Computationally efficient MIMO system identification using Signal Matched Synthesis Filter Bank
Binish~Fatimah and S.~D.~Joshi~
Department of Electrical Engineering
Indian Institute of Technology
New Delhi
India 110016
e-mail: binish.fatimah@ee.iitd.ac.in
sdjoshi@ee.iitd.ac.in% <-this % stops a space %\thanks{The authors are with the Department %of Electrical Engineering
Indian Institute of Technology
New Delhi 110016 India e-mail: (binish.fatimah@ee.iitd.ac.in
sdjoshi@ee.iitd.ac.in).
We propose a multi input multi output(MIMO) system identification framework by interpreting the MIMO system in terms of a multirate synthesis filter bank. The proposed methodology is discussed in two steps: in the first step the MIMO system is interpreted as a synthesis filter bank and the second step is to convert the MIMO system into a SISO system ``without any loss of information", which re-structures the system identification problem into a SISO form. The system identification problem, in its new form, is identical to the problem of obtaining the signal matched synthesis filter bank (SMSFB) as proposed in \cite{PartII. Since we have developed fast algorithms to obtain the filter bank coefficients in \cite{PartII, for ``the given data case" as well as ``the given statistics case", we can use these algorithm for the MIMO system identification as well. This framework can have an adaptive as well as block processing implementation. %The MIMO system is converted into a SISO system without any loss of information. %%In this paper, we present a multi input multi output (MIMO) system identification framework,which %The proposed algorithm can have an adaptive as well as block processing implementation. The concept of signal matched synthesis filter bank, as presented in our previous paper, is re-worked to obtain the proposed identification scheme. The algorithms, used here, involve only scalar computations, unlike the conventional MIMO system identification algorithms where one requires matrix computations. These order recursive algorithm can also be used to obtain approximate smaller order model for large order systems without using any model order reduction algorithm. The proposed identification framework can also be used for SISO LPTV system identification and also for a SIMO or MISO system. The efficacy of the proposed scheme is validated and its performance in the presence of measurement noise is illustrated using simulation results.
conf_MIMO.tex
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\begin{document}
\title{Computationally efficient MIMO system identification using Signal Matched Synthesis Filter Bank}
\author{Binish~Fatimah and
S.~D.~Joshi~\\
Department
of Electrical  Engineering, Indian Institute of Technology, New Delhi, India  110016 \\
e-mail: binish.fatimah@ee.iitd.ac.in, sdjoshi@ee.iitd.ac.in% <-this % stops a space

%\thanks{The authors are with the Department
%of Electrical  Engineering, Indian Institute of Technology, New Delhi 110016 India e-mail: (binish.fatimah@ee.iitd.ac.in, sdjoshi@ee.iitd.ac.in).}
}% <-this % stops a space

\markboth{}%
{}
\maketitle

\begin{abstract}
We propose a multi input multi output(MIMO) system identification framework by interpreting the MIMO system in terms of a multirate synthesis filter bank. The proposed methodology is discussed in two steps: in the first step the MIMO system is interpreted as a synthesis filter bank and the second step is to convert the MIMO system into a SISO system ``without any loss of information", which re-structures the system identification problem into a SISO form. The system identification problem, in its new form, is identical to the problem of obtaining the signal matched synthesis filter bank (SMSFB) as proposed in \cite{PartII}. Since we have developed fast algorithms to obtain the filter bank coefficients in \cite{PartII}, for ``the given data case" as well as ``the given statistics case", we can use these algorithm for the MIMO system identification as well. This framework can have an adaptive as well as block processing implementation. 
%The MIMO system is converted into a SISO system without any loss of information. 
%%In this paper, we present a multi input multi output (MIMO) system identification framework,which 
%The proposed algorithm can have an adaptive as well as block processing implementation. The concept of signal matched synthesis filter bank, as presented in our previous paper, is re-worked to obtain the  proposed identification scheme.
The algorithms, used here, involve only scalar computations, unlike the conventional MIMO system identification algorithms where one requires matrix computations. These order recursive algorithm can also be used to obtain approximate smaller order model for large order systems without using any model order reduction algorithm. The proposed identification framework can also be used for SISO LPTV system identification and also for a SIMO or MISO system. The efficacy of the proposed scheme is validated and its performance in the presence of measurement noise is illustrated using simulation results.


\end{abstract}
%\textbf{\emph{Index Terms}- Least squares, signal matched multirate filter bank, lattice algorithm.

\begin{IEEEkeywords} 
MIMO system identification, multirate filter bank, signal matched multirate synthesis filter bank, SIMO system identification, SISO LPTV system identification, Least squares, adaptive algorithm.
\end{IEEEkeywords}






\section{Introduction}
This paper addresses the problem of MIMO system identification, which is required in various applications like human control behavior, modeling of unmanned aerial vehicles, robotic assemblies, chemical plants, power systems, multiuser channel estimation in communication systems, stock market analysis etc.\cite{MIMO_commu.,MIMO_Power_SS,MIMO_OFDM,MIMO_aerial,MIMO_chemical,MIMO_human}. The objective of MIMO identification is to estimate the parameters of the system, which can either be described by a state space model \cite{MIMO_SS,MIMO_SS_1} or an FIR model \cite{MIMO_FIR}, given the input and output signals or only with the information of the output signals.

 In this paper, given the observed input and output signals, we estimate the FIR model structure of a MIMO system using a variate form of signal matched synthesis filter bank(SMSFB) as defined \cite{PartII}. An analogy is established between the MIMO system identification problem and the problem of estimating the SMSFB. Since we have developed fast algorithms to estimate the filter coefficients, in \cite{PartII}, we use the algorithm in this work for system identification. Most of the MIMO system identification methods, existing in the literature, suffer from two main drawbacks: 1) development of adaptive algorithms becomes cumbersome, and 2) parameter estimation involves matrix computations \cite{MIMO_1,MIMO_2,MIMO_3}. In contrast, the system identification algorithm used in this work involves only scalar computations and can be implemented in a block processing mode as well as in an adaptive manner. 
 
 In control theory we come across various large scale complex systems which are approximated using smaller order models for computational efficiency or for design simplification. One of the existing approaches decompose the large order system into a number of decoupled sub-systems or small scale systems. Each sub-system is individually modelled and then using a coordinator the subsystems are adjusted such that the required response is achieved. Since the order recursive algorithm used in this paper have a lattice-like implementation, they can be used to approximate large order systems using a smaller order model in a fast and computationally efficient manner. Also, the decentralized systems can be naturally modelled using the proposed scheme. 

Control theory community have used theories and results from signal processing literature for example ARMA models. It has also been discussed by authors like Ljung \cite{Ljung_1991} that a more extensive relationship between the two areas should be explored which will lead to promising results. This work is an example of a similar attempt. Here we use the results as well as algorithms existing in the multirate filter bank literature to propose computationally efficient system identification framework.
 
 This paper is organized as follows: in section II we present some preliminaries from the multirate filter bank theory and present a MIMO form of synthesis filter bank. This representation is, then, used to interpret a MIMO system in terms of a synthesis filter bank. In section III we convert a MIMO system into a SISO form and section IV discusses the MIMO system identification problem and re-structured the problem into a SISO form. The estimation problem, in this case, is identical to obtaining signal matched synthesis filter bank parameters, proposed in \cite{PartII} which can be exploited to obtain the required MIMO system. Since fast algorithms for SMSFB have already been developed in \cite{PartII,given_stat} for ``given data" as well as ``given statistics" cases, these algorithms can be used to compute the MIMO parameters in a computationally efficient manner. Thus, the MIMO parameters can be obtained if one realization of the process is available, also in real time and also for the cases where the statistics of the signal are known or can be estimated. The efficacy of the proposed algorithm is validated using simulation results in section V and conclusions are discussed in section VI.   

