Introduction MIMO

8/7/2019 Introduction MIMO

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Introduction

Recent research on wireless communication systems has shown that using mul-

tiple antennas at both transmitter and receiver offers the possibility of wireless

communication at higher data rates compared to single antenna systems. The

information-theoretic capacity of these multiple-input multiple-output (MIMO)

channels was shown to grow linearly with the smaller of the numbers of transmit

and receiver antennas in rich scattering environments, and at sufficiently high

signal-to-noise (SNR) ratios [1].

Some special detection algorithms have been proposed in order to exploit the high

spectral capacity offered by MIMO channels. One of them is the V-BLAST

(Vertical Bell-Labs Layered Space-Time) algorithm which uses a layered structure

[2]. This algorithm offers highly better error performance than conventional linear

receivers and still has low complexity. In this thesis, we offer a new symbol

detection algorithm called V-BLAST/MAP that has a layered structure as V-

BLAST, but uses a modified detecting algorithm that yields a better error

performance than V-BLAST at slightly higher complexity.

In this chapter, we state the MIMO channel model that will b e used throughout

this thesis, state the MIMO symbol detection problem, present some brief

description of previous detection algorithms and brie y compare their error

performance with that of V-BLAST/MAP. These topics are considered in detail in

The MIMO Channel Model

Throughout this thesis, we use the MIMO channel model depicted in Fig. 1.1 with

M transmitter and N receiver antennas.


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Figure 1: Multiple Input Multiple Output (MIMO) channel mo del. TX and RX

stand for transmitter and receiver antennas, respectively.

In each use of the MIMO channel, a vector

= (

1,

2,

, )

of complex

numbers is sent and a vector = (1, 2, , ) of complex numbers isreceived. We assume an input-output relationship of the form

= + Where H is a NxM matrix representing the scattering effects of the channel and

= (1,2,) is the noise vector. Throughout we assume that is arandom matrix with independent complex Gaussian elements with mean 0and unit variance, denoted (0,1). We also assume throughout that v is acomplex Gaussian random vector with i.i.d elements (0,). It isassumed that H and v ar independent of each other and of the data vector a.


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We will assume that the receiver has perfect knowledge of the channel

realization H, while the transmitter has no such channel state information (CSI).

Receivers possession of CSI is justified in cases where the channel is a relatively

slowly time-varying random process; see [3] for a discussion of this point.

The Symbol Detection Problem

The symbol detection problem considered in this thesis is the problem of estimat-

ing the MIMO channel input vector a given the received vector r assuming that

the receiver has perfect knowledge of H. This decision is made on a symbol by

symbol basis without taking into account any statistical dependencies that may

be present in the sequence of vectors a. In other words, we exclude coding across

the time dimension and consider only the modulation-demodulation problem as

depicted in Fig. 2. The goal is to minimize the probability of decision error

= { }Where = (1,2, , ) is the demodulators estimate of a

Figure 2: Modulation, transmission and decision in MIMO wireless systems.

We study the above detection problem under the additional assumptions on

the input vector that:

(i) Each element of a belongs to a common modulation alphabet.

,

, = 1, ,, . Typically, will be a QAM alphabet suchas = AS IN THE CASE OF 4-QAM

(ii) We will assume that symbols in have equal priori probabilities.(iii) The vector a is a random vector over such that


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= Where is a constant, is the identity matrix of size M, is the expectationoperator and denotes Hermitian transpose of a. Assumption implies that theelements of a are uncorrelated and each has energy.

2 = Yield a total average transmitted energy of per symbol, combined over allantennas.

The parameter also has the significance of being the average received energy persymbol at each receiver antenna, as can be seen by computing the energy atreceiver antenna i:

Using above equation, the average received energy per bit at each receiver antenna

may be computed as

b = 2and receiver signal to noise ratio (SNR) is defined as


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b0 = 2

0

While designing a receiver structure for this MIMO system, two main

considerations that should b e taken into account are the error performance and the

implementation complexity. The aim of this thesis is to design a receiver structure

that is powerful in terms of error performance and is practical to implement.

Some Symbol Detection Techniques

The detection strategy is one of the prime criteria to determine the effectiveness of

a communication system. The best performance is obtained by using ML estimate.

