Contribution

In this thesis, we will combine SDM and OFDM technique that can improve SNR performances as well as data rates for the practical channels are frequency-selective fading.

That is shown in the chapter 4. From that we understand the system architecture is robust in the frequency-selective fading channel.

Chapter 2 MIMO Systems

The material in this Chapter is largely taken from [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], and [17].

2.1 Introduction to MIMO Systems

In wireless communication demand high data rate and high link quality access, hence we employ the multiple-input-multiple-output (MIMO) systems architectures to obtain that.

We can employ different space-time code in the MIMO systems architectures to obtain high data rate and high link quality access. The high spectral efficiency due to spatial multiplex (SM), which transmit multiple data streams simultaneously by multiple antennas, and the high link quality access due to space diversity, which transmit the same multiple data streams simultaneously by multiple antennas, both at the transmitter and receiver. MIMO systems provide the ability to turn multipath propagation, which is traditionally the impairment because it can causes signal fading in the wireless transmission, into a benefit but the channel state is not correlative. Since MIMO systems effectively take advantage of random fading and multipath delay spread, the signals transmitted from each transmit antenna appear highly uncorrelated at each receive antenna and the signals travel through different spatial channels.

Then the receiver can exploit these different spatial channels and separate the signals transmitted from different antennas at the same frequency band simultaneously.

MIMO is a promising technology that is suite for high-speed broadband wireless communications. Through space division multiplexing, MIMO technology can transmit multiple data streams in independent parallel spatial channels, thereby increasing total system transmission rate. Considering an arbitrary wireless communication system, a link is considered for that the transmitter is equipped with Nt transmit antennas and the receiver is equipped with Nr receive antennas. Such a setup is illustrated in Figure. 2.1. considered at some assumptions.

Consider this system some important assumptions are made first:

1. Channels are constant during the transmission of a packet. It means the communication is carried out in the some packets period that are shorter than the coherence time of the channels. The channel state is assumed that is time invariance.

2. Channels are memoryless. It means that an independent channel realization is drawn for each use of the channels.

3. The channel is flat fading. It means that constant fading over the bandwidth is desired in the case of narrowband transmissions. It also indicates that the channel gains can be described by complex numbers.

4. The received signal is corrupted by additive white Gaussian noise (AWGN).

5. At all time the receiver can perfectly know the channel matrix which is also known as the channel state information (CSI) and the CSI can be obtained by channel estimation based on the transmission of a training sequence.

With these assumptions, it is common to represent the input/output relations of a narrowband, single-user MIMO link by the complex baseband vector notation and transmit signal vector is transmitted at each instant time.

= +

r Ha w (2.1.1)

where a=[ ,a₁"a_Nt]^T is the Nt×1 transmitted signal vector in \^Nt or^ whose entries ^Nt

are chosen from some complex constellation A (e.g.16-QAM etc.), r ^∈ ^Nr is the received signal vector is the Nr×1 received vector, H=[ ,h₁"h_Nt]is ^^{Nr Nt×} the Rayleigh flat

fading channel matrix whose ith column is h , and where_i w=[ ,w₁"w_Nr]^Tis ^^Nrzero-mean complex Gaussian noise vector at some instant time. We assume that the columns of H are linearly independent (e.g Nr ≧ Nt). We assume that the noise components are independent and identically distributed (i.i.d.) complex Gaussian random variable with E[ww^H]=N₀I

that is additive white Gaussian noise (AWGN). We assume that the complex inputs are uncorrelated and chosen from the same unit-energy discrete alphabet, so that [E aa^H]=I . All the coefficients hij comprise the channel matrix H and represent the complex gain of the channel between the jth transmit antenna and the ith receive antenna. The channel matrix can be written as

Those coefficients {hij} describe the unknown channel properties of the medium that is usually Rayleigh distributed in a rich scattering environment without line-of-sight (LOS) path.

