■ In a MIMO set-up how one can efficiently

obtain soft information, e.g., log-likelihood ratio (LLR) values, of the data bits?

The material presented here are protected by a patent application owned by the university of Utah.

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 5

doc.: IEEE 802.11-05/0790r1

Submission

Channel ModelWe consider: A frequency selective channel. OFDM is used to convert the frequency selective channel to a

number of parallel flat fading channels. Accordingly, each subcarrier channel has the following model:

y = Hd+n

where

d is a vector of transmit symbols

y is a vector of received signal

H is the channel gain matrix

n is an additive noise vector.

Page 6: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 6

doc.: IEEE 802.11-05/0790r1

Submission

Receiver Structure

Channel Decoder

MIMO Detector

y

ˆ d

(soft)

Page 7: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 7

doc.: IEEE 802.11-05/0790r1

Submission

Receiver Structure

Channel Decoder

MIMO Detector

y

ˆ d

(soft)

We address an implementation of this block

Page 8: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 8

doc.: IEEE 802.11-05/0790r1

Submission

Receiver Structure

Channel Decoder

MIMO Detector

y

ˆ d

(soft)

Feedback from the output of the channel decoder to MIMO detector, allows near capacity performance.

Page 9: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 9

doc.: IEEE 802.11-05/0790r1

Submission

Soft Information: Log-likelihood ratio (LLR) values

Symbol vector d is obtained from a vector b=[b1 b2 … bN] of information bits through some mapping.

Page 10: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 10

doc.: IEEE 802.11-05/0790r1

Submission

Soft Information: Log-likelihood ratio (LLR) values

Symbol vector d is obtained from a vector b=[b1 b2 … bN] of information bits through some mapping.

We wish to calculate

where

is calculated in the same way.

k lnP(bk 1 | y)

P(bk 1 | y)ln

P(bk 1 | y,b k )b k

P(bk | y)

P(bk 1 | y,b k )b k

P(bk | y)

b k [b1 b2 bk 1 bk1 bN ],bk [b1 b2 bk 1 1 bk1 bN ]

P(bk 1 | y)

and bk [b1 b2 bk 1 1 bk1 bN ]

Page 11: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 11

doc.: IEEE 802.11-05/0790r1

Submission

Soft Information: Log-likelihood ratio (LLR) values

Symbol vector d is obtained from a vector b=[b1 b2 … bN] of information bits through some mapping.

We wish to calculate

where

is calculated in the same way. Problem: the number of combinations that b-k takes is 2N-1!

k lnP(bk 1 | y)

P(bk 1 | y)ln

P(bk 1 | y,b k )b k

P(bk | y)

P(bk 1 | y,b k )b k

P(bk | y)

b k [b1 b2 bk 1 bk1 bN ],bk [b1 b2 bk 1 1 bk1 bN ]

P(bk 1 | y)

and bk [b1 b2 bk 1 1 bk1 bN ]

Page 12: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 12

doc.: IEEE 802.11-05/0790r1

Submission

Soft Information: Log-likelihood ratio (LLR) values

Symbol vector d is obtained from a vector b=[b1 b2 … bN] of information bits through some mapping.

We wish to calculate

where

is calculated in the same way. Problem: the number of combinations that b-k takes is 2N-1!

k lnP(bk 1 | y)

P(bk 1 | y)ln

P(bk 1 | y,b k )b k

P(bk | y)

P(bk 1 | y,b k )b k

P(bk | y)

b k [b1 b2 bk 1 bk1 bN ],bk [b1 b2 bk 1 1 bk1 bN ]

P(bk 1 | y)

and bk [b1 b2 bk 1 1 bk1 bN ]

The key point here is that most of the terms in the numerator and denominator are insignificant.

Thus, a handful of the significant terms may be sufficient for accurate estimation of k.

Page 13: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 13

doc.: IEEE 802.11-05/0790r1

Submission

Log-likelihood ratio values:max-log algorithm

k lnmax

b k

P(bk 1 | y,b k )P(bk | y)

maxb k

P(bk 1 | y,b k )P(bk | y)

Page 14: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 14

doc.: IEEE 802.11-05/0790r1

Submission

Log-likelihood ratio values:max-log algorithm

k lnmax

b k

P(bk 1 | y,b k )P(bk | y)

maxb k

P(bk 1 | y,b k )P(bk | y)

This incurs an insignificant loss (in the order of a a fraction of 1 dB) in performance.

