% This is part of the TFTB Reference Manual.
% Copyright (C) 1996 CNRS (France) and Rice University (US).
% See the file refguide.tex for copying conditions.


\markright{ambifunb}
\section*{\hspace*{-1.6cm} ambifunb}

\vspace*{-.4cm}
\hspace*{-1.6cm}\rule[0in]{16.5cm}{.02cm}
\vspace*{.2cm}


{\bf \large \sf Purpose}\\
\hspace*{1.5cm}
\begin{minipage}[t]{13.5cm}
Narrow-band ambiguity function.
\end{minipage}
\vspace*{.5cm}


{\bf \large \sf Synopsis}\\
\hspace*{1.5cm}
\begin{minipage}[t]{13.5cm}
\begin{verbatim}
[naf,tau,xi] = ambifunb(x)
[naf,tau,xi] = ambifunb(x,tau)
[naf,tau,xi] = ambifunb(x,tau,N)
[naf,tau,xi] = ambifunb(x,tau,N,trace)
[naf,tau,xi] = ambifunb(...,'plot')
\end{verbatim}
\end{minipage}
\vspace*{.5cm}


{\bf \large \sf Description}\\
\hspace*{1.5cm}
\begin{minipage}[t]{13.5cm}
        {\ty ambifunb} computes the narrow-band ambiguity function of a
        signal, or the cross-ambiguity function between two signals. Its
        definition is given by
\[A_x(\xi,\tau)=\int_{-\infty}^{+\infty} x(s+\tau/2)\ x^*(s-\tau/2)\
e^{-j2\pi \xi s}\ ds.\] 

\hspace*{-.5cm}
\begin{tabular*}{14cm}{p{1.5cm} p{8.5cm} c}
Name & Description & Default value\\ 
\hline 
{\ty x} & signal if auto-AF, or {\ty [x1,x2]} if cross-AF ({\ty length(x)=Nx})&\\ 
{\ty tau} & vector of lag values &{\ty (-Nx/2:Nx/2)}\\ 
{\ty N} & number of frequency bins &{\ty Nx}\\ 
{\ty trace} & if non-zero, the progression of the algorithm is shown&{\ty 0}\\ 
\hline
{\ty naf} & doppler-lag representation, with the doppler bins stored in the rows
and the time-lags stored in the columns&\\ 
{\ty xi} & vector of doppler values\\\hline
\end{tabular*}
\vspace*{.5cm}

This representation is computed such as its 2D Fourier transform equals the
Wigner-Ville distribution.  When called with 'plot' as additional argument, {\ty
ambifunb} displays the squared modulus of the ambiguity function by means
of {\ty contour}.

The ambiguity function is a measure of the time-frequency correlation of a
signal $x$, i.e. the degree of similarity between $x$ and its translated
versions in the time-frequency plane.
\end{minipage}
\vspace*{1cm}

\newpage

{\bf \large \sf Examples}\\
\hspace*{1.5cm}
\begin{minipage}[t]{13.5cm}
Consider a BPSK signal (see {\ty anabpsk}) of 256 points, with a keying
period of 8 points, and analyze it with the narrow-band ambiguity
function\,:
\begin{verbatim}
         sig=anabpsk(256,8);
         ambifunb(sig,'plot');
\end{verbatim}
The resulting function presents a high thin peak at the origin of the
ambiguity plane, with small sidelobes around. This means that the
inter-correlation between this signal and a time/frequency-shifted version
of it is nearly zero (the ambiguity in the estimation of its arrival time
and mean-frequency is very small).\\

Here is an other example that checks the correspondance between the WVD and
the narrow-band ambiguity function by means of a 2D Fourier transform\,:
\begin{verbatim}
         N=128; sig=fmlin(N); amb=ambifunb(sig);
         amb=amb([N/2+1:N 1:N/2],:);
         ambi=tftb_ifft(amb).';
         tdr=zeros(N,N); 		// Time-delay representation
         tdr(1:N/2,:)=ambi(N/2:N-1,:);
         tdr(N:-1:N/2+2,:)=ambi(N/2-1:-1:1,:);
         wvd1=real(tftb_fft(tdr));

         wvd2=tfrwv(sig);
         diff=max(max(abs(wvd1-wvd2)))
         diff = 
                1.5632e-13
\end{verbatim}
\end{minipage}
\vspace*{.5cm}


{\bf \large \sf See Also}\\
\hspace*{1.5cm}
\begin{minipage}[t]{13.5cm}
\begin{verbatim}
ambifuwb.
\end{verbatim}
\end{minipage}