\subsection{Notations}
In this paper we denote random variables by italic lowercase letters, vectors of random variable by italic, boldfaced, lowercase letters and matrices of random variables with italic capital letters. The norm of a vector  $\textit{\textbf{x}}$ is denoted by  $\mid \mid \textit{\textbf{x}} \mid \mid$, which is the positive square-root of the inner-product of $\textit{\textbf{x}}$ with itself, where $\textit{\textbf{x}}$ belongs to a Hilbert space.

\section{MIMO system and multirate synthesis filter bank}
In this section we interpret a MIMO system in terms of a multirate synthesis filter bank. First we briefly discuss the theory of synthesis filter bank and some of its results.
 \subsection{Preliminaries}
In this section we briefly review some relevant results from the filter bank theory which are required for better understanding and appreciation of the results presented in this paper. An extensive treatment of the theory of multirate filter banks is available in \cite{Vaidyanathan}.
\vspace*{-1cm}
\begin{figure}[H]
%\hspace{1cm}
%\centering
\includegraphics[width=1\textwidth,height=0.5\textwidth]{synthesis_filter_bank_conf}
\vspace{-3.8cm}
\caption{M-channel multirate synthesis filter bank} \label{fig:2.multirate_filter_bank}
\end{figure}
%\vspace*{-1cm}
A set of digital filters represented by a multi input single output system, with each filter preceded by an upsampler, is called a synthesis filter bank, shown in Figure \ref{fig:2.multirate_filter_bank}. An M-fold upsampler increases the sampling rate by  a factor of M by appending M-1 zeros between every two samples of the input signal.

Let us consider the i-th filter, of the M-band synthesis filter bank, denoted as $F_{i}(z)$. Assuming the filter to be FIR and filter order to be multiple of M, without any loss of generality, we can write :
%\begin{small}
\begin{eqnarray} 
&&\rm {F}_{i}(z)= \sum_{n=0}^{N-1}{f_{i}(n)z^{-n}}=\begin{small}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM)z^{-Mp}}\right)
+z^{-1}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM+1)z^{-Mp}}\right)\end{small}
\nonumber\\ 
&&\hspace{1cm}+\cdots+\rm  z^{-(M-1)}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM+M-1)z^{-Mp}}\right)\nonumber\\
&&\hspace{1cm}\rm \equiv{ F_{i,M-1}(z^M)+z^{-1}F_{i,M-2}(z^M)}+
 \cdots + z^{-(M-1)}F_{i,0}(z^M),\hspace{0.5cm} 0 \leq i\leq M-1.\nonumber
%&&= \sum_{k=0}^{M-1}{z^{M-1-k}F_{ik}(z^M)}
\end{eqnarray}
%\end{small}

The M-components, written in the braces, are called the type-II polyphase components. The k-th component for the i-th filter can be written as:\\
%these M-components, written in braces, corresponding to the i-th filter are called Type-I polyphase components and denoted as follows: %\begin{eqnarray*}
%\rm \tilde{{H}}_{il}(z^M)=\tilde{h}_{i}(l)+z^{-M}\tilde{h}_{i}(M+l)+\cdots+\\\rm z^{-(N-1)M}\tilde{h}_{i}((N-1)M+l)\nonumber
%\end{eqnarray*}
%or
\begin{eqnarray}
\rm {F}_{i,k}(z^M) \triangleq \sum_{p=0}^{(N/M-1)}{{f}_{i}(pM+M-1-k)z^{-Mp}}, \hspace{0.5cm} 0 \leq i\leq M-1.
\end{eqnarray}


The synthesis filter bank can be represented using the polyphase components, as shown in Fig. \ref{fig:synthesis_poly_fig1}.
\begin{figure}[H]
\hspace{3.5cm}
%\centering
\includegraphics[width=0.6\textwidth,height=0.4\textwidth]{synth_poly_fig1_conf}
%\vspace{-0.5cm}
\caption{Synthesis Filters in terms of type-II polyphase components} \label{fig:synthesis_poly_fig1}
\end{figure}
%Therefore the synthesis filter bank given in Figure \ref{fig:2.multirate_filter_bank} can also be represented as Figure \ref{fig:4.poly}, where $\rm {F}_{i,k}(z^M)$ is the k-th polyphase component of i-th filter, as defined above. 

\vspace*{-4cm}
\begin{figure}[H]
%\hspace{1cm}
%\centering
\includegraphics[width=1\textwidth,height=0.6\textwidth]{noble_identities_conf}
\vspace{-5cm}
\caption{Noble Identity for Multirate systems} \label{fig:noble_identities}
\end{figure}
%\vspace*{-1cm}

The polyphase components of synthesis filters and upsamplers, given in Fig.{\ref{fig:synthesis_poly_fig1}}, can be re-arranged using the noble identity (given in Fig.\ref{fig:noble_identities}) to obtain a computationally efficient structure for M-channel synthesis filter bank, as shown in Fig.\ref{fig:4.poly}. 
%% Fig 2
%\vspace{-0.5cm}
\begin{figure}[H]
\hspace{5cm}
%\centering
%\begin{center}
\includegraphics[width=0.6\textwidth, height=0.4\textwidth]{synth_poly_fig2_conf}
%\vspace{-10pt}
\caption{Polyphase decomposition of Synthesis filter bank} \label{fig:4.poly}
%\end{center}
\end{figure}

%%%%Fig 2
%\vspace{1cm}
%\vspace{-0.8cm}
%\begin{figure*}
%\setcounter{tempfigcounter2}{2}

%\setcounter{figure}{\value{tempfigcounter2}}
%\hrulefill

%\end{figure*}
%\vspace{0.4cm}
% From the basic filter bank theory and the polyphase decomposition \cite{Vaidyanathan}, mentioned above and also discussed in section 2.2.3, the synthesis filter bank can be re-structured into a multi input multi output system, as shown in Figure \ref{fig:4.poly}. 
From Figure \ref{fig:4.poly} we can write the output $y(n)$ as follows:
\begin{align*}
y(n)=\left\{ \begin{array}{l l} y_{0}(n)\hspace{0.5cm}\mbox{ if }{n=iM} ,\mbox{where $i$ is an integer}\\
 y_{1}(n)\hspace{0.5cm}\mbox{ if }{n=iM-1},\\
 \vdots\\
 y_{M-1}(n)\hspace{0.25cm}\mbox{ if }{n=iM-M+1} \end{array}\right.
\end{align*}
From Figure \ref{fig:4.poly} the outputs, $y_{i}(n)$, can be written in the following form (using convolution):
%From the multirate filter bank theory \cite{Vaidyanathan}, using polyphase decomposition and noble identities, 
%\vspace*{-1.5cm}
%The synthesis filter bank, as shown in Figure (\ref{fig:msmfb}), reconstructs the input signal delayed by $n_0$. For the sake of simplicity of notations we denote $x(n-n_0)$ as $d(n)$ and $\hat{d}(n)$ as its estimated value. This delay ensures that the filter bank is causal.
%From Figure (\ref{fig:poly}), output of the synthesis filter bank can easily be written as:
%\begin{eqnarray}
%{{\textit d}}(n)=\rm \sum_{j=0}^{M-1}{\sum_{p=0}^{N/M-1}{w_{j}(p)\textit{f}_{j}(n-Mp)}}\label{eq:poly}
%\end{eqnarray}
%In order to convert the multi input single output system, as represented by the above equation, into a multi input multi output system, we can easily write:
\begin{align}
& {{ y}}_{i}(n)= \sum_{j=0}^{M-1}{\sum_{k=0}^{N-1}{x_{j}(n-k)\rm{f}_{i,j}(k)}}, \hspace{0.5cm} 0 \leq i\leq M-1,\label{eq:4.poly}\\
 & \hspace*{0.8cm} = \sum_{j=0}^{M-1}{\sum_{k=0}^{N-1}{x_{j}(n-k)\rm{f}_{j}(Mk+M-1-i)}}, \hspace{0.5cm} 0 \leq i\leq M-1. 
\end{align}