Page 15: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 15

doc.: IEEE 802.11-05/0790r1

Submission

Zero-forcing / MMSE / VBLAST detectors

Zero-forcing detector: ■ Estimate of d = Q[(H*H)-1H*y]

Page 16: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 16

doc.: IEEE 802.11-05/0790r1

Submission

Zero-forcing / MMSE / VBLAST detectors

Zero-forcing detector: ■ Estimate of d = Q[(H*H)-1H*y]

MMSE detector:■ Estimate of d = Q[(H*H+2I)-1H*y]

Page 17: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 17

doc.: IEEE 802.11-05/0790r1

Submission

Zero-forcing / MMSE / VBLAST detectors

Zero-forcing detector: ■ Estimate of d = Q[(H*H)-1H*y]

MMSE detector:■ Estimate of d = Q[(H*H+2I)-1H*y]

VBLAST/Successive Interference Canceller (SIC) detector:■ Detects the strongest symbol first, subtract the detected

symbol, and continue with the successive detection and cancellation of the rest of the symbols.

Page 18: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 18

doc.: IEEE 802.11-05/0790r1

Submission

Zero-forcing / MMSE / VBLAST detectors: computation of LLR values

Starting with the detected d, for a chosen bit bk, it is identified that bk belongs to which element of d, say di.

Page 19: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 19

doc.: IEEE 802.11-05/0790r1

Submission

Zero-forcing / MMSE / VBLAST detectors: computation of LLR values

Starting with the detected d, for a chosen bit bk, it is identified that bk belongs to which element of d, say di.

All the elements of d, except di, are kept fixed. The symbol di is then given all possible values from the symbol constellation, and from all these choices, the maximum values of P(bk=+1|y,d-i) and P(bk=-1|y,d-i) are found and substitute in the max-log LLR formula

k lnmax

b k

P(bk 1 | y,b k )P(bk | y)

maxb k

P(bk 1 | y,b k )P(bk | y)

Page 20: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 20

doc.: IEEE 802.11-05/0790r1

Submission

Our solution to LLR computation:Markov chain Monte Carlo (MCMC) simulation technique

y=Ad+n,

d d1

d2

d3

State d3 d2 d1

S0 -1 -1 -1

S1` -1 -1 +1

S2 -1 +1 -1

S3 -1 +1 +1

S4 +1 -1 -1

S5 +1 -1 +1

S6 +1 +1 -1

S7 +1 +1 +1

Page 21: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 21

doc.: IEEE 802.11-05/0790r1

Submission

Our solution to LLR computation:Markov chain Monte Carlo (MCMC) simulation technique

y=Ad+n,

d d1

d2

d3

State d3 d2 d1

S0 -1 -1 -1

S1` -1 -1 +1

S2 -1 +1 -1

S3 -1 +1 +1

S4 +1 -1 -1

S5 +1 -1 +1

S6 +1 +1 -1

S7 +1 +1 +1

•This procedure gives us a set of selections of d that result in small distances |y-Ad(n)|.

•These may be viewed as important samples of d that correspond to significant terms in the LLR equation

or its max-log version.

k ln

P(bk 1 | y,b k )b k

P(bk | y)

P(bk 1 | y,b k )b k

P(bk | y)

Page 22: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 22

doc.: IEEE 802.11-05/0790r1

Submission

Our solution to LLR computation:Markov chain Monte Carlo (MCMC) simulation technique

y=Ad+n,

d d1

d2

d3

State d3 d2 d1

S0 -1 -1 -1

S1` -1 -1 +1

S2 -1 +1 -1

S3 -1 +1 +1

S4 +1 -1 -1

S5 +1 -1 +1

S6 +1 +1 -1

S7 +1 +1 +1

•This procedure gives us a set of selections of d that result in small distances |y-Ad(n)|.

•These may be viewed as important samples of d that correspond to significant terms in the LLR equation

or its max-log version.

k ln

P(bk 1 | y,b k )b k

P(bk | y)

P(bk 1 | y,b k )b k

P(bk | y)

If implemented in some clever way, the number of samples that is required for estimation of each k is in the order of 10 to 30, even though the size of the state space can be in the order of billions.

Page 23: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 23

doc.: IEEE 802.11-05/0790r1

Submission

How Complex is MCMC? It turns out that MCMC can be implemented VERY

efficiently.

