\subsection{Rectifier Behavior}\label{sec:rectifier-behavior}
A rectenna receives electromagnetic power via antenna and convert it to electric power with rectifier. Diverse configurations are available for energy harvesting, such as \textit{Schottky} \cite{Akkermans2005, Boaventura2013}, \textit{CMOS} \cite{Stoopman2014, Valenta2014}, \textit{series} \cite{Georgiadis2011, Collado2013}, \textit{shunt} \cite{McSpadden1998, Guo2012}. It is worth noting that those models favor different input power levels. As reported in \cite{Valenta2014, Costanzo2016}, low barrier Schottky diodes are commonly used for input power between \SI{1}{\uW} and \SI{1}{\mW}. Specifically, single diode is preferred for a power below \SI{500}{\uW} while multiple diodes are more suitable for power above \SI{500}{\uW} \cite{Clerckx2019}. Hybrid designs as \cite{Sun2013} may be employed to maintain a high efficiency for a wide power range.

Besides the rectenna model, the shape of the received signal also influences the RF-to-DC efficiency ${e_3}$. It was first demonstrated in \cite{Trotter2009} that multisine waveform i.e. \textit{Power-Optimized waveform} (POW) outperforms the single-tone waveform i.e. \textit{Continuous Wave} (CW) in both operation range and power efficiency. Therefore, multisine is commonly employed in WPT waveform design.

The expression of a multisine waveform with $N$ subcarriers writes as a summation of $N$ sine waves

\begin{equation}\label{eqn:multisine}
  {V_{{\text{multisine}}}}(t) = \sum\limits_{n = 0}^{N - 1} {\frac{1}{{\sqrt N }}} \sin \left( {2\pi \left( {{f_{\text{0}}} + n\Delta f} \right)t} \right)
\end{equation}

where ${{f_{\text{0}}}}$ is the minimum frequency and ${\Delta f}$ is the spacing. Fig. \ref{fig:waveform_comparison} \cite{Trotter2009} illustrates a three-subcarrier multisine and single sine in time and frequency domains. It can be observed that multisine provides a higher PAPR of ${\sqrt N }$ and occupies a bandwidth of $(N - 1) \Delta f$. Compared with the single sine, it has the same average power but equally distributed to the components. The thick lines indicate typical rectifier output voltage.

\begin{figure}[ht]
  \centering
  \subfigure[Frequency domain]{
    \includegraphics[width=0.48\textwidth]{waveform_frequency_domain}\label{fig:waveform_frequency_domain}}
  \subfigure[Time domain]{
    \includegraphics[width=0.48\textwidth]{waveform_time_domain}\label{fig:waveform_time_domain}}
  \caption{A typical 3-subcarrier multisine and single sine \cite{Trotter2009}}
  \label{fig:waveform_comparison}
\end{figure}

The advantage of multisine is that high PAPR can be exploited to increase the peak output voltage of the rectifier. With a proper signal and circuit design, the high output voltage may be preserved during the cycle if discharging is slow enough (as indicated by the thick blue line in Fig. \ref{fig:waveform_time_domain}). To enhance the harvested power, a large number of tones may be used to increase PAPR, and the multisine signal will appear as pulses with a period of $1/\Delta f$. Most of the signal power will be concentrated in those pulses to trigger the diode and charge the capacitor. However, more subbands can lead to smaller frequency gaps and longer charging cycle when the bandwidth is fixed.

It can be hard to derive an accurate expression of the RF-to-DC efficiency ${e_3}$ on the power and shape of the rectifier input signal, as practical energy harvesting circuits consists of various nonlinear components like diodes, capacitors, and inductors. It is also sensitive to parasitic sources, impedance matching, and harmonic generation \cite{Strassner2013, Valenta2014}. In this article, we employ the \textit{diode linear model} and \textit{diode nonlinear model} proposed in \cite{Clerckx2016} based on the diode current-voltage (I-V) characteristics to capture the fundamental pattern of the rectifier and investigate its impact on resource allocation and system design. A superposed waveform containing modulated information and multisine power components is optimized according to CSI on top of both models.