%In order to obtain multi input multi output form, for the system under consideration,
The above set of equations, for $0 \leq i \leq M-1$, can be written in a matrix form as follows:\\

%\newcounter{tempfigcounter2}
%\newcounter{tempequationcounter1}
%\begin{figure*}
%\setcounter{tempequationcounter1}{2}
%\vspace{-0.5cm}
\renewcommand{\arraystretch}{1.2}
\begin{footnotesize}
%\hrulefill
\begin{eqnarray} &\hspace*{-0.5cm}
\left[\begin{array}{c}
{y}_{0}(n)\\
{y}_{1}(n)\\
 \vdots \\
{y}_{M-1}(n)
\end{array}\right] =\left[\begin{array}{cccccc}
\rm f_{0}(M-1) & \rm f_{1}(M-1) & \ldots & \rm f_{M-1}(M-1)\\
\rm f_{0}(M-2) & \rm f_{1}(M-2) & \ldots & \rm f_{M-1}(M-2)\\
\vdots & \vdots & \ldots & \vdots\\
\rm f_{0}(0) & \rm f_{1}(0) & \ldots & \rm f_{M-1}(0)
\end{array}\right]
\left[\begin{array}{c}
\textit{x}_{0}(n)\\
\textit{x}_{1}(n)\\
\vdots \\
\textit{x}_{M-1}(n)
\end{array}\right]+...+
% +\left[\begin{array}{cccccc}
%f_{0}(M) & f_{1}(M) & \ldots & f_{M-1}(M)\\
%f_{0}(M+1) & f_{1}(M+1) & \ldots & f_{M-1}(M+1)\\
%\vdots & \vdots & \ldots & \vdots \\
%f_{0}(2M-1) & f_{1}(2M-1) & \ldots & f_{M-1}(2M-1)
%\end{array}\right]\nonumber \\
% \left[\begin{array}{c}
%\textit{w}_{0}(n-1)\\
%\textit{w}_{1}(n-1)\\
%\vdots \\
%\textit{w}_{M-1}(n-1)
%\end{array}\right]
\nonumber \\&\hspace*{0.5cm}
\left[\begin{array}{cccccc}
\rm f_{0}(N-M) & \rm  f_{1}(N-M) & \ldots & \rm f_{M-1}(N-M)\\
 \rm f_{0}(N-M-1) & \rm  f_{1}(N-M-1) & \ldots & \rm f_{M-1}(N-M-1)\\
\vdots & \vdots & \ldots & \vdots \\
 \rm f_{0}(N-1) & \rm  f_{1}(N-1) & \ldots &  \rm f_{M-1}(N-1)
\end{array}\right]
\left[\begin{array}{c}
\textit{x}_{0}(n-N+1)\\
\textit{x}_{1}(n-N+1)\\
\vdots \\
\textit{x}_{M-1}(n-N+1)
\end{array}\right].
\label{eq:4.2}
\end{eqnarray}
\end{footnotesize}\renewcommand{\arraystretch}{1}
%\setcounter{equation}{\value{tempequationcounter1}}
%\hrulefill
%\vspace*{4pt}
%\end{figure*}
Equation (\ref{eq:4.2}) is a multi-input multi-output (MIMO) form of the synthesis filter bank. It is pertinent to note that i-th channel output, i.e. ${y}_{i}(n)$, makes use of all the past inputs of all the input channels, i.e. $w_{i}(k)$ for $0 \leq i \leq M-1$ and  $n-N+1\leq k \leq n$. 
%The MISO equation can be easily converted to a single input single output form. This would be accomplished by serializing the inputs as follows:
%
%% The multi input multi output system represented by(\ref{eq:2}) can now be converted into the single input single output form, (Figure \ref{fig:poly_synth}), with the following steps:
%%\begin{enumerate}
%%\item Serializing the inputs:\\
%% A multivariate (vector) AR process was represented as a scalar periodic AR process in \cite{pagano}. Later, this representation was used in \cite{sakai} to present a Levinson-Durbin like algorithm for the multivariate AR processes. The recursions, obtained in the algorithm, lead to a circular lattice structure. In the same spirit,
%%Motivated by an earlier work \cite{pagano}, which is for a different problem but similar in spirit.
% We define
%%   a process $z$ such that instead of M-input signals we have one periodic signal with period M:
%%
%% From (\ref{eq:2}), we observe that the input is a set of M processes. Define
%  a scalar process, $z$, which is a scalarized form of the vector process 
%   $\textbf{\textit{w}}(n)=[ w_{0}(n)\hspace{0.25cm}
%    \hspace{0.25cm} w_{1}(n)$
%    $  \cdots \hspace{0.25cm} w_{M-1}(n)]$, as follows:
%\begin{equation}
%\textit{z}(Mn-i)=\textit{w}_i(n), \hspace{1cm} 0 \leq i\leq M-1.\label{eq:4.2(a)}
%\end{equation}
%
%The SISO form of the synthesis filter bank can now be written as: 
%\renewcommand{\arraystretch}{1.2}
%\begin{footnotesize}
%%\hrulefill
%\begin{eqnarray} &\hspace*{-0.5cm}
%\left[\begin{array}{c}
%{y}(Mn)\\
%{y}(Mn-1)\\
% \vdots \\
%{y}(Mn-M+1)
%\end{array}\right] =\left[\begin{array}{cccccc}
%f_{0}(M-1) & f_{1}(M-1) & \ldots & f_{M-1}(M-1)\\
%f_{0}(M-2) & f_{1}(M-2) & \ldots & f_{M-1}(M-2)\\
%\vdots & \vdots & \ldots & \vdots\\
%f_{0}(0) & f_{1}(0) & \ldots & f_{M-1}(0)
%\end{array}\right]
%\left[\begin{array}{c}
%\textit{z}(Mn)\\
%\textit{z}(Mn-1)\\
%\vdots \\
%\textit{z}(Mn-M+1)
%\end{array}\right]+...+
%% +\left[\begin{array}{cccccc}
%%f_{0}(M) & f_{1}(M) & \ldots & f_{M-1}(M)\\
%%f_{0}(M+1) & f_{1}(M+1) & \ldots & f_{M-1}(M+1)\\
%%\vdots & \vdots & \ldots & \vdots \\
%%f_{0}(2M-1) & f_{1}(2M-1) & \ldots & f_{M-1}(2M-1)
%%\end{array}\right]\nonumber \\
%% \left[\begin{array}{c}
%%\textit{w}_{0}(n-1)\\
%%\textit{w}_{1}(n-1)\\
%%\vdots \\
%%\textit{w}_{M-1}(n-1)
%%\end{array}\right]
%\nonumber \\&\hspace*{0.5cm}
%\left[\begin{array}{cccccc}
%f_{0}(N-M) & f_{1}(N-M) & \ldots & f_{M-1}(N-M)\\
%f_{0}(N-M-1) & f_{1}(N-M-1) & \ldots & f_{M-1}(N-M-1)\\
%\vdots & \vdots & \ldots & \vdots \\
%f_{0}(N-1) & f_{1}(N-1) & \ldots & f_{M-1}(N-1)
%\end{array}\right]
%\left[\begin{array}{c}
%\textit{z}(M(n-N+1))\\
%\textit{z}(M(n-N+1)-1)\\
%\vdots \\
%\textit{z}(M(n-N+1)-M+1)
%\end{array}\right].
%\label{eq:4.2(b)}
%\end{eqnarray}
%\end{footnotesize}\renewcommand{\arraystretch}{1}

 With these pre-requisites at our disposal we are now in a position to discuss the interpretation of a MIMO system in terms of multirate synthesis filter bank. 