If αij and βij are independent and Gaussian distributed random variables, then |hij| is a Rayleigh distributed random variable. Actually, those coefficients {hij} are likely to be subject to varying degrees of fading and change over time. Therefore, determination of the channel matrix is an important and necessary aspect of MIMO techniques. If all these coefficients are known, there will be sufficient information for the receiver to eliminate interference from other transmitters operating at the same frequency band. Although the introduced MIMO transmission requires flat-fading channels, and it is limited to applications with narrowband transmissions, in real broadband transmission systems, channel conditions are often frequency-selective fading. In wireless transmission, we demand a technique to alleviate the severe effect of frequency-selective fading. Therefore the OFDM technique is a good solution for this purpose in wireless transmission owing to its advantages.

DEMUX

Figure 2-1 Model of MIMO systems

2.2 Maximum Likelihood (ML) Detection Methods

First, we will employ the Maximum Likelihood (ML) Detection for the MIMO systems and it is given by

From the transmitted vector symbols, A is the complex-valued modulating constellation and ANt is the entire set of the possible transmitted vector symbols. We know that find the entire set of the possible transmitted vector symbols so that the complexity is huge due to Nt and A.

We know that Nt is the transmit antennas and A is the complex-valued modulating constellation, so Nt and A is huge such that spend much complexity to find the solution from (2.2.1). From the optimal Maximum Likelihood (ML) Detection in the MIMO systems know the complexity increases when Nt and A increases, so find the suboptimal detection for the MIMO systems.

2.3 The Linear Detector Methods

We could employ these linear detectors for the MIMO systems. The received signal vector r is multiplied with a filter matrix G and then followed by a parallel decision on all layers. Zero-forcing means that the mutual interference between the layers shall be perfectly suppressed. This is accomplished by the Moore-Penrose pseudo-inverse (denoted by (·)+ ) of the channel matrix

( ^H ) 1 ^H

ZF = + = −

G H H H H (2.3.1)

where we assume that H has full column rank. The decision step consists of mapping each element of the filter output vector

( ^H ) 1 ^H

ZF = ZF = + −

a G r a H H H w (2.3.2)

into an element of the symbol alphabet by a minimum distance quantization. The estimation errors of the different layers correspond to the main diagonal elements of the error covariance matrix

2 1

{( )( ) }^H ( ^H )

ZF E ZF ZF σ ⁻

Φ = a −a a −a = _w H H (2.3.3)

which equals the covariance matrix of the noise after the receive filter. It is obvious that small eigenvalues of H^HH will lead to large errors due to noise amplification. This effect is especially observed in systems with the same number of transmit and receive antennas. We can use Linear MMSE detector to decrease the noise amplification. Minimizing the mean squared error (MSE) between the actually transmitted symbols and the output of a linear detector leads to the filter matrix

2 1

( ^H ) ^H

MMSE = +σ_w Nt ⁻

G H H I H (2.3.4)

The resulting filter output is given by

2 1

( ^H ) ^H

MMSE = MMSE = +σ_w Nt ⁻

a G r H H I H r (2.3.5)

and, after some manipulations, the error covariance matrix is found to be

2( ^H 2 ) 1

MMSE σ σ Nt ⁻

Φ = _w H H+ _wI (2.3.6)

With the definition of a (Nt+Nr)×Nt extended channel matrix H and a (Nt+Nr)×1 extended receive vector r through

We can write the output of the MMSE filter as

( ^H ) 1 ^H

MMSE = − = +

a H H H r H r (2.3.8)

Furthermore, the error covariance matrix becomes

2( ^H ) 1

MMSE σ ⁻

Φ = _w H H (2.3.9)

We compare that are the corresponding expression for zero-forcing that can find the only difference is that the channel matrix H has been replaced by H . We can use the QR

decomposition of the channel matrix for ZF or MMSE. For ZF, we can do the QR decomposition of the channel matrix H=QR that we can rewrite the a filter matrix as

1 H

Figure 2-2 QR decomposition algorithm

Table 2-1 Complexity of QR decomposition algorithm

No. Multiplication Nr=4, Nt=4

4. Mult:2Nr 8

7. Mult:3Nr 12

ZF

total complex

Mult:2*Nr+3*Nr*(Nt)²-3Nr*Nt 152

MMSE total complex

Mult:3(Nt)³-3Nr(Nt)²-3(Nt)² -3NrNt+2Nr+2Nt 304

For MMSE, we can do the QR decomposition of the extended channel matrix that we can write as

where the (Nt+Nr)×Nt matrix Q with orthonormal columns was partitioned into the Nr ×Nt matrix Q1 and the Nt ×Nt matrix Q2. From that equation we get the relation as

holds. The filtered receive vector becomes

1 1

H H σ ⁻H H

= = = − _w +

a Q r Q r Ra R a Q w (2.3.14)

From the filtered receive vector we know that have the remaining interference that can not be removed in the detected procedure.

