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\title{Applications of Compressive Sensing in Communications and Signal Processing}
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\author{%
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{ Mohan Krishna N (14104101), Srujan Teja T (14104178) }%
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% 20080211 CAUSAL PRODUCTIONS
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Department of Electrical Engineering, Indian Institute of Technology, Kanpur, India.
\\*Email: nmohank@iitk.ac.in, srujant@iitk.ac.in.
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\begin{document}
\maketitle
\begin{abstract}
Compressive Sensing is a Signal Processing technique, which gave a break through in 2004. The main idea of CS is, by exploiting the sparsity nature of the signal (in any domain), we can reconstruct the signal from very fewer samples than required by Shannon-Nyquist sampling theorem. Reconstructing a sparse signal from fewer samples is equivalent to solving a under-determined system with sparsity constraints. Least square solution to such a problem yield poor `results because sparse signals cannot be well approximated to a least norm solution. Instead we use l1 norm(convex) to solve this problem which is the best approximation to the exact solution given by l0 norm(non-convex). In this paper we plan to discuss three applications of CS in estimation theory. They are, CS based reliable Channel estimation assuming sparsity in the channel is known for TDS-OFDM systems[1]. Indoor location estimation from received signal strength (RSS) where CS is used to reconstruct the radio map from RSS measurements[2]. Identifying that subspace in which the signal of interest lies using ML estimation, assuming signal lies in a union of subspaces which is a standard sparsity assumption according to CS theory[3].
Index terms : Compressive Sensing, Indoor positioning, fingerprinting, radio map, Maximum likelihood estimation, union of linear subspaces, subspace recovery.
\end{abstract}
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\begin{IEEEkeywords}
Compressive Sensing, Indoor positioning, fingerprinting, radio map, Maximum likelihood estimation, union of linear subspaces, subspace recovery.
\end{IEEEkeywords}
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% "Appearance-Based Obstacle Detection with Monocular Color Vision",Proceedings of the AAAI National Conference %on Artificial Intelligence, Austin, TX, July/August 2000,Iwan Ulrich and Illah Nourbakhsh
%The Robotics Institute, Carnegie Mellon University
\begin{thebibliography}{20}
\bibitem{b_0}
Zhenkai Fan ; Zhaohua Lu ; Yuting Hu , “Reliable channel estimation based on Bayesiancompressive sensing for TDS-OFDM systems” ,\emph{
IEEE International Conference on DOI: 10.1109/ICCS.2014.7024877, Communication Systems (ICCS), 2014.}
\bibitem{b_1}
Chen Feng ; Au, W.S.A. ; Valaee, S ; Zhenhui Tan, “Received-Signal-Strength-Based Indoor Positioning Using Compressive Sensing” ,\emph{IEEE Transactions on Volume: 11 , Issue: 12 DOI: 10.1109/TMC, Mobile Computing, 2011.}
\bibitem{b_2}
Wimalajeewa, T. ; Eldar, Y.C. ; Varshney, P.K, “Subspace Recovery From Structured Union of Subspaces” ,\emph{
IEEE Transactions on Volume: 61, Issue: 4, Information Theory, 2015.}
%\bibitem{b_3}
%StephenBoyd, and LievenVandenberghe, “ConvexOptimization,” \emph{Cambridge UniversityPress,2009.}
%\bibitem{similar}
%Bo Li, Edward Smith, Huosheng Hu, and Libor Spacek, A Real-time Visual Tracking System in the Robot Soccer Domain, Proceedings of EUREL Robotics-2000, Salford, England (12th - 14th April 2000)
%\bibitem{avoidance}
%Ching-Chang Wong, Chi-Tai Cheng, Kai-Hsiang Huang, Yu-Ting Yang,Design and Implementation of Humanoid Robot for Obstacle Avoidance,
%Department of Electrical Engineering, Tamkang University
%\bibitem{obstacle}
%Mohd. Shanudin Zakaria, Khairuddin Omar,
%Md. Jan Nordin, Shahnorbanun Sahran, Siti Norul Huda Sheikh Abdullah and Anton Heryanto,Position and Obstacle Avoidance Algorithm in Robot Soccer,Journal of Computer Science,2010,Center for Artificial Intelligence Technology, Faculty of Information Science and Technology,
%University Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia
\end{thebibliography}
\end{document}
%
\section{Introduction}
%
Block diagonalization (BD) is a technique for multiuser MIMO (MU-MIMO) system with multiantenna users, where the transmit precoding matrix of each user is designed such that its subspace lies on the null space of all other remaining users, so that multiuser interference is completely canceled. For realizing the BD algorithm, the base station should know the accurate channel state information (CSI) of users. However, the CSI fed back to the base station becomes outdated due to the time varying nature of the channels. Channel prediction is an efficient approach to combat the performance degradation due to feedback delay. Although channel prediction
can reduce the performance degradation due to feedback delay, the unpredictable CSI error still remains, and leads to multiuser interference, so robust design becomes important.