\subsection{Interpretation of MIMO system as multirate synthesis filter bank}
In this sub-section we represent a MIMO system in terms of a synthesis filter bank. This interpretation is used in the next section to propose the system identification algorithm. 
%In this section we first convert a MIMO system, N-inputs and M-outputs, into a SISO system, which will, then, be used to re-define the system identification problem.   
%%%%%  Figure  1
\vspace*{-1cm}
\begin{figure}[H]
\hspace{4cm}
%\centering
%\begin{center}
\includegraphics[width=12cm, height=9cm]{conf_MIMO}
\vspace{-3cm}
\caption{N-input M-output system} \label{fig:6.MIMO}
%\end{center}
\end{figure}

A MIMO system with P inputs and M outputs, Fig. \ref{fig:6.MIMO}, can be represented using an FIR model given as follows:
\begin{align}
\left[\begin{array}{c}
Y_{0}(z)\\
Y_{1}(z)\\
\vdots\\
Y_{M-1}(z)
\end{array}\right]=
\left[\begin{array}{c}
 \rm H_{0}(z)\\
 \rm H_{1}(z)\\
\vdots\\
 \rm H_{M-1}(z)
\end{array}\right]
\left[\begin{array}{c}
X_{0}(z)\\
X_{1}(z)\\
\vdots\\
X_{P-1}(z)
\end{array}\right]
\end{align}
where, $Y_{i}(z)=\sum_{n=-\infty}^{\infty}{y_{i}(n)z^{-n}}$, $X_{i}(z)=\sum_{n=-\infty}^{\infty}{x_{i}(n)z^{-n}}$ and $ \rm H_{i}(z)=\sum_{n=0}^{N'-1}{h_{i}(n)z^{-n}}$(without any loss of generality we can assume N' to be a multiple of P). In time domain, the above equation can be written as: 
\begin{align}
&\left[\begin{array}{c}
 {y}_{0}(n)\\
{y}_{1}(n) \\
\vdots \\
{y}_{M-1}(n)
\end{array}
\right]=
\left[\begin{array}{cccc}
 \rm h_{0}(0) &  \rm h_{0}(1) & \cdots &  \rm h_{0}(P-1)\\
 \rm h_{1}(0) &  \rm h_{1}(1) & \cdots &  \rm h_{1}(P-1)\\
\vdots & \vdots & \cdots & \vdots \\
 \rm h_{M-1}(0) &  \rm h_{M-1}(1) & \cdots &  \rm h_{M-1}(P-1)
\end{array}
\right]
\left[\begin{array}{c}
x_{0}(n)\\
x_{1}(n)\\
\vdots \\
x_{P-1}(n)
\end{array}
\right]+\nonumber\\
&\cdots +
\left[\begin{array}{cccc}
 \rm h_{0}(N'-P) &  \rm h_{0}(N'-P-1) & \cdots &  \rm h_{0}(N'-1)\\
 \rm h_{1}(N'-P) & \rm  h_{1}(N'-P-1) & \cdots &  \rm h_{1}(N'-1)\\
\vdots & \vdots & \cdots & \vdots \\
 \rm h_{M-1}(N'-P) & \rm  h_{M-1}(N'-P-1) & \cdots &  \rm h_{M-1}(N'-1)
\end{array}
\right]
\left[\begin{array}{c}
x_{0}(n-N'+1)\\
x_{1}(n-N'+1)\\
\vdots \\
x_{P-1}(n-N'+1)
\end{array}
\right]
%\mathbf{y}(n)=H\mathbf{x}(n)
\label{eq:6.11}
\end{align}
%where
%\\
%$\mathbf{y}(n)=\left[\begin{array}{cccc}
% {y}_{0}(n)&
%{y}_{1}(n) &
%\cdots &
%{y}_{M-1}(n)
%\end{array}
%\right]^T$
%\\
%\\
%$H=\left[\begin{array}{cccc}
%h_{0}(0) & h_{0}(1) & \cdots & h_{0}(N-1)\\
%h_{1}(0) & h_{1}(1) & \cdots & h_{1}(N-1)\\
%\vdots & \vdots & \cdots & \vdots \\
%h_{M-1}(0) & h_{M-1}(1) & \cdots & h_{M-1}(N-1)
%\end{array}
%\right]$
%\\
%\\
%$\mathbf{x}(n)=\left[\begin{array}{cccc}
%x_{0}(n)&
%x_{1}(n)&
%\cdots &
%x_{N-1}(n)
%\end{array}
%\right]^T$
%\\
%\\
%\\

Comparing (\ref{eq:4.2}) and (\ref{eq:6.11}), 
%the SISO system, shown in Fig. \ref{fig:conf.poly_synth}, when N=M, is equivalent to a synthesis filter bank, Fig. \ref{fig:5.synthesis filter bank}.
 we observe that the MIMO system, can be interpreted using a synthesis filter bank where the synthesis filter coefficients and the MIMO system parameters are related as:
%\begin{align}
%\rm h_{j}(k-Q+1+i+j)\triangleq f_{i}(k), \hspace{1cm} 0 \leq i\leq Q-1,
%\end{align}
%for $j={k+1 \_mod \_ (Q)}$ and Q is the LCM(P,M).
\begin{align}
\rm H_{i}(z)=\sum_{j=0}^{Q-1}{\sum_{k=0}^{N'-1}{{f}_{j}(Qk+Q-1-i)}z^{j+Qk}}
\end{align}
where Q is the LCM(P,M) and N'=N.

Using this analogy between the multirate filter bank and MIMO system the algorithms, results and tools available for filter banks can be easily used for MIMO systems and vice-verse. As mentioned in the abstract the next step in the proposed framework is to convert the MIMO system into a SISO form which will be latter used to re-define the MIMO system identification problem as a SISO form.    

\section{SISO form of a MIMO system}
In this section we convert the MIMO system into a SISO system. In order to use the existing theories of SISO directly for MIMO systems researchers diagonalize the matrix $\rm H$, resulting into M-SISO systems. However, this operation leads to loss of information which increases the error between the estimated system and the true system and these techniques can only be used when H is a square matrix. However, the methodology proposed here converts the MIMO system into a SISO form ``without any loss of information". 
The conversion can be achieved using the following steps:
\\
\\
\textbf{Step 1: Serializing the input signals using a commutator switch.}\\
We define
%   a process $z$ such that instead of M-input signals we have one periodic signal with period M:
%
% From (\ref{eq:2}), we observe that the input is a set of M processes. Define
  a scalar process, $z$, which is a scalarized form of the vector process $\boldsymbol{x}(n)$, given as follows:
%   $\textbf{\textit{w}}(n)=\left[\begin{array}{cccc} w_{0}(n) & w_{1}(n) & \cdots w_{M-1}(n)
% 
% \end{array}\right]$, as follows: a signal $z(n)$ can be defined as follows:
 \begin{equation}
\textit{z}(Pn-i)=\textit{x}_i(n), \hspace{1cm} 0 \leq i\leq P-1.
\end{equation} 