MCMC simulator for a MIMO channel with 4 transmit antenna and 16 QAM symbols has a complexity that is comparable or lower than that of a 16 bit-by-16 bit multiplier.

An implementation of this MCMC simulator on FPGA requires 600 slices.

Page 24: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 24

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results MIMO-OFDM

■ FFT size: 64■ Cyclic prefix length: 16■ Channel is estimated through pilot symbols transmitted at

the beginning of each frame■ Channel: convolutional code with polynomials [133,

171], R = 3/4

Page 25: Doc.: IEEE 802.11-05/0790r1 Submission July 2005 Behrouz Farhang-Boroujeny, Univ of UtahSlide 1 A Novel Soft MIMO Detector for MIMO-OFDM (802.11n) Receivers

July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 25

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 26

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 27

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 28

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 29

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 30

doc.: IEEE 802.11-05/0790r1

Submission

Simulation Results

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 31

doc.: IEEE 802.11-05/0790r1

Submission

Wireless Communications Lab at ECE Dept of Univ of Utah

We are actively involved in development of MIMO detection techniques

In collaboration with L-3 Communication West in Salt Lake City, we have developed a MIMO testbed with 4 transmit and 4 receive antennae

A new version of our testbed that facilitates our research on MIMO detectors is under development.

We are open and seeking collaboration with industry. In particular, we are looking forward to any collaboration with IEEE 802.11n consortia.

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 32

doc.: IEEE 802.11-05/0790r1

Submission

Conclusions The problem of soft estimation of information bits in a MIMO setup was addressed.

Using Markov chain Monte Carlo simulation technique, in the Wireless Communications lab of UofU, we have developed a very efficient detector for this task.

The proposed method could be used along with any conventional detector (ZF/MMSE/VBLAST-SIC) to improve its performance.

Gains in the order of 6 dB or more have been observed.

The proposed method is an excellent choice in systems that employ advanced channel coding, i.e., turbo and LDPC codes.

The proposed technology is extremely hardware friendly. The complexity of the MCMC simulator is not greater than a 16 bit-by-16 bit multiplier. Therefore, in a MIMO-OFDM where many subcarrier channels have to be examined in parallel, a number of MCMC simulators can be

run in parallel at a minimum cost.

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July 2005

Behrouz Farhang-Boroujeny, Univ of Utah

Slide 33

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Submission

Publications[1] B. Farhang-Boroujeny, H. Zhu, and Z. Shi, “Markov chain Monte Carlo algorithms for CDMA and MIMO

communication systems,” IEEE Trans. Signal Processing, Accepted for publication.

[2] H. Zhu, B. Farhang-Boroujeny, and R-R. Chen, “On performance of sphere decoding and Markov chain Monte Carlo detection methods,” IEEE Signal Processing Letters, Accepted for publication.

[3] R-R. Chen, B. Farhang-Boroujeny and A. Ashikhmin, “Capacity-approaching LDPC codes based on Markov chain Monte Carlo MIMO detection,” Submitted to IEEE Communications Letters, March 2005.

[4] H. Zhu, Z. Shi, and B. Farhang-Boroujeny, “MIMO detection using Markov chain Monte Carlo techniques for near-capacity performance,” Int. Conf. Acoustics, Speech and Signal Processing, ICASSP’05, Philadelphia, March 18 – 23, 2005.

[5] Z. Shi, Haidong Zhu, and B. Farhang-Boroujeny, Markov chain Monte Carlo techniques in iterative detectors: a novel approach based on Monte Carlo integration, IEEE Global Telecommunications Conference, GLOBECOM'04., vol. 2 , 29 Nov.-3 Dec., 2004, pp. 325 – 329.

[6] H. Zhu, B. Farhang-Boroujeny, and R-R. Chen, “On performance of sphere decoding and Markov chain Monte Carlo detection methods,” SPAWC 2005, the sixth IEEE International Workshop on Signal Processing Advances for Wireless Communications, June 5-8, 2005, Invited.

[7] R-R. Chen, B. Farhang-Boroujeny and A. Ashikhmin, “Capacity-approaching LDPC codes based on Markov chain Monte Carlo MIMO detection,” SPAWC 2005, the sixth IEEE International Workshop on Signal Processing Advances for Wireless Communications, June 5-8, 2005.