Rapid Prototyping of a Fixed-Throughput Sphere Decoder … · Title: Microsoft PowerPoint - fsd_april2009 Author: jst Created Date: 4/23/2009 2:21:00 AM

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IDCOM, University of Edinburgh

April 2009© 2009 Copyright John Thompson

Rapid Prototyping of a Fixed-Throughput Sphere Decoder for MIMO Systems

Luis G. Barbero +, John S. Thompson and Xiang Wu

Institute for Digital CommunicationsUniversity of Edinburgh

+ Now with Queen’s University, Belfast

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Overview of Presentation

• Introduction

• MIMO System

• Rapid Prototyping

• Sphere Decoder (SD)

• Fixed-Sphere Decoder (FSD)

• Diversity Order Analysis

• Ongoing Work and Conclusions

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Introduction

• The use of multiple antennas at both transmitter and receiver (MIMO) significantly increases the capacity and spectral efficiency of wireless communication systems.

• The sphere decoder gives optimal maximum-likelihood performance in MIMO detection with reduced complexity compared to the maximum-likelihood detector.

• However, it has a variable complexity, depending on the noise level and the channel conditions, that hinders its integration into a complete communication system.

• A fixed-sphere decoder is proposed to achieve quasi-ML performance with fixed-complexity resulting in a fully-pipelined hardware implementation.

• Rapid prototyping is a design methodology where a system-level design, specified in a high-level description is quickly translated to a hardware implementation.

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MIMO Wireless Systems

• Wireless communication system equipped with M transmit and N receive antennas, denoted M×N

• Capacity increase compared to single-antenna systems if the different paths between TX/RX are independent– Improve link quality (BER) → Space-Time Coding

– Increase data rate (bps) → Spatial Multiplexing

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MIMO in Wireless Standards

• MIMO is now being integrated into many wireless standards

• Wireless Local Area Networks:

– IEEE 802.11N (Wifi)

• Cellular/Mobile Communications:

– IEEE 802.16 (Wimax)

– 3GPP Long Term Evolution (LTE)

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The Detection Problem

• Each receive antenna observes the random sum of M transmit signals

• If the transmitted digital constellation contains Ppoints (P-QAM, P-PSK)– Received noise free constellation PM points

– Exponential increase in constellation size

TX1: 4-QAM

RX Constellation

TX2: 4-QAM

+ =

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• System equation is written as:

• The optimum detector compares the received signal vector r to every possible noise free constellation point Hs

• The optimum decision on the transmitted data is the closest constellation point in Euclidean distance

• Complexity of this approach is usually too high for complex constellations and large numbers of antennas → evaluate PM constellation points

Maximum Likelihood Decoding

vHsr += Receiver noise

Noise free signal

constellation

Received signal

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MIMO Sphere Decoder

• Maximum-likelihood performance with reduced complexity.

• Search only noiseless received points (defined as the lattice Hs) that lie within a hypersphere of radius R around the received signal r

• Real SD: widely used, only for QAM constellations. Equivalent real decomposition of the system.

• Complex SD: more recent version, can be applied to any constellation. More optimized hardware implementation.

• Initial radius R is selected according to the

noise variance per antenna σ2.

• Complexity of algorithm depends on the noise level and the channel conditions.

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• Where U is the Cholesky decomposition of the Gram matrix HHH

• is the Least Squares estimate of s i.e.

• The solution can be obtained recursively-for each antenna we have:

Sphere Decoder Algorithm

• The sphere constraint can be written as:

rHHHsHH 1)(ˆ −

=s

Where:

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Sphere Decoder Algorithm [2]

• Algorithm can be seen as a tree search through M levels with a metric constraint R2. Each node on the tree has P branches

• Di+1 is the accumulated Euclidean metric down to antenna level i+1

• di is the partial Euclidean metric from antenna level i

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Enumeration

• Two main modulation point enumeration methods:– Pohst Enumeration: work through constellation points in arbitrary

order

– Schnorr-Euchner Enumeration: work through by increasing distance of si from zi

• Schnorr-Euchner used here as it reduces complexity which turns out to be independent of radius R

Schnorr-Euchner tries most

likely path first in each part of

tree search to reduce

complexity

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Antenna Ordering

Antennas can also be ordered before sphere

decoding:

– VBLAST-ZF ordering: order antennas by increasing

noise amplification (reducing SNR)

– VBLAST-MMSE ordering: order antennas by

decreasing signal-to-interference & noise ratio

• MMSE reduces complexity compared to ZF but

suffers slight performance degradation

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Antenna Ordering Results

Complexity Performance

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VBLAST path

Sphere Decoder

Tree Structure

Comparison of Sphere Decoding and V-BLAST

• V-BLAST decoding works as follows:

– Decode each antenna in turn, then subtract out decoded data from

received signal (a.k.a. Successive Interference Cancellation)

– Often the antennas are ordered by decreasing signal-to-noise ratios, as

per ZF/MMSE ordering

• V-BLAST can be interpreted as finding the first path in the Schnorr-Euchner enumeration sphere decoder

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Hardware in the Loop

• Prototyping system that concentrates on the analysis of the MIMOalgorithm.