Therefore, equation(\ref{eq:6.11}) can be written as:
%\begin{small}
\begin{align}
\boldsymbol{y}(n)= \sum_{i=0}^{N'/P-1}{\rm{ H(i)}\boldsymbol{z}(Pn-i)}
%&&\left[\begin{array}{c}
%\hat {y}_{0}(n)\\
%\hat {y}_{1}(n)\\
%\vdots\\
%\hat {y}_{M-1}(n)
%\end{array}
%\right]=\left[\begin{array}{cccc}
%h_{0}(0) & h_{0}(1) & \cdots & h_{0}(N-1)\\
%h_{1}(0) & h_{1}(1) & \cdots & h_{1}(N-1)\\
%\vdots & \vdots & \cdots & \vdots \\
%h_{M-1}(0) & h_{M-1}(1) & \cdots & h_{M-1}(N-1)
%\end{array}
%\right]\nonumber\\
%&&
 \label{eq:6.22}
\end{align}
%\end{small}
where\\
$\boldsymbol{z}(Pn-i)=\left[\begin{array}{cccc}
z(Pn-i)&
z(Pn-1-i)&
\cdots &
z(Pn-P+1-i)
\end{array}
\right]^T$,
\\
\\
$ \rm H(i)=\left[\begin{array}{cccc}
 \rm h_{0}(P*i) & \rm  h_{0}(P*i+1) & \cdots &  \rm h_{0}(P*i+P-1)\\
 \rm h_{1}(P*i) &  \rm h_{1}(P*i+1) & \cdots &  \rm h_{1}(P*i+P-1)\\
\vdots & \vdots & \cdots & \vdots \\
 \rm h_{M-1}(P*i) &  \rm h_{M-1}(P*i+1) & \cdots &  \rm h_{M-1}(P*i+P-1)
\end{array}
\right]$ and
\\
\\
$\boldsymbol{y}(n)=\left[\begin{array}{cccc}
y_{0}(n)&
y_{1}(n)&
\cdots &
y_{M-1}(n)
\end{array}
\right]^T$.
\\
\\
\textbf{Step 2: Serializing the outputs using a commutator switch.} \\
Using a commutator switch, the M-outputs of the single input multi output(SIMO) system, given in (\ref{eq:6.22}), are interleaved to give a single output, shown in Fig.\ref{fig:6.SISO}, given as follows:
\begin{align}
{\textit{w}}(Mn-i)={\textit{{y}}}_{i}(n), \hspace{1cm} 0 \leq i\leq M-1. \label{eq:6.4.3}
\end{align}

Therefore, we can write:
%\begin{small}
\begin{eqnarray}
\boldsymbol{w}(Mn)=\sum_{i=0}^{N'/P-1}{\rm{ H(i)}\boldsymbol{z}(Pn-i)}
%&&\left[\begin{array}{c}
%\hat {y}(Mn)\\
%\hat {y}(Mn-1)\\
%\vdots\\
%\hat {y}(Mn-M+1)
%\end{array}
%\right]=\left[\begin{array}{cccc}
%h_{0}(0) & h_{0}(1) & \cdots & h_{0}(N-1)\\
%h_{1}(0) & h_{1}(1) & \cdots & h_{1}(N-1)\\
%\vdots & \vdots & \cdots & \vdots \\
%h_{M-1}(0) & h_{M-1}(1) & \cdots & h_{M-1}(N-1)
%\end{array}
%\right]\nonumber\\
%&&
%\left[\begin{array}{c}
%z(Mn)\\
%z(Mn-1)\\
%\vdots\\
%z(Mn-N+1)
%\end{array}
%\right]
 \label{eq:6.22(a)}
\end{eqnarray}
%\end{small}
where,\\
$\boldsymbol{w}(Mn)=\left[\begin{array}{cccc}
{w}(Mn)&
{w}(Mn-1) &
\cdots &
{w}(Mn-M+1)
\end{array}
\right]^T$

%%%%  Figure  1
\vspace*{-2cm}
\begin{figure}[H]
\hspace{3cm}
%\centering
\includegraphics[width=10cm, height=9.5cm]{SISO}
\vspace{-100pt}
\caption{SISO form of the MIMO system} \label{fig:6.SISO}
\end{figure}
The MIMO system of Fig.\ref{fig:6.MIMO} can also be represented as Fig.\ref{fig:conf.poly_synth}.
%The two steps, as discussed above, can also be represented by Fig.\ref{fig:conf.poly_synth}.
\begin{figure}[H]
%\begin{center}
\vspace*{-1.5cm}
\hspace{3cm}
\includegraphics[width=0.6\textwidth, height=0.4\textwidth]{conf_poly_synth}
\vspace{-40pt}
\caption{SISO interpretation of the MIMO system} \label{fig:conf.poly_synth}
%\end{center}
\end{figure}