The MLD is the optimal approach for the MIMO decoder because it achieves the best BER performance as compare to the others such as Linear MMSE, LRA-MMSE, etc. However, the complexity of the MLD increases exponentially with the number of constellation points of the modulation and with the number of the spatial streams. Therefore, the MLD system is applicable for only low order MIMO system.
This paper analyzes the effects of the imperfect CSI due to feedback delay on BD MIMO downlink system with CSI prediction and by using the BD technique, the proposed high order MIMO receiver can perfectly decompose CSI matrix into sub-matrices of MIMO channel matrix. This means that the hardware circuit area of the proposed MIMO decoder becomes smaller than the conventional one. This paper also shows the proposed technique achieves higher accuracy than the conventional MIMO decoder methods.
%
\section{Higher Order MIMO Decoder Issues}
%
Multi-user MIMO can leverage multiple users as spatially distributed transmission resources, at the cost of somewhat more expensive signal processing. In comparison, conventional, or single-user MIMO considers only local device multiple antenna dimensions. Multi-user MIMO algorithms are developed to enhance MIMO systems when the number of users, or connections, numbers greater than one (admittedly, a useful concept). Multi-user MIMO can be generalized into two categories: MIMO broadcast channels (MIMO BC) and MIMO multiple access channels (MIMO MAC) for downlink and uplink situations, respectively. Single-user MIMO can be represented as point-to-point, pairwise MIMO.
To remove ambiguity of the words receiver and transmitter, we can adopt the terms access point and users.An access point is the transmitter and a user is the receiver for downlink environments, whereas an access point is the receiver and a user is the transmitter for uplink environments.
The BER of the MLD is different from that of
Linear MMSE and LRA-MMSE, their BER simulation was taken into consideration as shown in the figure 1,2 and 3. We have taken these simulation graphs from reference [1].
\begin{figure}[h]
\centering
\includegraphics[width=6cm,height=6cm]{Fig1.jpg}
\caption {2x2 MIMO BER Characteristic.}
\label{fig:Fig1}
\end{figure}
In case of 2x2 MIMO system shown in Fig. 1, the LRAMMSE achieves the same BER performance as the MLD.
In case of 4x4 MIMO system shown in Fig. 2, the MLD achieves the best BER performance. The LRA-MMSE is worse than the MLD by 3.75 dB at BER = 10-4.
In case of 8x8 MIMO system shown in Fig. 3, the BER performance of the 8x8 LRA-MMSE is even worse than the 4x4 MLD by 7 dB at BER = 10-4. It means that the BER performance of the 8x8 LRA-MMSE is worse than the 8x8 MLD by more than 7 dB. The BER performance of the Linear MMSE is the worst in all cases.
