www.pudn.com > tftb2002toolbox.rar > correlmx.tex

\markright{correlmx} 
\hspace*{-1.6cm}{\Large \bf correlmx} 
 
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{\bf \large \fontfamily{cmss}\selectfont Purpose}\\ 
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computes an estimation of the autocorrelation matrix of a signal. 
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{\bf \large \fontfamily{cmss}\selectfont Synopsis}\\ 
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\begin{verbatim} 
Rx=correlmx(x,p,Rxtype) 
Rx=correlmx(x,p) 
\end{verbatim} 
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{\bf \large \fontfamily{cmss}\selectfont Description}\\ 
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{\ty correlmx} computes a correlation matrix of a signal.  
 
\begin{tabular*}{14cm}{p{1.5cm} p{8.5cm} c} 
Name & Description & Default value\\\hline 
{\ty x}      & analyzed signal &  \\  
{\ty p}      & highest autocorrelation lag. For a given $p$,  
               the size of the autocorrelation matrix will be  
               $(p+1)\times(p+1)$ & \\ 
{\ty Rxtype} & computation algorithm.  
               Possible values for  
               \texttt{Rxtype} are \texttt{'burg'},  
               \texttt{'hermitian'} and \texttt{'fbhermitian'} 
             & {\ty 'fbhermitian'}\\\hline  
{\ty Rx}     & computed correlation matrix, of size \hbox{$(p+1)\times(p+1)$} & \\\hline  
\end{tabular*} 
 
For a given signal $x[n]$, $n\,\in\,[0,N-1]$, this function 
computes \texttt{Rx} by different approaches~: 
 
If \texttt{Rxtype} equals \texttt{'burg'}, then \texttt{correlmx} 
first computes the autocorrelation function 
$$ 
R_x[\tau]= 
\left\{{ 
\begin{array}{lcl} 
{1\over N-\tau}\,\sum_{n=\tau}^{N-1}x[n]\,x[n-\tau]^* 
&\quad{\rm if}\quad& \tau\geq 0\\[0.2 cm] 
{1\over N+\tau}\,\sum_{n=0}^{N+\tau-1}x[n]\,x[n-\tau]^* 
&\quad{\rm if}\quad& \tau< 0 
\end{array} 
}\right. 
$$ 
for $\tau\,\in\,[-p,p]$, and then builds a toeplitz matrix 
\texttt{Rx} such that $(R_x)_{ij}=R_x[i-j]$. 
 
If \texttt{Rxtype} equals \texttt{'hermitian'} or  
\texttt{'fbhermitian'}, then \texttt{correlmx} computes 
$$ 
R_x^h={1\over N-p}\,\sum_{n=p}^{N-1}X[n]^*\,X[n]^T 
$$ 
with $X[n]^T=\left[{x[n]\ x[n-1]\ \ldots\ x[n-p]}\right]$. 
If \texttt{Rxtype} equals \texttt{'hermitian'}, then  
\texttt{Rxtype} returns $R_x^h$, and if \texttt{Rxtype} equals  
\texttt{'fbhermitian'}, then \texttt{Rxtype} returns  
$0.5\,(R_x^h+{R_x^h}^H)$. 
\end{minipage} 
 
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{\bf \large \fontfamily{cmss}\selectfont Example}\\ 
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\begin{verbatim} 
N=100; sig=real(fmconst(N,0.1))+0.4*randn(N,1);  
Rx=correlmx(sig,2,'burg'); [v,d] = eig(Rx) 
acos(-0.5*v(2,1)/v(1,1))/(2*pi) 
Rx=correlmx(sig,2,'hermitian'); [v,d] = eig(Rx) 
acos(-0.5*v(2,1)/v(1,1))/(2*pi) 
\end{verbatim} 
computes an estimation of the frequency of a sinusoid using the Pisarenko 
approach, with a correlation matrix computed by the Burg or the hermitian 
algorithm. 
 
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{\bf \large \fontfamily{cmss}\selectfont See Also}\\ 
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\begin{verbatim} 
parafrep, tfrparam 
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