%\textbf{Remark:} A MIMO system, with M-inputs and M-ouputs, can be interpreted as a M-channel synthesis filter bank. This is discussed as follow:
%
%% As proposed in \cite{PartII}, the SISO system, shown in Fig. \ref{fig:conf.poly_synth}, when N=M, is equivalent to a synthesis filter bank, Fig. \ref{fig:5.synthesis filter bank}.
%\vspace{-1cm}
%%%%% Figure 3
%\begin{figure}[H]
%%\begin{center}
%%\centering
%\includegraphics[width=0.6\textwidth,height=0.5\textwidth]{conf_synth_fb}
%\vspace{-4cm}
%\caption{M-channel Synthesis Filter Bank} \label{fig:5.synthesis filter bank}
%%\end{center}
%\end{figure}
%
%Consider the i-th filter of the M-band synthesis filter bank, denoted as $F_{i}(z)$. Assuming the filter to be FIR and filter order to be M-1, we can write :
%%\begin{small}
%\begin{eqnarray} 
%\rm {F}_{i}(z)= \sum_{n=0}^{M-1}{f_{i}(n)z^{-n}}
%%=\begin{small}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM)z^{-Mp}}\right)
%%+z^{-1}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM+1)z^{-Mp}}\right)\end{small}
%%\nonumber\\ 
%%&&\hspace{1cm}+\cdots+\rm  z^{-(M-1)}\left(\sum_{p=0}^{(N/M-1)}{{f}_{i}(pM+M-1)z^{-Mp}}\right)\nonumber\\
%%&&\hspace{1cm}={ F_{i,M-1}(z^M)+z^{-1}F_{i,M-2}(z^M)}+
%% \cdots + z^{-(M-1)}F_{i,0}(z^M),\hspace{0.5cm} 0 \leq i\leq M-1.\nonumber
%%&&= \sum_{k=0}^{M-1}{z^{M-1-k}F_{ik}(z^M)}
%\end{eqnarray}
%%\end{small}
%
%From Figure \ref{fig:5.synthesis filter bank}, the outputs, $y(Mn-i)$, can be easily written in the following form:
%%From the multirate filter bank theory \cite{Vaidyanathan}, using polyphase decomposition and noble identities, 
%%\vspace*{-1.5cm}
%%The synthesis filter bank, as shown in Figure (\ref{fig:msmfb}), reconstructs the input signal delayed by $n_0$. For the sake of simplicity of notations we denote $x(n-n_0)$ as $d(n)$ and $\hat{d}(n)$ as its estimated value. This delay ensures that the filter bank is causal.
%%From Figure (\ref{fig:poly}), output of the synthesis filter bank can easily be written as:
%%\begin{eqnarray}
%%{{\textit d}}(n)=\rm \sum_{j=0}^{M-1}{\sum_{p=0}^{N/M-1}{w_{j}(p)\textit{f}_{j}(n-Mp)}}\label{eq:poly}
%%\end{eqnarray}
%%In order to convert the multi input single output system, as represented by the above equation, into a multi input multi output system, we can easily write:
%\begin{eqnarray}
%&{{\textit y}}(Mn-i)=\rm {\sum_{p=0}^{M-1}{z(Mn-p)}}\nonumber
%\\
%&\textit{f}_{p}(Mn-i-Mp), \hspace{0.5cm} 0 \leq i\leq M-1.\label{eq:conf.poly}
%\end{eqnarray}
%In order to obtain multi input multi output form, for the system under consideration, we write the above set of equations, for $0 \leq i \leq M-1$, in a matrix form, as follows:\\
%
%%\newcounter{tempfigcounter2}
%%\newcounter{tempequationcounter1}
%%\begin{figure*}
%%\setcounter{tempequationcounter1}{2}
%%\vspace{-0.5cm}
%\renewcommand{\arraystretch}{1}
%\begin{footnotesize}
%%\hrulefill
%\begin{eqnarray} &&\hspace*{-0.5cm}
% \left[\begin{array}{c}
%{y}_{0}(Mn)\\
%{y}_{1}(Mn-1)\\
% \vdots \\
%{y}_{M-1}(Mn-M+1)
%\end{array}\right] = 
%%\sum_{k=0}^{N/M-1}
%{\left[\begin{array}{c}
%\textit{z}(Mn)\\
%\textit{z}(Mn-1)\\
%\vdots \\
%\textit{z}(Mn-M+1)
%\end{array}\right]}\nonumber\\
%&& {\left[\begin{array}{cccccc}
%f_{0}(M-1) & f_{1}(M-1) & \ldots & f_{M-1}(M-1)\\
%f_{0}(M-2) & f_{1}(M-2) & \ldots & f_{M-1}(M-2)\\
%\vdots & \vdots & \ldots & \vdots\\
%f_{0}(0) & f_{1}(0) & \ldots & f_{M-1}(0)
%\end{array}\right]}.\nonumber\\
%\label{eq:conf.2}
%%+...+
%%% +\left[\begin{array}{cccccc}
%%%f_{0}(M) & f_{1}(M) & \ldots & f_{M-1}(M)\\
%%%f_{0}(M+1) & f_{1}(M+1) & \ldots & f_{M-1}(M+1)\\
%%%\vdots & \vdots & \ldots & \vdots \\
%%%f_{0}(2M-1) & f_{1}(2M-1) & \ldots & f_{M-1}(2M-1)
%%%\end{array}\right]\nonumber \\
%%% \left[\begin{array}{c}
%%%\textit{w}_{0}(n-1)\\
%%%\textit{w}_{1}(n-1)\\
%%%\vdots \\
%%%\textit{w}_{M-1}(n-1)
%%%\end{array}\right]
%%\nonumber \\&&
%%\left[\begin{array}{cccccc}
%%f_{0}(N-M) & f_{1}(N-M) & \ldots & f_{M-1}(N-M)\\
%%f_{0}(N-M-1) & f_{1}(N-M-1) & \ldots & f_{M-1}(N-M-1)\\
%%\vdots & \vdots & \ldots & \vdots \\
%%f_{0}(N-1) & f_{1}(N-1) & \ldots & f_{M-1}(N-1)
%%\end{array}\right]\nonumber\\
%%&& \left[\begin{array}{c}
%%\textit{w}_{0}(n-N+1)\\
%%\textit{w}_{1}(n-N+1)\\
%%\vdots \\
%%\textit{w}_{M-1}(n-N+1)
%%\end{array}\right],\\
%%&& 0 \leq i\leq M-1.\nonumber
%\end{eqnarray}
%\end{footnotesize}\renewcommand{\arraystretch}{1}
%The above equation can be written in a concise form as:
%\begin{small}
%\begin{align}
%\mathbf{y}(n)=F\mathbf{z}(n),\label{eq:conf.synth}
%\end{align}
%\end{small}
%where \\
%\vspace{0.5cm}
%\renewcommand{\arraystretch}{0.8}
%\begin{small}$F={\left[\begin{array}{cccccc}
%f_{0}(M-1) & f_{1}(M-1) & \ldots & f_{M-1}(M-1)\\
%f_{0}(M-2) & f_{1}(M-2) & \ldots & f_{M-1}(M-2)\\
%\vdots & \vdots & \ldots & \vdots\\
%f_{0}(0) & f_{1}(0) & \ldots & f_{M-1}(0)
%\end{array}\right]}$.\end{small}
%\renewcommand{\arraystretch}{1}
%\vspace{0.5cm}
%Comparing (\ref{eq:6.22}) and (\ref{eq:conf.synth}), 
%%the SISO system, shown in Fig. \ref{fig:conf.poly_synth}, when N=M, is equivalent to a synthesis filter bank, Fig. \ref{fig:5.synthesis filter bank}.
% we observe that the MIMO system, for N=M, can be interpreted using a synthesis filter bank where the synthesis filter coefficients and the MIMO system parameters are related as:
%\begin{align}
%\rm h_{p}(k-M+1+i+p)\triangleq f_{i}(k), \hspace{1cm} 0 \leq i\leq M-1,
%\end{align}
%for $p={k+1 \_mod \_ (M)}$, as also noted in \cite{PartII}.

Equation (\ref{eq:6.22(a)}) represents a single input single output system which is equivalent to the MIMO system represented by equation(\ref{eq:6.11}). In the next section we first present the MIMO system identification problem, then using the equivalent SISO form, given by (\ref{eq:6.22(a)}), re-frame the problem. 

\section{MIMO system identification}
System identification refers to the operation of mathematical modeling of dynamic systems from the measured input and output data, as shown in Figure \ref{fig:6.MIMO_sys}. 
\begin{figure}[H]
\begin{center}
\vspace{-20pt}
\hspace{-1cm}
\includegraphics[width=10cm, height=6cm]{MIMO_system}
\vspace{-40pt}
\caption{MIMO system identification block diagram} \label{fig:6.MIMO_sys}
\end{center}\end{figure}

\begin{eqnarray}
&&\boldsymbol{e}(n)= \boldsymbol{d}(n)-\boldsymbol{y}(n)\\
&&\hspace{1cm}=\boldsymbol{d}(n)-H\boldsymbol{x}(n)
%&&\left[\begin{array}{c}
%e_{0}(n)\\
%e_{1}(n)\\
%\vdots\\
%e_{M-1}(n)
%\end{array}
%\right]=\left[\begin{array}{c}
%y_{0}(n)\\
%y_{1}(n)\\
%\vdots\\
%y_{M-1}(n)
%\end{array}
%\right]\nonumber\\
%&&-\left[\begin{array}{cccc}
%h_{0}(0) & h_{0}(1) & \cdots & h_{0}(N-1)\\
%h_{1}(0) & h_{1}(1) & \cdots & h_{1}(N-1)\\
%\vdots & \vdots & \cdots & \vdots \\
%h_{M-1}(0) & h_{M-1}(1) & \cdots & h_{M-1}(N-1)
%\end{array}
%\right]\nonumber\\
%&&
%\left[\begin{array}{c}
%z(Mn)\\
%z(Mn-1)\\
%\vdots\\
%z(Mn-N+1)
%\end{array}
%\right]\nonumber
, \label{eq:6.33}
\end{eqnarray}
%\end{small}
 
 where the error vector is defined as:
 
 $\boldsymbol{e}(n)=\left[\begin{array}{cccc}
e_{0}(n)&
e_{1}(n)&
\cdots &
e_{M-1}(n)
\end{array}
\right]^T$ and \\
the measured output vector is given as:\\
 $\boldsymbol{y}(n)=\left[\begin{array}{cccc}
y_{0}(n)&
y_{1}(n)&
\cdots &
y_{M-1}(n)
\end{array}
\right]^T$.
%\vspace{1cm}

 The system identification problem considered in this work, for an N-input and M-output system (Figure  \ref{fig:6.MIMO_sys}), can be stated as: ``given the P-inputs of the system, $x_{i}(n)$ for $ 0 \leq i \leq P-1$, and the M-outputs, $d_{i}(n)$ for $ 0 \leq i \leq M-1$, obtain the system parameters, $h_{i}(n)$, such that distance between the response of the estimated system, ${y}_{i}(n)$ and the true system, $d_{i}(n)$, for $0 \leq i \leq M-1$, is minimized". For dynamic systems, this is an adaptive identification problem. We are considering here a noiseless condition, however, the system can also be analyzed in the presence of noise.
%\begin{small}