But it has a flaw .The computational complexity is very high. It increases with the

increase in number of transmitting and receiving antennas and the constellation

size. Other alternative were then thought of which reduces complexity and gives

performance close to ML estimation scheme. Assumptions made in these

algorithms are

1. Frequency flat AWGN channel condition

2. Input signal and noise are uncorrelated

3. Number of receiving antenna is greater than number of transmitting antennas

(a) Maximum Likelihood Detector

The maximum likelihood detector (MLD) outperforms all the existing detection

methods but its computational complexity increases exponentially with the


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increase in number of transmitting antennas. In ML detection method minimization

is performed over all possible codeword vectors

X = arg min y Hx2

One of the method to reduce computational complexity of ML detector is sphere

decoding.

(b) Sphere Decoder

Sphere decoder restricts the range of search in ML estimation. The principle of the

sphere decoding algorithm is to search the closest lattice point to the received

signal within a sphere radius, where each codeword is represented by a lattice point

in a lattice field. The computational coding is reduced in sphere decoder by

searching over only those points that lie within a hemisphere of radius C around

the received signal y, rather than searching over the entire lattice. The steps

followed in sphere decoding are:

1. The ZF solution for the received signal is calculated is assigned as the centre of

the hemisphere.

2. According to the radius i.e. C ,the limits of the hemisphere are found in all the

dimensions.

3. The points within the previous limits are checked, starting from the lowest

values in each dimension.

4. When any point is found to be closer to the center than the surface of the

hemisphere, the radius is set to that distance. The algorithm jumps back to step2.


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5. The search stops when no points are found inside the hemisphere, and the point

that set the last radius is chosen as the ML solution. The low complexity detection

methods are discussed below:

(c) Zero forcing (ZF) detector

ZF forcing detector Zero forcing(ZF)[14] receiver is a low complexity linear

detection algorithm that outputs

x = Q(xZF)

Where

xZF = H+y

where H+

denotes the Moore-Penrose pseudo inverse of H. ZF is beneficial at high

SNR values but MMSE effective at low as well as high SNR values.

(d) Minimum mean square Estimation

The estimated output of MMSE receiver is

x = Q(xMMSE)

where xMMSE is a linear estimator given by

xMMSE = Gy

where G is chosen to minimize

E {Gy x 2}

The MMSE estimator matrix is given by

G = H

( HH

+ N0INr )

1


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Where is average received energy for each symbol and /M is the average energy

of the constellation point.N0 is noise power.

(e) V-BLAST/ZF detection

The V-BLAST detection is recursive algorithm that estimate the output values

according to certain ordering mechanism so as to minimize the noise power that is

added in non recursive ZF algorithm. The steps followed in V-BLAST ZF

detection process are:

5.5.1 Suppression and cancellation

The first step in ZF receiver is to create a suppression vector from the known

channel matrix such that applying this suppression vector to the received vector

will completely remove interference signals of all other sub streams except the sub

stream of interest. However, the additive noise vector will be enhanced by the

suppression vector as well.

G = (HHH)

1H

H= [g1, g2, . . . , gNt ]

T

be the pseudo-inverse of H and gi = [gi1, gi2, . . . , gNt ]T is the ith

row of G. gi will

be the suppression vector for data symbol xi since it satisfies

gihi= ij

where ij is the Kronecker delta function,

ij = 1 = 0 On applying gi on the received vector y yields decision statistic ai for symbol xi:

ai = giy = xi + giw


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The enhanced noise component giw is a complex Gaussian random variable with

zero mean and variancegi2

N0/2 per dimension. To obtain the estimation of x i,

xi, we need to apply the quantization or slicing function f(.) on a i, based on the the

maximum a posteriori (MAP) probability decision. For example, if xis are binaryphase shift keying (BPSK) modulated and Es are transmitted with equal

probability, the quantization function will give

xi = f(ai) = + 0 0

After detecting xi, the signal of this substream is cancelled from the received vector

y, producing a modified received vector y1

y1= y xi hi

and the channel matrix will be correspondingly modified, by removing the i th

column, as

H1 = [h1, . . . , hi1, hi+1, . . . , hm]

H1 will be used to calculate the suppression vector for the next sub stream.