2.4 BLAST Detection Methods

For get high data rate and performance in the MIMO systems, therefore employ Vertical – Bell Laboratories Layered Space-Time (V-BLAST)Architecture to implement that.

TX data

Figure 2-3 Block diagram of V-BLAST structure

Where the transmit antennas send a vector symbol of the size Nt over a rich-scattering wireless channel to the Nr receive antennas at each symbol time. At the transmitter, a single data stream is partitioned into Nt substreams, and each substream is encoded and sent through a different transmit antenna. During reception, each antenna receives signals transmitted from all the Nt transmit antennas. We are base on (V-BLAST) Architecture to find some detector.

We use successive interference cancellation (SIC) technique or ordered SIC (OSIC) based on zero-forcing criterion (ZF V-BLAST) that require the decision-feedback equalization (DFE) and detect sequentially transmitted signals with the smallest estimation error. On zero-forcing criterion find the filter matrix GZF. For get the smallest estimation error, so find the largest signal-to-noise ratio (SNR) and reduce noise enhancement. Find the row g_ZF of G_ZF that has the minimum norm and multiply the received signal.

ˆa_i =gⁱ_ZFr g= ⁱ_ZF(Ha w+ )= +a_i η_i (2.4.1)

where i is the order index a signal is detected. ˆa is quantized to get estimate of _i a and _i regenerate an estimate of signal then the received signal subtract the regenerate an estimate of signal to remove the interference of this signal. Sequential do Nulling and canceling process until all signals are detected. That is shown

1

Figure 2-4 ZF V-BLAST OSIC algorithm [2]

where gⁱ_ZF means the k_i-th row of Gⁱ_ZF , ordering do the repeated computations of a channel matrix pseudoinverse and spend much complexity with O Nt( ⁴), where Nt is the number of channel inputs. We find a low complexity algorithm to do the repeated computations of a channel matrix pseudoinverse and the ordering for the performance. We employ the decision-feedback (DF) detector that to do

nulling and canceling. We can know the risk of error propagation in the decision-feedback (DF) detector, so find out the best ordering to reduce the risk of error propagation. That is to find the max SNR at the first time which reduces the detection errors to do nulling and canceling. Find the low complexity algorithm or/and the best performance on the below when assume Nt =Nr =N. We will use the QR and the sorted QR decomposition in V-BLAST to reduce the complexity. Use the QR decomposition to decompose the H = QR that Q is the N×N unitary matrix and R is the N×N upper triangular matrix and we know the amplitudes of the entries of the matrix R are χ-distributed. We use the feedforward filter matrix Q^H for the received signal. That is shown.

( )

Since Q is unitary, the statistical properties of the noise term w Q w = ^H remain unchanged.

First, we can use the last row to solve the last equation and that is shown.

,

Form that we know the first time to solve the equation and it can affect the performance. If we can solve the equation at the first time is error then we can have much error at the second time.