\subsection{Antenna Model}\label{sec:antenna-model}

As illustrated in Fig. \ref{fig:single_diode_rectifier} \cite{Clerckx2016}, the rectifier consists of a single diode as the source of nonlinearity and a low-pass filter to store energy.

\begin{figure}[ht]
  \centering
  \subfigure[A single diode rectifier]{
    \includegraphics[width=0.48\textwidth]{single_diode_rectifier}\label{fig:single_diode_rectifier}}
  \subfigure[Antenna equivalent circuit]{
    \includegraphics[width=0.48\textwidth]{antenna_equivalent_circuit}\label{fig:antenna_equivalent_circuit}}
  \caption{Rectenna architecture \cite{Clerckx2016}}
  \label{fig:rectenna_architecture}
\end{figure}

Fig. \ref{fig:antenna_equivalent_circuit} illustrates the antenna equivalent circuit. It includes a voltage source ${v_{\text{s}}}(t)$ connected to a series antenna impedance ${Z_{{\text{ant}}}} = {R_{{\text{ant}}}} + j{X_{{\text{ant}}}}$, followed by a combined impedance of the rectifier and the matching network ${Z_{{\text{in}}}} = {R_{{\text{in}}}} + j{X_{{\text{in}}}}$. Assuming lossless, the perfect matching condition is

\begin{equation}\label{eqn:perfect_match}
  {R_{{\text{in}}}} = {R_{{\text{ant}}}},{X_{{\text{in}}}} =  - {X_{{\text{ant}}}}
\end{equation}

When \eqref{eqn:perfect_match} is satisfied, the rectifier input voltage equals

\begin{equation}\label{eqn:rectifier_input_voltage}
  {v_{{\text{in}}}}(t) = {v_{\text{s}}}(t)/2 = y(t)\sqrt {{R_{{\text{in}}}}}
\end{equation}

where ${y(t)}$ is the received signal. Therefore, the input power to the rectifier is

\begin{equation}\label{eqn:rectifier_input_power}
  P_{{\text{rf}}}^r = \mathbb{E}\left[ {y{{(t)}^2}} \right] = \mathbb{E}\left[ {{v_{{\text{in}}}}{{(t)}^2}} \right]/{R_{{\text{in}}}}
\end{equation}

It is also assumed that the noise is too small to be harvested.

\subsection{Diode Characteristics}\label{sec:diode-characteristics}
Consider the single diode rectifier presented in Fig. \ref{fig:single_diode_rectifier} for simplicity. Without loss of generality, the diode models can be employed for other circuits as voltage doubler and bridge rectifiers \cite{Clerckx2017}.

Denoting ${v_{{\text{in}}}}(t)$ and ${v_{{\text{out}}}}(t)$ as diode input and output voltages, the voltage across the diode is ${v_{\text{d}}}(t) = {v_{{\text{in}}}}(t) - {v_{{\text{out}}}}(t)$. It determines the current flowing through the diode

\begin{equation}\label{eqn:diode_characteristics}
  {i_{\text{d}}}(t) = {i_{\text{s}}}\left( {{e^{\frac{{{v_{\text{d}}}(t)}}{{n{v_{\text{t}}}}}}} - 1} \right)
\end{equation}

where ${i_{\text{s}}}$ is the reverse saturation current, $n$ is the ideality factor, and ${{v_{\text{t}}}}$ is the thermal voltage. With Taylor expansion around a quiescent point $a = {v_{\text{d}}}(t)$, \eqref{eqn:diode_characteristics} rewrites as

\begin{equation}\label{eqn:diode_current_expansion}
  {i_{\text{d}}}(t) = \sum\limits_{i = 0}^\infty  {k_i^\prime } {\left( {{v_{\text{d}}}(t) - a} \right)^i}
\end{equation}

with

\begin{equation}\label{eqn:diode_k_prime}
  k_i^\prime  = \left\{ {
  \begin{array}{*{20}{c}}
    {{i_{\text{s}}}\left( {{e^{\frac{a}{{n{v_{\text{t}}}}}}} - 1} \right),}&{i = 0} \\
    {{i_{\text{s}}}\frac{{{e^{\frac{a}{{n{v_{\text{t}}}}}}}}}{{i!{{\left( {n{v_{\text{t}}}} \right)}^i}}},}&{i \in {\mathbb{N}^ + }}
  \end{array}} \right.
\end{equation}

$k_i^\prime $ depends on the diode parameters and is a constant when $a$ is fixed. Note that the Taylor series expression is a small-signal model that only fits the nonlinear operation region of the diode. Therefore, \eqref{eqn:diode_current_expansion} is no longer accurate for a large input voltage ${v_{{\text{in}}}}(t)$, where the diode behavior is dominated by the series resistor and the I-V relationship is linear \cite{Boaventura2013}.