 Using the equivalent SISO form of the MIMO system, (\ref{eq:6.22(a)}), as obtained in the previous section, the above equation can be easily written as:

% Therefore, now we can state the objective as, ``given the signal $z(n)$, obtain the filter parameters, $h_{i}$'s, as defined in (\ref{eq:6.22(a)}), such that the square of the distance between the given output signal, $y(n)$'s and the estimated output signal, $\hat{y}(n)$ is minimized". The error in this case can, thus, be defined as:

%\begin{small}
\begin{eqnarray}
\boldsymbol{e}(n)=\boldsymbol{d}(n)-\rm{H}\boldsymbol{z}(Pn)
%&&\left[\begin{array}{c}
%e_{0}(n)\\
%e_{1}(n)\\
%\vdots\\
%e_{M-1}(n)
%\end{array}
%\right]=\left[\begin{array}{c}
%y_{0}(n)\\
%y_{1}(n)\\
%\vdots\\
%y_{M-1}(n)
%\end{array}
%\right]\nonumber\\
%&&-\left[\begin{array}{cccc}
%h_{0}(0) & h_{0}(1) & \cdots & h_{0}(N-1)\\
%h_{1}(0) & h_{1}(1) & \cdots & h_{1}(N-1)\\
%\vdots & \vdots & \cdots & \vdots \\
%h_{M-1}(0) & h_{M-1}(1) & \cdots & h_{M-1}(N-1)
%\end{array}
%\right]\nonumber\\
%&&
%\left[\begin{array}{c}
%z(Mn)\\
%z(Mn-1)\\
%\vdots\\
%z(Mn-N+1)
%\end{array}
%\right]\nonumber
 \label{eq:6.33}
\end{eqnarray}
%\end{small}
 
% where
% 
% $\mathbf{e}(n)=\left[\begin{array}{cccc}
%e_{0}(n)&
%e_{1}(n)&
%\cdots &
%e_{M-1}(n)
%\end{array}
%\right]^T$
%\\
%\\
%Comparing the above equation with equation(\ref{eq:4.8}), we observe that the parameter estimation of the SISO system, shown in Figure \ref{fig:6.SISO} , can be obtained using the proposed method for parameter estimation of signal matched synthesis filter bank in chapter 4.

Here, H can be obtained in the least squares sense, as given below:
\begin{align}
\rm{H}=\boldsymbol{d}(n)\boldsymbol{z}^{T}(Mn)\left[\boldsymbol{z}(Mn)\boldsymbol{z}^{T}(Mn)\right]^{-1}\label{eq:LS_sol}
\end{align}
where, $\boldsymbol{z}^{T}(Mn)$ denotes the transpose of vector $\boldsymbol{z}(Mn)$. Objective of the given system identification problem, therefore, is the solution of the above equation. However, the solution will involve matrix inversion and its complexity will increase as the number of inputs and outputs or the filter order increases. As mentioned in the signal processing literature, a recursive solution of equation (\ref{eq:LS_sol}) will not involve any matrix computations; thus yielding a computationally efficient and fast solution.

It can be easily observed that the MIMO system identification problem, as represented by (\ref{eq:6.33}), is structurally similar to the estimation problem proposed in \cite{PartII}, for the signal matched synthesis filter bank(SMSFB). The concept of SMSFB, as proposed in \cite{PartII}, can be easily re-worked to obtain the MIMO system parameters.

In \cite{PartII} we have also developed a fast, order and time recursive least squares algorithm to compute the optimum filter parameters for the ``given data case". Moreover, for application where the statistics of the signal are available or can be computed, an order recursive Levinson-type algorithm has been developed in \cite{given_stat}, to obtain the optimum SMSFB coefficients in the mean square sense. Due to the similarity between the proposed MIMO system and SMSFB, as discussed above, the fast algorithms proposed in \cite{PartII, given_stat} can be used to obtain the MIMO system parameters. The recursions, obtained in the algorithms, gives rise to lattice-like structures. Hence, we obtain a framework for MIMO system identification which involves only scalar computations and can be implemented in both block processing mode as well as adaptive mode. 
%The computation of $h_{i}$'s, i.e. parameter estimation involves matrix computation, to propose a computationally efficient algorithm required. Since, an order recursive algorithm for ``given statistics case" for the same, has been proposed in the previous chapter . For cases where no prior knowledge about the signal structure is available, a fast order and time recursive algorithm has also been proposed for the ``given data case". 
\\

\textbf{Remarks:}
\begin{enumerate}
\item In the proposed scheme we converted the given MIMO system into an equivalent SISO system and obtained the parameters using only scalar computations. It is easy to observe that the proposed framework can be used for a SIMO system as well as a MISO system identification problem.
\item The ``given data" as well as ``given statistics" algorithms developed in \cite{PartII} and [] are order recursive algorithms and can be implemented using lattice-like structures. Therefore large order systems can be estimated using smaller order models without using any existing model reduction algorithms.  
%\item As mentioned in \cite{PartII} the computations required to estimate an $M \times M$ synthesis filter bank are: additions and multiplications.
\item  Researchers have proposed that a SISO systems with linear periodically time varying property can be implemented with multiple filters and thus can be converted to MIMO systems \cite{MIMO_LPTV}. Thus, the LPTV SISO system parameters can be obtained using the MIMO system identification proposed above.
%\item In practical systems, all the input signals are not necessarily of the same bandwidth. In such cases, sampling every signal at the same rate does not seem to be a ``natural thing" to do. To avoid oversampling of some of the input signals, we propose to sample each input signal at a different rate depending on the individual bandwidths. 
\end{enumerate}
%1. The proposed MIMO system identification method can be used to obtain system parameters in a computationally efficient manner. Also, it gives an adaptive solution, unlike system identification methods involving matrix decomposition which cannot be used for adaptive processing [], which is often required.
 




% Consider an $M \times M$ system, where the sampling rates of the M inputs signals according to their bandwidths are, $L_{0}, L_{1}, \cdots, L_{M-1}$. As proposed above a multirate filter bank can be interpreted as the system identifier, in this case consider the filter bank shown in Figure \ref{fig:6.non-uniform} , where each input is sampled at a different rate.
%  From the multirate filter bank literature, an equivalent uniform filter bank can be obtained for the non-uniform filter bank under consideration. The parameters of the uniform system can, now, be obtained in a computationally efficient manner using the system identification methodology proposed above.
% Where $L=\left(L_{0}, L_{1}, \cdots, L_{M-1}\right)$, $n_{i}={L}/{L_{i}}$ and $L,L_{i},n_{i} \in I$.
%%%  Figure  1
%\vspace*{-2cm}
%\begin{figure}[h!]
%\hspace{-2cm}
%%\centering
%\includegraphics[width=16cm, height=14.5cm]{non-uniform_synth_fb_1}
%\vspace{-180pt}
%\caption{Non-uniform filter bank} \label{fig:6.non-uniform}
%\end{figure}

\section{Simulations}
In this section, the efficacy of the MIMO system identification framework, as discussed in the previous section, is validated using simulation results. The following experiment is designed for this purpose.