%
\section{Multi-User MIMO Transmission Model}
%
We consider a downlink of a multiuser MIMO system with $N_{k}$ transmit antennas at the base station and $N_{k},k=1,2,....K$, receive antennas at the $k^{th}$ user, where $K$ is the number of multiantenna users. The total number of receive antennas is $N_{r}=\sum_{k=1}^{K}N_{k}$. In this paper we only consider the simplest case that $N_{t} \geq N_{r}$. Let $H_{k}$ be a $N_{t} \times N_{r}$ complex matrix, and $H_{k}$ denotes the downlink channel matrix of the $k^{th}$ user. We assume a flat Rayleigh fading spatially uncorrelated channel model so that the elements of $H_{k},k=1, 2,k=1,2,....K,K$ can be modeled as independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance.
Transmitted signal vector $x_{k}=M_{k}s_{k}$ ,where $M_{k}$ is the precoding matrix and $s_{k}$ data vector of $k^{th}$ user.
The received signal vector of the $k^{th}$ user can be expressed as
%$\mathbf{y}_{k}=\mathbf{H}_{k}\sum_{j=1}^{K}\mathbf{M_{j}}\mathbf{d}_{j}+\mathbf{n_{k}}$
%$\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{M}_{k}\mathbf{d}_{k}+\mathbf{H}_{k}\sum_{i=1,i\neq k}^{K}\mathbf{M}_{i}\mathbf{d}_{i}+\mathbf{n}_{k}$
\begin{equation}
\label{eqn_1}
\mathbf{y}_{k}=\mathbf{H}_{k}\sum_{j=1}^{K}\mathbf{M_{j}}\mathbf{s}_{j}+\mathbf{n_{k}}
\end{equation}
%\begin{equation}
%\label{eqn_1}
%P_{total}= \zeta P_{trans}+P_{C}
%\end{equation}
\begin{equation}
\label{eqn_2}
\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{M}_{k}\mathbf{s}_{k}+\mathbf{H}_{k}\sum_{i=1,i\neq k}^{K}\mathbf{M}_{i}\mathbf{s}_{i}+\mathbf{n}_{k}
\end{equation}
%where,$\mathbf{n}_{k}$ is a vector in $\mathbb{C}^{N_{k}}$ and denotes the zero mean additive Gaussian noises with the same variance of $\sigma _{n}^{2}$ When the multiuser interference is eliminated with proper design of the precoding matrices $M_{k},k=1,2,...,K(i.e.\mathbf{H}_{k}\mathbf{M_{j}=\mathbf{0} \forall k \neq j)$,the multiuser MIMO system is decomposed into parallel single user MIMO channels.
%
\subsection{Block Diagonalization}
%
Define a $(N_{r}-N_{k})\times N_{t}$ aggregate channel matrix $\widetilde{\mathbf{H}_{k}}= \left [\mathbf{H}_{1}^{T}\: ...\: \: \mathbf{H}_{k-1}^{T}\: \mathbf{H}_{k}^{T}\: \mathbf{H}_{k+1}\: ...\: \mathbf{H}_{K}^{T} \right ]^{T}.$ Zero multiuser interference constraint requires that the precoding matrix $\mathbf{M}_{k}$ of the user $k$ lies in the null space of $\widetilde{\mathbf{H}_{k}}$, which requires the null space of $\widetilde{\mathbf{H}_{k}}$ to have dimension greater than 0. Denote the SVD as
\begin{equation}
\label{eqn_3}
\tilde{\mathbf{H}}_{k}= \tilde{\mathbf{U}}_{k}\begin{bmatrix}
\tilde{\mathbf{\sum}_{k}} & \mathbf{0}
\end{bmatrix}\begin{bmatrix}
\tilde{\mathbf{V}}_{k}^{(1)} & \tilde{\mathbf{V}}_{k}^{(0)}
\end{bmatrix}^{H}
\end{equation}
where, $\tilde{\mathbf{U}}_{k}$ holds the left singular vectors of $\tilde{\mathbf{H}}_{k}$ and $\tilde{\mathbf{\sum}_{k}}$ is an $\left ( N_{r}-N_{k} \right )\times \left ( N_{r}-N_{k} \right )$ diagonal matrix with diagonal entries
that are the singular values of $\tilde{\mathbf{H}}_{k}$. $\tilde{\mathbf{V}}_{k}^{(1)}$ holds the first $\left ( N_{r}-N_{k} \right )$ right singular vectors and $\tilde{\mathbf{V}}_{k}^{(0)}$ holds the last $\left ( N_{t}-N_{r}+N_{k} \right )$ right singular vectors. Thus, $\tilde{\mathbf{V}}_{k}^{(0)}$ forms an orthogonal
basis for the null space of $\tilde{\mathbf{H}}_{k}$.