• Analysis and hardware implementation of novel algorithms.• Requirements:

– Reconfigurable hardware platform

– Methodology that does not require detailed knowledge of the underlying hardware.

– Uniform testing environment.

– Real-time testing → Hardware in the loop.

– Simple and flexible interface between the hardware platform and the host.

• Hardware Platform– Provided by Alpha Data Ltd

– Xilinx Virtex II (XC2V4000)

– Xilinx Virtex II Pro (XC2VP70)

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Rapid Prototyping System

1. MATLAB MIMO evaluation system

2. Simulink testbench and Xilinx DSP System Generator.

3. Xilinx’s Integrated Software Environment to generate FPGA bitstream.

4. MATLAB/ Hardware-in-the-loop real time co-simulation (4×4).

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• Sphere decoder uses sequential tree search

• Difficult to implement efficiently in hardware

Hardware Implementation

Sphere Decoding

Algorithm Flowchart

Conditional

branches

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Sphere Decoder Implementation

• We have implemented 4 parallel sphere decoders

• Use ~50% of FPGA resources

• Data processing speed similar to other state-of-art circuits (more later)

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Implementation Performance

• Circuit throughput depends on clock frequency and No of cycles

• f = 50 MHz and 25 cycles →max throughput 128 Mbps

• Relatively poor performance due to complexity of sphere decoding algorithm

• Design cannot be fully pipelined and optimised

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Motivation for Fixed Sphere Decoder

• Disadvantages of the sphere decoder:– Variable complexity poses problem in actual communication

systems where data needs to be processed in a fixed number of operations.

– Its sequential search results in a hardware implementation that is not fully pipelined, affecting the achievable throughput.

• Modifications of the sphere decoder to marginally reduce the average complexity require additional operations or calculation of limiting thresholds.– It results in a more complex hardware implementation without

achieving a fixed complexity.

• The K-best lattice decoder (based on the sequential M-algorithm) provides a fixed complexity.– It does not take into consideration the MIMO system model

resulting in high complexity.

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Fixed Sphere Decoder Algorithm

• Consideration of the Cholesky coefficients uii shows they are Chi-

squared distributed with 2(N−i+1) degrees of freedom

• The FSD assigns a fixed number of candidate modulation points, ni with 1 ≤ ni ≤ P, to be searched per level.

– More candidates in first levels due to interference from later levels.

– Decision-feedback equalization performed on zi and the increase in

value of E[u2ii] reduce the number of candidates in the last levels.

• The total number of M-dimensional points checked is

• Objective is to achieve maximum likelihood performance but with greatly reduced search space Ns << PM

∏=

=M

k

ks nN1

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Fixed Sphere Decoding• FSD is equivalent to performing tree search following predefined

paths and selecting path with the smallest metric as the solution.

– The ni candidates on each level i are selected according to increasing

distance to zi, following the Schnorr-Euchner enumeration.

• Hypothetical Example for 4-QAM modulation and 4 TX antennas:

)3,2,1,1(=sn

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Fixed Sphere Decoding [2]

• Determine the detection ordering of si according to the distribution of candidates:

• Where for any level the max value of ni is P (constellation size)

Fixed Sphere Decoder Ordering

• If ni = P we select the antenna with the highest noise amplification (or lowest SNR)

• If ni < P we select the antenna with the lowest noise amplification (or highest SNR)

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FSD Ordering Performance

• The FSD ordering has a dominant effect on distribution of points

• No exact analytical study of the FSD ordering on the distributions seems to be feasible when M > 2

• Effect of the FSD ordering on cumulative distribution of the signals si

Cum

ula

tiv

e D

istr

ibuti

on F

unct

ion

Cum

ula

tiv

e D

istr

ibuti

on F

unct

ion

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Fixed Sphere Decoder Performance

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Fixed Sphere Decoder Implementation

• Resource usage of both circuits generally similar

• FSD requires significantly less control logic as the tree search is now straight-forward

• FSD architecture is fully pipelined, providing significant speed-up

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FSD Hardware Implementation

• FSD algorithm can be fully pipelined unlike sphere decoder case

• f = 100 MHz and 4 cycles →max throughput 400 Mbps

• Throughput is fixed and is independent of Eb/No

• FSD algorithm provides much better hardware performance than sphere decoder

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FPGA Performance Evaluation• Our sphere decoder (SD) did not implement efficiently on FPGA

due to sequential tree search

• Fixed sphere decoder (FSD) performs much better

• Much shorter design time for FPGA than ASIC

* - Number in brackets shows throughput for optimized FPGA design

*

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Diversity Order Analysis

• With Jalden/Ottersten from KTH Stockholm we have recently derived diversity order results

• The FSD algorithm splits into two different parts

• Full Enumeration (FE) mimics Maximum Likelihood Detection

• Single Evaluation (SE) is similar to the VBLAST algorithm

• Qu: Under what conditions does the FSD detector perform like the ML detector?