Experiment: For the purpose of simulation studies, we generate a signals using two AR processes, with poles at $0.5e^{-j\pi/{3}}$ and $0.9e^{-j\pi/{3}}$ respectively, and a $2 \times 2$ system, with filter order as 5. Now, given the input signals and the generated output signals the algorithm, proposed in this paper, is used to estimate the MIMO system parameters. And as mentioned earlier, the scheme proposed in this work can be analyzed in the presence of noise. For this purpose, we estimate the MIMO system parameters in the presence of measurement noise, which is white and independent to the input signals. The parameters computed for different SNR values are given in Table \ref{table:SNR_1}.
\renewcommand{\arraystretch}{1.6}
\begin{table}[H]
\begin{small}
\caption{Parameter values for different SNR} \label{table:SNR_1}
\begin{center}\begin{tabular}{|llllll|}
\hline
SNR & $h_{0}(0)$& $h_{0}(1)$ & $h_{0}(2)$ & $h_{0}(3)$  & $h_{0}(4)$ \tabularnewline
\hline
actual parameters & 1.0183 &  -1.9500  &  2.2364  & -1.7274  &  0.6192 \tabularnewline
1 &   1.0218 &  -1.9626  &  2.2733  & -1.7285   & 0.6547    \tabularnewline
0.1 & 0.9988  & -1.9238  &  2.2806  & -1.7740  &  0.6843  \tabularnewline
0.001 &  0.9903 &  -1.9183 &   2.2690 &  -1.7709  &  0.6918\tabularnewline
\hline
%\end{tabular}
%\end{center}
%\end{small}
%\end{table}
%\begin{table}[H]
%\begin{small}
%\caption{Parameter values for different SNR} \label{table:SNR_2}
%\begin{center}\begin{tabular}{|llllll|}
%\hline
SNR & $h_{1}(0)$& $h_{1}(1)$ & $h_{1}(2)$ & $h_{1}(3)$  & $h_{1}(4)$ \tabularnewline
\hline
actual parameters & 0.9872  & -2.3314  &  3.4243 &  -3.0962   & 1.7222\tabularnewline
1 & 0.9453  & -2.3106  &  3.4250   & -3.1156  &  1.7222\tabularnewline
0.1 & 0.9062  & -2.3276  &  3.4055 &  -3.1027  &  1.7338 \tabularnewline
0.001 & 1.0030 &  -2.3336  &  3.4322 &  -3.0888 &   1.7059\tabularnewline
\hline
\end{tabular}
\end{center}
\end{small}
\end{table}
%\begin{figure}
%\begin{center}
%\includegraphics[width=1\textwidth]{coeff2_1}
%%%%        \caption{}
%%%%        \label{rfidtest_xaxis}
%\end{center}
%\end{figure}

% The MIMO system parameters, used to generate the output signals, and the parameters estimated are plotted in Fig..
%% The generated signal is used as the input of signal matched whitening filter bank, to obtain M-white processes. With these M-white outputs as inputs, we use the signal matched synthesis filter bank, proposed in this chapter, to reconstruct the given signal. 
%%The synthesis filter, so obtained, provides the required modeling parameters and are given in the second column of table \ref{Table:4.sim}, these are recorded after the values converged. 
%
%
%%%%% Figure  6 
%\begin{figure}[H]
%%    \centering
%%    \begin{subfigure}[b]{0.3\textwidth}
%%        \includegraphics[width=\textwidth]{min_phase_whitening}
%%%        \caption{}
%%%        \label{rfidtest_xaxis}
%%    \end{subfigure}
%    \begin{subfigure}[b]{0.5\textwidth}
%        \includegraphics[width=8cm, height=2.5cm]{coeff2_1}
%         \caption{}
%%        \label{rfidtest_yaxis}
%    \end{subfigure}   
%    \begin{subfigure}[b]{0.5\textwidth}
%%       \vspace{1cm} 
%%       \hspace{-1.8cm}
%        \includegraphics[width=8cm, height=2.5cm]{Coeff_2}
%        \caption{}
%%        \label{rfidtest_zaxis}
%    \end{subfigure}
%    \caption{For input signal of Experiment 2(i), Parameter trajectory of filter coefficients (a)$\rm H_{0}$ and (b)$\rm H_{1}$.}
%    \label{fig:4.convergence_propoerty(i)}
%\end{figure}
%%
%%
%
%%
%%Figure \ref{fig:4.convergence_propoerty(i)} shows that the synthesis filters which are used to generate the desired signals and the filters obtained using the proposed algorithm matches closely as well as non-minimum phase.\\
%
%Experiment 2: As mentioned in the previous section, the signal matched synthesis filter bank is re-worked to obtain the MIMO system parameters. In this experiment using the fast algorithm given in \cite{PartII}  MIMO coefficients, of the system designed in experiment 1, are computed for the ``the given data case" and their convergence is plotted in Fig. and Fig. .
%
%%%% Figure  6 
%\begin{figure}[H]
%%    \centering
%%    \begin{subfigure}[b]{0.3\textwidth}
%%        \includegraphics[width=\textwidth]{min_phase_whitening}
%%%        \caption{}
%%%        \label{rfidtest_xaxis}
%%    \end{subfigure}
%%    \begin{subfigure}[b]{0.5\textwidth}
%        \includegraphics[width=8cm, height=2.5cm]{convergence_H0_1}
%%         \caption{}
%%%        \label{rfidtest_yaxis}
%%    \end{subfigure}   
%%    \begin{subfigure}[b]{0.5\textwidth}
%%%       \vspace{1cm} 
%%%       \hspace{-1.8cm}
%%        \includegraphics[width=8cm, height=2.5cm]{convergence_H1_1}
%%        \caption{}
%%%        \label{rfidtest_zaxis}
%%    \end{subfigure}
%    \caption{For input signal of Experiment 2(i), Parameter trajectory of filter coefficients (a)$\rm H_{0}$ and (b)$\rm H_{1}$.}
%    \label{fig:3.convergence_propoerty(i)}
%\end{figure}
%
%
%%Experiment 3: 
%
%% The system designed in experiment 1 is estimated here and the rms value of the error, between the given output signal and the reconstructed signal, is observed for different values of SNR.  
%
%%%%%% Figure  5 Coding gain 
%%\begin{figure}[H]
%%%\hspace*{2cm}
%%\begin{center}
%%        \includegraphics[width=8cm, height=4cm]{pole_position}
%%        \caption{Coding gain as $\rho$ goes towards unity}
%%        \label{fig:3.pole_position}
%%        \end{center}
%%    \end{figure}
%%
%
%%\textbf{Discussion on simulation results}\\
%%%\begin{enumerate}
%%
%%2. Table \ref{Table:4.sim} illustrates that the proposed modeling scheme can be used for filters with minimum as well as non-minimum phase.

\section{Conclusions} 
In this work, a given MIMO system has been converted to a SISO system ``without any loss of information". This methodology can be used in various applications involving MIMO systems, for example in the design of MIMO control systems.  We have also proposed that a MIMO system can be interpreted as the synthesis side of a filter bank, which helps to establish that the MIMO system identification problem is structurally identical to the parameter estimation problem of Signal Matched Synthesis Filter Bank \cite{PartII}. This analogy is exploited to obtain a computationally efficient solution for system identification.
%% A computationally efficient algorithm for MIMO system identification has been proposed, discussing only linear time invariant and linear periodically time invariant systems. However, systems with slowly time varying characteristics can also be estimated using the proposed framework by simple exponential windowing technique conventionally used in the development of fast least squares lattice algorithms \cite{Friedlander}.   
%In this paper we presented a MIMO system identification framework using a signal matched synthesis filter bank, as proposed in \cite{PartII}. Since fast algorithms have been developed for the case of signal matched synthesis filter bank, for ``the given data case" as well as ``the given statistics case", the present scheme for MIMO identification can have an adaptive as well as block processing implementation. The algorithm, developed here, involves only scalar computations.  Simulation results validate the theory presented and also illustrates performance of the proposed framework in the presence of measurement noise.
%\section{Reference}
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\end{document}
conf_MIMO.bbl
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\bibitem{PartII}
\BIBentryALTinterwordspacing
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\BIBentrySTDinterwordspacing

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