This structure allows the SVD to be determined individually
for each user. Define the SVD of ${\mathbf{H}}_{k}'$
\begin{equation}
\label{eqn_4}
\mathbf{H}_{k}'=\mathbf{H}_{k}\tilde{\mathbf{V}}_{k}^{(0)}=\mathbf{U}_{k}'\begin{bmatrix}
\mathbf{\sum }'_{k} & \mathbf{0}
\end{bmatrix}\begin{bmatrix}
{\mathbf{V}_{k}}'^{(1)} & {\mathbf{V}_{k}}'^{(0)}
\end{bmatrix}^H
\end{equation}
where, ${\mathbf{U}_{k}}'$ holds the left singular vectors of ${\mathbf{H}_{k}}'$ and ${\mathbf{\sum }_{k}}'$ is an $N_{k}\times N_{k}$ diagonal matrix with diagonal entries, $\sqrt{\lambda_{i}'^{(k)}},i=1,2,...,N_{k}$,are the singular values of $\mathbf{H}_{k}'$. ${\mathbf{V}_{k}}'^{(1)}$ holds the first $N_{k}$ right singular vectors and ${\mathbf{V}_{k}}'^{(0)}$ holds the last $ \left ( N_{t}-N_{r} \right )$ right singular vectors.The product of $\tilde{\mathbf{V}}_{k}^{(0)}$ and ${\mathbf{V}_{k}}'^{(1)}$ now
produces an orthogonal basis of dimension $N_{k}$ and represents
the transmission vectors that maximize the information rate
for user $k$ subject to producing zero multiuser interference.
Thus, the encoding matrix for $k^{th}$ user is defined as
\begin{equation}
\label{eqn_5}
\mathbf{M}_{k}=\tilde{\mathbf{V}}_{k}^{(0)}{\mathbf{V}_{k}}'^{(1)}
\end{equation}
and we substitute (5) into (1), the received signal vector of
user $k$ becomes
\begin{equation}
\label{eqn_6}
\mathbf{r}_{k}=\mathbf{H}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)}\mathbf{s}_{k}+\mathbf{H}_{k}\sum _{j=1,j\neq k}^{K}\tilde{\mathbf{V}}_{k}^{(0)}{\mathbf{V}_{k}}'^{(1)}\mathbf{s}_{j}+\mathbf{n}_{k}
\end{equation}
where the second term will become zero. Then the $k^{th}$ user
can decode the received signal with decoding matrix $ \mathbf{U}_{k}'$. The
decoded signal vector of user $k$ becomes
\begin{align*}
\label{eqn_7}
\mathbf{y}_{k}&= {\mathbf{U}_{k}}'^{H}\mathbf{r}_{k}
= {\mathbf{U}_{k}}'^{H}\mathbf{H}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)}\mathbf{s}_{k}+{\mathbf{U}_{k}}'^{H}\mathbf{n}_{k} \\
&= {\mathbf{\sum}_{k}}'\mathbf{s}_{k}+{\mathbf{n}_{k}}'
\end{align*}\begin{equation}\label{eqn_1}\end{equation}
where ${\mathbf{n}_{k}}'={\mathbf{U}_{k}}'^{H}\mathbf{n}_{k}$ and the MIMO channel of user $k$
becomes $N_{k}$ independent subchannels with power gains are the eigenvalues, $\lambda_{i}'^{(k)}$.