Also, we assume an ideal rectifier with steady-state response to deliver a constant output voltage ${v_{{\text{out}}}}$, whose amplitude is a function of the peaks of the input voltage ${v_{{\text{in}}}}(t)$ \cite{Curty2005}. Based on those assumptions, a proper choice of voltage drop would be

\begin{equation}\label{eqn:diode_voltage_drop}
  a = \mathbb{E}\left[ {{v_{\text{d}}}(t)} \right] = \mathbb{E}\left[ {{v_{{\text{in}}}}(t) - {v_{{\text{out}}}}} \right] =  - {v_{{\text{out}}}}
\end{equation}

On top of \eqref{eqn:diode_voltage_drop} and \eqref{eqn:rectifier_input_voltage}, the diode current \eqref{eqn:diode_current_expansion} can be further expressed as

\begin{equation}\label{eqn:diode_current}
  {i_{\text{d}}}(t) = \sum\limits_{i = 0}^\infty  {k_i^\prime } {v_{{\text{in}}}}{(t)^i} = \sum\limits_{i = 0}^\infty  {k_i^\prime } R_{{\text{ant}}}^{i/2}y{(t)^i}
\end{equation}

It reveals an explicit relationship between the received waveform $y(t)$ and the diode current ${i_{\text{d}}}(t)$. Nevertheless, the waveform varies at every symbol period due to the randomness of the input distribution. Hence, the diode current ${i_{\text{d}}}(t)$ also fluctuates with time. By taking an expectation over the symbol distribution, the harvested DC current can be modeled as

\begin{equation}\label{eqn:diode_current_expectation}
  {i_{{\text{out}}}} = \mathbb{E}\left[ {{i_{\text{d}}}(t)} \right]
\end{equation}

and the available power is

\begin{equation}\label{eqn:harvested_power}
  P_{{\text{dc}}}^r = i_{{\text{out}}}^2{R_{\text{L}}} = \mathbb{E}{\left[ {{i_{\text{d}}}(t)} \right]^2}{R_{\text{L}}}
\end{equation}

To investigate the fundamental dependency of harvested power on waveform design, a practical strategy is to approximate \eqref{eqn:diode_current} with truncation to the ${n_o}$-th order

\begin{equation}\label{eqn:output_current_truncation}
  {i_{{\text{out}}}} \approx \sum\limits_{i = 0}^{{n_o}} {k_i^\prime } R_{{\text{ant}}}^{i/2}\mathbb{E}\left[ {y{{(t)}^i}} \right]
\end{equation}

The contribution of odd terms is indeed zero as $\mathbb{E}\left[ {y{{(t)}^i}} \right] = 0$ for odd $i$. Therefore, the approximated rectifier output DC current equals

\begin{equation}\label{eqn:output_current_function}
  {i_{{\text{out}}}} \approx \sum\limits_{i{\text{ even,i}} \geqslant 0}^{{n_o}} {k_i^\prime \left( {{i_{{\text{out}}}}} \right)} R_{{\text{ant}}}^{i/2}\mathbb{E}\left[ {y{{(t)}^i}} \right]
\end{equation}

Recall from \eqref{eqn:diode_k_prime} that the diode parameter $k_i^\prime $ is a function of $a =  - {v_{{\text{out}}}} =  - {i_{{\text{out}}}}{R_{\text{L}}}$. Therefore, ${i_{{\text{out}}}}$ occur in both sides of \eqref{eqn:output_current_function}. \cite{Clerckx2016} suggested an approach to decouple the correlation. Denote

\begin{equation}\label{eqn:diode_k_prime_prime}
  k_0^{\prime \prime } = {e^{\frac{a}{{n{v_{\text{t}}}}}}} = {e^{ - \frac{{{R_{\text{L}}}{i_{{\text{out}}}}}}{{n{v_{\text{t}}}}}}}
\end{equation}

such that $k_0^\prime  = {i_{\text{s}}}(k_0^{\prime \prime } - 1)$ and \eqref{eqn:output_current_function} rewrites as