FE

SE

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Diversity Order Analysis [2]

• More formally, the probability of error can be bounded by two terms:

• peML is the probability of error of the maximum likelihood decoder, which has diversity order N (number of receiver antennas)

– The FSD considers the correct vector s but another erroneous vector

minimises Euclidean distance instead

• peSE is the probability that the FSD set does not contain the maximum likelihood solution vector

– The exact distribution of the error event is difficult to obtain

– Resort instead to asymptotic diversity order analysis

– Use Zheng/Tse’s 2003 paper on diversity-multiplexing tradeoff

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Diversity Order Analysis [3]

• In order to investigate peSE we partition our ordered channel matrix into two parts as follows

Single Evaluation (SE) Full Enumeration (FE)

• In FE Stage all possible transmit signals are considered

• For peSE we only need to consider errors made by VBLAST decoding in this stage of decoding

• Consider simplified signal model

• Vector is the transmit data for the SE part of decoding

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Error Analysis of SE Stage

• Diversity order of detection of SE stage depends on distribution of smallest eigenvalue of SE Gram matrix

• This problem may look extremely hard to solve!

• However, it turns out that we can make a link to the complete channel matrix H and its ordered (permuted) version Ho

• The main result of the analysis is a relation between the eigenvalues of H and Ho as follows:

• Here we assume p FE levels and (M-p) SE levels in FSD detector

• Interpretation: smallest eigenvalue of has the same diversity order as that of the (p+1)th sorted eigenvalue of

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Recap of Diversity Order Analysis

• From Zheng/Tse 2003, we have that the asymptotic diversity order of sorted eigenvalue (p+1) of is given by:

2

1 )1()1)(()}({ +++−=+ ppMNdH

p HHλ

• So now we can return to our error probability analysis again:

• Diversity order of PeML is N (No of RX antennas)

• So FSD will asymptotically approach ML performance when:

NppMN >+++−2)1()1)((

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Theoretical Justification...

For M=N=4, we solve:

(p+1)2 > 4 ⇒ p=1

Matches our simulation choice

For M=N=8, we solve:

(p+1)2 > 8 ⇒ p=2

Matches our simulation choice

• So the theory seems to support our empirical algorithmic settings!

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The Islay Project

• In Summer 2008, we started a £1.2M EPSRC multi-disciplinary project in the area of efficient algorithm design and prototyping

– Algorithm Design: John Thompson (UoE) and Andrew Wallace (HWU)

– Hume Language Development: Greg Michelson (HWU) and Kevin

Hammond (StA)

– FPGA Tool Development: John McAllister and Roger Woods (QUB)

• Seek closer links between algorithm development & implementation

• Two representative algorithms for demonstration

– Sphere Decoding for MIMO Systems

– Monte Carlo Markov Chain Detection Algorithms for LIDAR

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Current Research

• Improved K-Best Algorithm– Combining K-Best approach with an MMSE metric

appears to reduce complexity a lot– Hardware implementation runs at same speed as

FSD (less multipliers, more logic operations)

• Thresholding FSD Tree Search– Basic FSD computes tree in parallel fashion– Instead check quality of first few tree branches

– If we have found a likely solution, do not need to computer remaining tree branch metrics

– Can reduce power consumption/improve data throughput

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Conclusions

• We have seen that the sphere decoder does not give an efficient hardware implementation due to tree search

• Fixed sphere decoder algorithm overcomes this problem by fixing the number of points searched per antenna– Allows for an efficient, pipelined hardware implementation

– Achieves close to maximum likelihood performance

• Implemented a list fixed sphere decoder to compute soft metrics for decoding of convolutional/turbo codes– Need to modify number of points per level for good soft metrics

– Optimised iterative decoding with list decoder

• Hard/soft decoding work will be further developed in the Islay project

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Web Sites

• Fixed Sphere Decoder Homepage:

http://www.eng.ed.ac.uk/~jst/sphere/

• The Islay Project Homepage:

http://www.eng.ed.ac.uk/~idcomislay/

Documents

Rapid Prototyping of a Fixed-Throughput Sphere Decoder … · Title: Microsoft PowerPoint - fsd_april2009 Author: jst Created Date: 4/23/2009 2:21:00 AM