%
\subsection{BD for IUI Canceling Method }
%
In the MU-MIMO, a block diagonalization method is well
known as the Inter User Interference (lUI) canceling technique.
We define channel matrix $\mathbf{H}$
transmitted signal vector
x, received signal vector $ \mathbf{r}$, noise vector $ \mathbf{n}$. Received signal
vector $ \mathbf{r}$ is defined as
\begin{equation}
\label{eqn_8}
\mathbf{r}=\mathbf{H}\mathbf{x}+\mathbf{n}
\end{equation}
MU-MIMO AP multiples transmitted signal x by BD weight
matrix $\mathbf{W}_{BD}$.Then, received signal $ \mathbf{r}$ is rewritten by using
matrix $\mathbf{B}$ as
\begin{equation}
\label{eqn_9}
\mathbf{r}=\mathbf{H}\mathbf{W}_{BD}\mathbf{x}+\mathbf{n}=\mathbf{B}\mathbf{x}+\mathbf{n}
\end{equation}
The block diagonalized matrix $\mathbf{B}$ is shown in
\begin{equation}
\label{eqn_10}
\mathbf{B}= \begin{bmatrix}
\mathbf{B1}& \mathbf{0} &... & \mathbf{0}\\
\mathbf{0}& \mathbf{B2} &... & \mathbf{0} \\
:& :& ... &: \\
\mathbf{0}& \mathbf{0} &... & \mathbf{B_{k}}
\end{bmatrix}
\end{equation}
Where $k$ is the index of a block diagonalized sub-matrices
%$\mathbf{B_{k}}$, and the index $k$ is bounded as $2\leqslant$ $k$ $\leqslant M$.
and
M is
the number of transmit antenna. According to above equation,
matrix $\mathbf{B}$ consists of several non-zero sub-matrix $\mathbf{B_{k}}$ and zero
sub-matrix $\mathbf{0}$.
%
\subsection{BD for Complexity Reduction Method }
%
The proposed technique reduces the complexity of high
order MIMO decode by using BD weight matrix. The proposed
technique's received signal is also represented by (8).
we define that the hermitian transported channel matrix $\mathbf{H}$ is matrix $\mathbf{H}^{H}$.We represent the proposed technique's block
diagonalized matrix $\mathbf{B}_{prop}$
\begin{equation}
\label{eqn_11}
\mathbf{B}_{prop}=\mathbf{H}^{H}\mathbf{W}_{prop}
\end{equation}
We take an hermitian transpose the $\mathbf{B}_{prop}$ matrix, then we get
\begin{equation}
\label{eqn_12}
\mathbf{B}_{prop}^{H}=\mathbf{W}_{prop}^{H}\mathbf{H}
\end{equation}
Eq. (12) shows that it is possible to obtain the block diagonalized
channel matrix $\mathbf{B}_{prop}^{H}$ by using BD weight on only the
receiver side. Then we multiply the received signal $\mathbf{r}$ by the
proposed BD weight matrix $\mathbf{W}_{prop}^{H}$ from left side.
\begin{equation}
\label{eqn_13}
\mathbf{W}_{prop}^{H}\mathbf{r}=\mathbf{W}_{prop}^{H}\mathbf{H}\mathbf{x}+\mathbf{W}_{prop}^{H}\mathbf{n}=\mathbf{B}_{prop}^{H}\mathbf{x}+\mathbf{W}_{prop}^{H}\mathbf{n}
\end{equation}
After this process, the proposed high order MIMO system
demodulates per the block diagonalized sub-matrix $\mathbf{B_{k}}$ of
$\mathbf{B}_{prop}^{H}$ In the proposed technique, a noise enhancement is not
caused by multiplying $\mathbf{B}_{prop}^{H}$ matrix to noise $\mathbf{n}$.