We call that is error propagation. So, we will use the sorted QR decomposition to choose which columns of H at the first time. That can get the optimum R to solve the equation. We

can use the complexity O N( ²/ 2) in the QR decomposition of permutations of H. We can use the sorted QR decomposition that use an extension of the modified Gram-Schmidt (MGS) algorithm by ordering the columns of H in each orthogonalisation step. That algorithm is shown.

exchange columns i and k in r

Figure 2-5 The Sorted-QR decomposition algorithm [6]

Table 2-2 Complexity of the Sorted-QR decomposition algorithm

No. Multiplication Nr=Nt=N=4

3. Mult:2Nr*Nt 32

5. Mult:2Nr 8

8. Mult:3Nr 12

ZF

total complex

Mult:3.5*Nr*(Nt)²+0.5Nr*Nt 232

MMSE total complex

Mult: 3.5*Nr*(Nt)²+3.5(Nt)³+0.5Nr*Nt+0.5(Nt)² 464

We find the permutation vector p that store the used reordering of H that minimises each r_{k k,}

with k running from 1 to N.. We consequently compute the diagonal elements that are calculated from r1,1to rN,N and it would be optimal to maximise the r_{k k,} in every decoding step, that means from rN,N to r1,1 .That can reduce the risk of error propagation beacause we have the huge SNR gain in the Nth subchannel. We will know the performance is limited by the Nth subchannel. The performance of V-BLAST is limited by the worst subchannel, i.e., subchannel N. Basically this is due to the error propagation which is inherent in a DFE, and the distribution of the upper triangular matrix R. The amplitudes of the entries of R have x distribution with different degrees of freedom, and furthermore, rN,N has the least degree of freedom. Therefore, the Nth subchannel has the worst statistics, and it is crucial to improve its statistics in order to improve the overall performance of the V-BLAST. So we propose to combine ML decoding with the DFE procedure.

2.4.1 Combine ML and DFE Scheme

On the below when assume Nt =Nr =N. For the worst p subchannels, we perform ML decoding and then use a DFE for the remaining subchannels. In order to do this, we do not completely triangularize the channel matrix H. That is shown.

0

×p. To get the above decomposition, we follow the usual Gram-Schmidt orthogonalization procedure for

(

h h1, , ,2 " h_{N p−}

)

which yields Gaussian with zero mean and unit variance. Using this decomposition, we first detect

(

^{aN p− +1,aN p− +2, ," aN}

)

^Tjointly by ML decoding of size p, cancel the interferences caused by

these symbols, and then detect

(

a a1, , ,2 " a_{N p−}

)

^T by the usual DFE procedure. For the decomposition of H we use that for the received signal and show that.

= H

We perform ML decoding with r_b =H a_{b b} +w_b to jointly decode

(

^{aN p− +1,aN p− +2, ,aN}

)

^T

b =

a " and employ the DFE procedure using R to decode

(

a a1, , ,2 a_{N p−}

)

^T

a =

a " .

2.4.2 Parallel Detection (PD) Scheme

We can propose a new parallel detection (PD) frame work which is a compromise between the low complexity schemes and the maximum likelihood estimation (MLE). The parallel detection (PD) frame empoly the optimally ordered decision feedback equalizer (OO-DFE) act as the subdetector. We will describe the optimally OO-DFE. The received signal in complex baseband representation can be then written as

−1

= =

r HP Pa + w Ha + w  (2.4.2.1)

where P is a permutated matrix representing the detection order and H = HP ^-1,a = Pa represent the permutated channel matrix and the substream vector respectively. Substreams are detected recursively in the order from a to ₁ a . The i-th detection on substream symbol _Nt a is explained in the following three steps: cancelling, nulling and ordering.For the i

cancelling considered : all the proceeding detected substream symbolsaˆ₁,"aˆ_i−1 are cancelled out from the received signal,r = r - h a′ _{1 1} "h a_i−1_i−1where h i represents the i-th column of the _i choose the optimally ordered row from F⁺as the nulling vector and make a hard decision. If

the row with smallest norm provides the largest signal-to-noise power ratio (SNR), then it can make the most reliable hard decision. We discuss the block error rate (BLER) for OO-DFE , MLE, and zero-forcing equalizer at a given channel H .Firstly, we discuss the block error rate (BLER) for OO-DFE. Denote the BLER of OO-DFE algorithm by POO

2

For a given H, d is different if the different ordered P is used. The optimal order gives the _oo² largest d . The free distance for a maximum likelihood estimation (MLE) detector where its _oo² BLER is

= H H where subscript (i,i) represents the diagonal element in the i-th row and the i-th column. Intuitively, the relationship between performances of MLE, OO-DFE and zero-forcing is PMLE ≦ Poo ≦ PZF, which suggests d²_free ≥d_oo² ≥d_ZF² . We will use the OO-DFE act as subdetector in parallel detector and show that.