\begin{equation}\label{eqn:output_current_rewritten}
  {e^{\frac{{{R_{\text{L}}}{i_{{\text{out}}}}}}{{n{v_{\text{t}}}}}}}\left( {{i_{{\text{out}}}} + {i_{\text{s}}}} \right) \approx {i_{\text{s}}} + \sum\limits_{i{\text{ even}},i \geqslant 2}^{{n_o}} {\frac{{k_i^\prime }}{{k_0^{\prime \prime }}}} R_{{\text{ant}}}^{i/2}\mathbb{E}\left[ {y{{(t)}^i}} \right]
\end{equation}

Note the r.h.s. of \eqref{eqn:output_current_rewritten} is independent of ${i_{{\text{out}}}}$. On the other hand, the l.h.s. is a monotonic increasing function of ${i_{{\text{out}}}}$. Therefore, we can further write

\begin{equation}\label{eqn:diode_k}
  {k_i} = \frac{{k_i^\prime }}{{k_0^{\prime \prime }}} = \frac{{{i_{\text{s}}}}}{{i!{{\left( {n{v_{\text{t}}}} \right)}^i}}}
\end{equation}

Therefore, maximizing ${i_{{\text{out}}}}$ is equivalent to maximizing the target function

\begin{equation}\label{eqn:target_function}
  {z_{DC}} = \sum\limits_{i{\text{ even,i}} \geqslant 2}^{{n_o}} {{k_i}} R_{{\text{ant}}}^{i/2}\mathbb{E}\left[ {y{{(t)}^i}} \right]
\end{equation}


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

Up to this point, we have examined the main solutions proposed to
the problem of representing a non-stationary signal in the
time-frequency plane. We now consider the problem of the
interpretation of the time-frequency image which describes the
evolution with time of the frequency content of the signal. Even if they
all tend to the same goal, each representation has to be interpreted
differently, according to its own properties. For example, some of
them present important interference terms, other are only positive,
other are perfectly localized on particular signals\ldots So the
extraction of information has to be done with care, from the knowledge
of these properties. We give in the following some general guide lines
to profit from a time-frequency image.

\section{Moments and marginals}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  The moments and marginals of some representations provide important
information about the signal, like its amplitude modulation or its
instantaneous frequency, for example with the aim of demodulating the
signal.

\subsection{Moments}
%'''''''''''''''''''
\index{moments}
  The first and second order moments, in time and in frequency, of a
time-frequency energy distribution tfr are defined as
\begin{eqnarray*}
f_m(t) &=& \frac{\int_{-\infty}^{+\infty} f\ \mbox{tfr}(t,f)\ df}
{\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ df}\
  \\
B^2(t)  &=& \frac{\int_{-\infty}^{+\infty} f^2\ \mbox{tfr}(t,f)\ df}
{\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ df}\  - f_m(t)^2;
\end{eqnarray*}
for the {\it time moments}, and as
%where $E=\int\int_{-\infty}^{+\infty}\mbox{tfr}(t,f)\ dt\ df$ is the energy
%of the signal,
\begin{eqnarray*}
t_m(f) &=& \frac{\int_{-\infty}^{+\infty} t\ \mbox{tfr}(t,f)\ dt}
{\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ dt}\\
T^2(f)  &=& \frac{\int_{-\infty}^{+\infty} t^2\ \mbox{tfr}(t,f)\ dt}
{\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ dt}\ - t_m(f)^2;
\end{eqnarray*}
for the {\it frequency moments}. They describe the averaged positions and
spreads in time and in frequency of the signal. For some particular
distributions, if the signal is considered in its analytic form, the first
order moment in time also corresponds to the instantaneous frequency, and
the first order moment in frequency to the group delay of the signal. These
moments can be obtained numerically thanks to the functions
\index{\ttfamily momttfr}{\ttfamily momttfr.m} and \index{\ttfamily
momftfr}{\ttfamily momftfr.m}.