%
\subsection{Channel Ordering Method }
%
The state of after block diagonalized matrix $\mathbf{B}_{prop}^{H}$ is
different if we change the order of the before block diagonalized
matrix $\mathbf{H}$'s row vectors. The proposed technique selects
channel ordering to obtain lowest BER performance as in Fig.5.
%
\subsection{Channel Prediction}
%
With minimum mean square error (MMSE) optimization
criteria, the optimal prediction filter weithts are derived as
\begin{align*}
{w}_{ij}^{(k)}&= (E[\mathbf{h}_{ij}^{(k)}\left [ n\right ]\mathbf{h}_{ij}^{(k)H}\left [ n\right ]])^{-1}(E[\mathbf{h}_{ij}^{(k)*}\left [ n+q\right ]\mathbf{h}_{ij}^{(k)}\left [ n\right ]]) \\
&= \mathbf{R}_{k}^{-1}\mathbf{u}_{k}
\end{align*}\begin{equation}\label{eqn_1}\end{equation}
where the entries of matrix $mathbf{R}_{k}$ and vector $mathbf{u}_{k}$ are denoted
by $\mathbf{r}_{ij}^{(k)}$ and $\mathbf{u}_{i}^{(k)}$ respectively, and $i,j=,
, ...,L$.As we
assume the values of $\mathbf{h}_{ij}^{(k)}\left [ n\right ]$, $i=1,2..N_{k},j=1,2...,N_{t}$, are i.i.d. complex Gaussian random variables with zero mean
and unit variance.
the channel prediction error for the MSE
becomes
\begin{align*}
\label{eqn_22}
\sigma _{k}^{2} &= E\begin{bmatrix}
\begin{vmatrix}
\mathbf{\varepsilon }_{ij}^{(k)}\left [ n\right ]\end{vmatrix}^{2}\end{bmatrix}\\
&= E\begin{bmatrix}\begin{vmatrix}\mathbf{w}_{ij}^{(k)}\mathbf{h}_{ij}^{(k)H}\left [ n\right ]-\mathbf{h}_{ij}^{(k)}\left [ n+q\right ]\end{vmatrix}^{2}\end{bmatrix}\\
&= 1-\mathbf{u}_{k}^{H}\mathbf{R}_{k}^{-1}\mathbf{u}_{k}
\end{align*}
\begin{equation}
\label{eqn_1}
\end{equation}
where , $\mathbf{\varepsilon }_{ij}^{(k)}\left [ n+q\right ]$ is defined as channel predection error.
In the following,$\hat{\mathbf{H}}_{k}$ denotes the predicted channel matrix
of $\bar{\mathbf{H}}_{k}$.Then the channel predition error matrix of user k is
defined as
\begin{equation}
\label{eqn_23}
\mathbf{\rho }_{k} =\bar{\mathbf{H}}_{k}-\hat{\mathbf{H}}_{k}
\end{equation}
The ergodic average
of $\mathbf{\rho }_{k}\mathbf{\rho }_{k}^{H}$ becomes
\begin{equation}
\label{eqn_1}
E\begin{bmatrix}\mathbf{\rho }_{k}\mathbf{\rho }_{k}^{H}\end{bmatrix}=\sigma _{k}^{2}N_{t}\mathbf{I }_{N_{k}}
\end{equation}
%
\subsection{MULTIUSER INTERFERENCE AND ROBUST DESIGN }
%
We consider the same BD MIMO system again with feedback
delay and channel prediction, based on the
predicted channel matrix $\hat{\mathbf{H}}_{k}$,i.e. we let $\mathbf{H}_{k}=\hat{\mathbf{H}}_{k}$ and then $\hat{\mathbf{H}}_{k}\tilde{\mathbf{V}_{k}}^{(0)}= 0$
then the
received signal vector of user k becomes
\begin{equation}
\label{eqn_14}
\mathbf{r}_{k}=\bar{\mathbf{H}}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)} \mathbf{s}_{k}+\mathbf{\rho}_{k}\sum _{j=1,j\neq k}^{K}\tilde{\mathbf{V}}_{j}^{(0)}{\mathbf{V}_{j}}'^{(1)}\mathbf{s}_{j}+\mathbf{n}_{k}
\end{equation}
If the prediction error is zero, the second term of (18) will be zero, then the receiving becomes normal BD receiving, if the prediction error is not zero, the interference from the
data streams for other users will not be zero. In this section, we
will analyse this multiuser interference and propose a robust
design for such kind of systems.