Figure 2-6 Parallel detection

We can understand when the receiver antennas Nr > the transmiter antennas Nt, OO-DFE can perform quite well.However, in the case Nr =Nt, its performance is quite far from that of the MLE. An explanation is given in this section.

The nulling vectors

{

g₁^{, ,}" gNt

}

defined in the zero-forcing based OO-DFE algorithm are orthogonal to each other.

This can be shown easily in the following. Since the nulling vector g is the first row of the ₁^H

pseudo-inverse matrix ⎡⎣h₁, ," h_Nt⎤⎦⁺, thus g1 must be orthogonal to h₂, ," h_Nt .Again

orthogonal to each other. The algorithm of OO-DFE is actually a process of the constructing an orthogonal set

{

^g₁^{, ," g}_Nt

}

with Nt basis vectors in an Nr-dimensional space for the given channel H . these vectors hi^,

(

i=^{1, ,}" Nt

)

are then projected onto gi. It is not difficult to see that doo is only the shortest projection timed by Δ.Therefore, a channel H is a poor channel for OO-DFE algorithm if there exits a column whose projection is small. We show an example of H with three columns, where

Since the three h vectors are almost co-planar, the shortest projection is also small. In other _i words, since g ₁^H is a row of matrix

( )

^{F FH −1FH}, its norm certainly becomes large when matrix F is near singular. We propose a new algorithm which makes the square channel matrix into a tall matrix by making hypotheses on a substream and apply the low complexity detectors on the tall channel matrix to improve the overall performance. We make hypotheses on a and assume it is correctly subtracted from the received signal. The remaining ₁ submatrixH_(2:3)= ⎣⎡h h _{1 2}⎤⎦ becomes a better channel where ^doo2

(

^H( )^2;3

)

^=d2free

(

^H( )^2;3

)

^{= Δ2.}

We make all Mc hypotheses on the first substream a and leave the remaining Nt-1 ₁ substreams to be detected by using Mc subdetectors. Therefore, the PD algorithm consists of Mc branches each with a subdetector. In the qth branch, hypothesis a = x₁ q is made where xq

represents the q-th point in the signal constellation. After subtracting h_{1 q}x from the received signal, the q-th subdetector makes a hard decision bq on a_{(2: )Nt} . For these Mc branches in the PD algorithm, each branch outputs a different hard-decision { , }x b_{q q} on a. Finally, a final decision ˆa is made by selecting the branch with the smallest error r - H_{(2: )Nt}b_q −h₁x_q . Since the subdetectors are now functioning on a Nr-by-(Nt-1) matrix, the diversity is higher and they are expected to perform better. Additionally, we can further improve the performance by properly selecting the substream a on which hypotheses are made. We analyze the BLER ₁ performance of the PD algorithm that employ OO-DFE as its subdetectors and illustrate the

method to select the optimal a . The BLER of the PD algorithm can be written as ₁ that select the best submatrix of the channel to be detected by the subdetectors.

2.5 Chase Detector

We already know the large gap in both performance and complexity between the maximum-likelihood (ML) and the other existed detectors, which are linear detectors or BLAST-ordered decision-feedback (BODF) [15] detectors, hence we have the motivated search for find out a favorable performance-complexity trade-off and a unified framework which is the chase family of detection. In the chase family of detection, there is an important class of reduced-complexity detectors called list-based detectors that adopt a two-step approach of first creating a list of candidate decision vectors, and second choosing the best

We already know the large gap in both performance and complexity between the maximum-likelihood (ML) and the other existed detectors, which are linear detectors or BLAST-ordered decision-feedback (BODF) [15] detectors, hence we have the motivated search for find out a favorable performance-complexity trade-off and a unified framework which is the chase family of detection. In the chase family of detection, there is an important class of reduced-complexity detectors called list-based detectors that adopt a two-step approach of first creating a list of candidate decision vectors, and second choosing the best

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