Detection and cancellation based on y1 will be similar, and we perform this SIC

procedure for every sub stream until all the sub streams have been detected.

It is clear that for the kth sub stream, applying suppression vector to the received

vector will suppress the remaining Nt k sub streams and combine Nr Nt + k

diversity paths to generate the decision statistic. Indeed, by reducing the number ofcolumns of the channel matrix by one every time we cancel a sub stream and

assuming all sub streams are perfectly detected and cancelled, the kth sub stream

will have a diversity order of Nr Nt + k.


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When processing the kth sub stream, we need to lower the noise enhanced in ZF

detection

xZF = H+(Hx + w) = x + H

+w

So, this method increases the effect of noise. The SNR at kth stream ZF detector

output is

SNRk= |xk|2 / Hk2N0

The modified received vector will be affected and the detection of the remaining

Ntk substreams will be influenced, resulting an effect called error propagation.

Hence, a proper ordering in detecting substreams is desired for SIC, to ensure that

the error propagation is minimized.

(f) Ordering

The ordering determined based on SNR . Since the error probability decreases with

increasing SNR and the sub stream with the highest SNR introduces the largest

interference on the remaining substreams, the substreams are detected and

cancelled in order or largest SNR, i.e. at stage k of SIC, the substream with the

highest SNR among all remaining Nt k + 1 substreams is detected and cancelled

first.

V-BLAST ZF detection algorithm

Initialization:

G1 = H

i = 1


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Recursion:

ki = arg max j {k1,...,ki1} (Gi)j 2aki = (Gi)kiyi

yki = Q(aki)

yi+1 = yi yki(H)ki

Gi+1 = H ki

Repeat above equations until all transmitted symbols are detected.

G = H

is the Moore penrose pseudo inverse of the channel matrix and is the

nulling matrix in this algorithm. The lower script ki refers to zeroing columns k1, . .

. , ki because the corresponding components are already canceled. Q is for

quantization of the symbol to the nearest constellation point.

(g) V-BLAST/MMSE detection

The steps followed for detection in V-BLAST/MMSE are same as V-BLAST/ZF

algorithm. Only difference lies in the calculation of inverse channel matrix.

V-BLAST MMSE detection algorithm

Initialization:

G1 = H

( HH

+ N0INr )

i = 1

Recursion:


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ki = arg max j {k1,...,ki1} (Gi)j 2aki = (Gi)kiyi

xki = Q(aki)

yi+1 = yi xki(H)ki

Gi+1 = H

ki (

H ki H

ki + N0INr )


(h) V-BLAST/ZF/MAP algorithm

Initialization:

G1 = H

i = 1

Recursion:

ai = Giyi

si = Q(ai)

pij = fij(aij |sij) / s A fij(aij|s) , j k1, ..., ki1

ki = arg max j{k1,...,ki1} pij xki = siki

yi+1 = yi xki(H)ki


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Gi+1 = H ki


fij is a density function given by

fij(aij /sij) = (1/ j2) exp { - (1/ j

2) aij - sij 2}

where j2

= N0(Gi)j 2.

V-BLAST/ZF/MAP is identical to V-BLAST /ZF except for the ordering in which

symbols are detected. In this method the set of all potential symbol decisions are

ranked with respect to their a-posteriori probabilities of being correct, as estimated

by pij . Thus, pijs are not true MAP probabilities but approximations to how

probable it is that sij = aj .

(i) V-BLAST/MMSE/MAP algorithm

V-BLAST MMSE algorithm is same as V-BLAST ZF algorithm only the method

for channel matrix computation varies.

Initialization:

i = 1

Gi = H

( HH

+ N0INr )

Recursion:

ai = Giyi

si = Q(ai)


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pij = fij(aij |sij)/s2A fij(aij|s) , j k1, ..., ki1

ki = arg maxj {k1,...,ki1} pij

xki = siki

yi+1 = yi xki(H)ki

Gi+1 = H

ki (

H ki H

ki + N0INr )


In methods above ZF and MMSE uses MAP detector to perform demodulation.

MAP demodulation uses log likelihood ratio to decide whether a 0 or 1 is sent.

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Introduction MIMO