%
\subsection{Analysis of Multiuser Interference}
%
We employ zero forcing (ZF) receiving to cancel the interference
from the data streams for user. If there was no
multiuser interference, with ZF receiving the performance will
almost be the same as if there is no prediction error. The
decoded signal vector of user k becomes
\begin{equation}
\label{eqn_15}
\mathbf{y}_{k}=(\bar{\mathbf{H}}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)})^{-1}\mathbf{r}_{k}
\end{equation}
\begin{multline*}
\mathbf{y}_{k}=\mathbf{s}_{k}+(\bar{\mathbf{H}}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)})^{-1}\mathbf{\rho}_{k}\sum _{j=1,j\neq k}^{K}\mathbf{x}_{j}\\
+(\bar{\mathbf{H}}_{k}\tilde{\mathbf{V}_{k}}^{(0)}{\mathbf{V}_{k}}'^{(1)})^{-1}\mathbf{n}_{k}
\end{multline*}\begin{equation}\label{eqn_1}\end{equation}
Now we define the SVD
\begin{equation}
\label{eqn_17}
\bar{\mathbf{H}}_{k}'=\bar{\mathbf{H}_{k}}\tilde{\mathbf{V}}_{k}^{(0)}=\bar{\mathbf{U}}'\begin{bmatrix}\bar{\mathbf{\sum }_{k}}' & \mathbf{0}
\end{bmatrix}\begin{bmatrix}\bar{\mathbf{V}_{k}}'^{(1)} & \bar{\mathbf{V}_{k}}'^{(0)}
\end{bmatrix}^{H}
\end{equation}
The inverse marix in (19) becomes
\begin{equation}
\label{eqn_18}
(\bar{\mathbf{U}}'\bar{\mathbf{\sum}_{k}}'\bar{\mathbf{V}_{k}}'^{(1)H}\bar{\mathbf{V}_{k}}'^{(1)})^{-1}\approx\bar{\mathbf{\sum}}'^{-1}_{k}\bar{\mathbf{U}}'^{H}\approx\mathbf{\sum}'^{-1}_{k}\mathbf{U}'^{H}
\end{equation}
Then the signal to interference and noise power ratio (SINR)
of i’th subchannel of user k becomes
\begin{equation}
\label{eqn_18}
SINR_{i}^{(k)}\approx \frac{\mathbf{\lambda_{i}}'^{k}}{\frac{N_{r}-N_{k}}{N_{t}}{\mathbf{d}_{i}}'^{(k)H}{\mathbf{d}_{i}}'^{(k)}+N_{r}/\gamma }
\end{equation}
%
\subsection{Robust Design}
%
Not considering the interference will result in high received
symbol error rate.From the result of (18), we define the
estimation of SINR of the k’th user’s
i’th subchannel as
\begin{equation}
\label{eqn_19}
\hat{SINR_{i}^{(k)}}=\frac{\mathbf{\lambda_{i}}'^{k}}{\left ( \alpha \left ( N_{r}-N_{k} \right )\sigma _{k}^{2}+N_{r}/\gamma \right )}
\end{equation}
where $\alpha$ is the robustness parameter.If $\alpha$ is set larger, that means interference power is
estimated to be large, the system will be robust in interference,
but lower transmit data rate. And if $\alpha$ is set smaller, that means
interference power is estimated to be small, the system will
be sensitive to interference, but higher transmit data rate with
more received symbol error.
%\begin{equation}
%\label{eqn_20}
%\hat{h}_{ij}^{(k)}\left [ n+q \right ]=\sum_{l=1}^{L}w_{ij}^{(k,l)}^{\ast }h_{ij}^{(k)}\left [ n-l+1 \right ]
%=\mathbf{w}_{ij}^{(k)H}\mathbf{h}_{ij}^{(k)}\left [ n\right ]
%\end{equation}
%
\section{SIMULATION RESULTS}
%
we show the simulation results with different
durations of feedback delay, and compare the performances of
systems with and without robust design.The feedback delay durations are set as 2, 4 and 8 time slots, and the associated $\sigma_{k}$ and $\mu _{k}$ are shown in below Table
\begin{table}[h!]
\centering
\begin{tabular}{|c || c |c |c|}
\hline
Delay (time slots) & 2 & 4 & 8 \\ [0.5ex]
\hline
$\sigma _{k}^{2}$ & $7.5\times 10^{-4}$ & $5.6\times 10^{-3}$ & $5.4\times 10^{-2}$ \\
\hline
$\mu _{k}^{2}$ & $32.5dB$ & $23.8dB$ & $13.9dB$ \\ [1ex]
\hline
\end{tabular}
\caption{MSE OF CSI With Feedback Delay and $\mu _{k}$}
\label{table:1}
\end{table}
Below figure shows the average throughput of
4 users BD MIMO downlink system over average SNR from
0 dB to 35 dB with different delay durations.when SNR is larger than $\mu _{k}$ the throughput
decreases due to the interference becoming stronger than
the noise, but when SNR is smaller than $\mu _{k}$ the throughput decrease is not so large.
Fig. 7 shows the average SINR of 4 users BD MIMO
downlink system from simulation and the lower bound of average SINR.
We also evaluate the BER performance of the proposed technique by using the computer simulation.The BER comparison between the conventional MLD and the proposed one without ordering, is shown in Fig. 8.$4\times4$ Proposed denotes that $4\times4$ MIMO matrix is separated into two $2\times2$-MIMO matrices by using the proposed technique, and then we decode with $2\times2$ MLD decoder. The $8\times8$ Proposed denotes that $8\times8$ MIMO matrix is separated into two $4\times4$-MIMO matrices by using the proposed technique.
According to this result we notice that the proposed $4\times4$ MIMO decode performance degrades around 3 dB from the conventional $2\times2$ MLD's one. In case of the $8\times8$ proposed has also similarly trend, its BER performance degrades around 3dB from the conventional $4\times4$ MLD's one.
Finally, we represent a $8x\times8$ MIMO BER comparison in Fig.
9. This simulation results show that $8\times8$ proposed technique obtains the best BER performance of other MIMO decoding, and it improves around 4 dB of BER performance at $BER = 10-4.$ If we can get the optimum channel ordering, a $8\times8$ proposed with ordering has improvement of around 9dB BER performance at $BER = 10-4.$
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\section{CONCLUSION}
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In this paper, we analysed the effects of the imperfect
CSI due to feedback delay on BD MIMO downlink system with channel prediction and have proposed a complexity reduction
technique for higher order MIMO system by using same technique
in the receiver.Although channel prediction can reduce the performance degradation due to feedback delay, the
unpredictable CSI error still remains, and leads to multiuser interference.But BD technique forces all inter-user interference to zero.The results of our analysis is that the interference is proportional to the MSE of predicted CSI, and we obtained
a lower bound of ergodic average SINR. Based on the result,
we also proposed a robust scheme for this kind of systems.By using the proposed technique, high
order MIMO system can significantly reduce the complexity
of MIMO decode.If we can find the optimum channel ordering, the
proposed technique becomes close high order MLD original
BER performance.Proposed technique allows a tradeoff between performance and complexity.
By simulated results, we observed that the lower bound of ergodic average SINR and the performance improvement of the robust scheme.
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