\subsection{Marginals}
%'''''''''''''''''''''
\index{marginals} It can also be interesting to consider the {\it marginal
distributions} of a time-frequency representation. These marginals are
defined as\,:
\begin{eqnarray*}
m_f(t)=\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ df && \mbox{\it time
marginal}\\
m_t(f)=\int_{-\infty}^{+\infty} \mbox{tfr}(t,f)\ dt && \mbox{\it frequency
marginal}
\end{eqnarray*}
and express, by integrating the representation along one variable, the
repartition of the energy along the other variable. A natural
constraint for a time-frequency distribution is that the time marginal
corresponds to the instantaneous power of the signal, and that the
frequency marginal corresponds to the energy spectral density\,:
\begin{eqnarray*}
m_f(t)=|x(t)|^2\ \ \mbox{ and }\ \ m_t(f)=|X(f)|^2.
\end{eqnarray*}
The M-file \index{\ttfamily margtfr}{\ttfamily margtfr.m} computes the
marginal distributions of a given time-frequency representation.

\section{More on interferences\,: information on phase}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  The interference terms present in any quadratic time-frequency
representation, even if they disturb the readability of the
representation, contain some information about the analyzed
signal. The precise knowledge of their structure and construction rule
is useful to interpret the information that they contain.

  For instance, the interference terms contain some information about the
phase of a signal. Let us consider the pseudo WVD of the superposition of
two constant frequency modulations, with a phase shift between the two
sinusoids. If we compare the pseudo WVD for different phase shifts, we can
observe a time-sliding of the oscillating interferences. The M-file
\index{\ttfamily movpwdph}{\ttfamily movpwdph.m} illustrates this property
(see fig. \ref{Ex1fig1})\,:
\begin{verbatim}
     >> M=movpwdph(128); movie(M,10);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig1.eps}}
\caption{\label{Ex1fig1}Two simultaneous complex sinusoids analyzed by the
pseudo-WVD\,: the position of the interferences depends on the phase-shift
between the two components. These phase-shifts are respectively $\pi/4,\
3\pi/4,\ 5\pi/4$ and $7\pi/4$}
\end{figure}
Each snapshot corresponds to the pseudo WVD with a different phase
shift between the two components.

  A second example of signature of the phase is given by the influence of a
jump of phase in a signal analyzed by the (pseudo) Wigner-Ville
distribution\,: for instance, if we consider a constant frequency
modulation presenting a jump of phase in its middle (see
fig. \ref{Ex1fig2})\,:
\begin{verbatim}
     >> M=movpwjph(128,'C'); movie(M,10);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig2.eps}}
\caption{\label{Ex1fig2}Complex sinusoid presenting a jump of phase in its
middle, analyzed by the pseudo-WVD : the shape of the PWVD-pattern changes
with the importance of the jump. These jumps of phase are respectively $\pi/4,\
\pi/2,\ 3\pi/4$ and $pi$}
\end{figure}
the pseudo WVD presents a pattern around the jump position which is
all the more important since this jump of phase is close to $\pi$. This
characteristic can be used to detect a jump of phase in a signal.

\section{Renyi information}
%~~~~~~~~~~~~~~~~~~~~~~~~~~
\index{Renyi information}
  Another interesting information that one may need to know about an
observed non-stationary signal is the number of elementary signals
composing this observation. This also leads us to the following question\,:
how much separation between two elementary signals must one achieve in
order to be able to conclude that there are two signals present rather than
one ?

  A solution to this problem is given by applying an information measure to
a time-frequency distribution of the signal. Unfortunately, the well known
Shannon information, defined as
\begin{eqnarray*}
I_x = -\int_{-\infty}^{+\infty} f(x)\ \log_2 f(x)\ dx
\end{eqnarray*}
where $f(x)$ is the probability density function of $x$, can not be applied
to some time-frequency distributions due to their negative values. The
generalized form of information, which admits negative values in the
distribution, will then be used. This information, known as {\it Renyi
information}, is given by
\begin{eqnarray*}
R_x^{\alpha} =  \frac{1}{1-\alpha}\ log_2\left\{\int_{-\infty}^{+\infty}
f^{\alpha}(x)\ dx\right\}
\end{eqnarray*}
in the continuous case, where $\alpha$ is the order of the
information. First order Renyi information ($\alpha=1$) reduces to Shannon
information. Third order Renyi information, applied to a time-frequency
distribution $C_x(t,nu)$, is defined as
\begin{eqnarray*}
R_C^3 = -\frac{1}{2}\
log_2\left\{\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} C_x^3(t,\nu)\
dt\ d\nu\right\}.
\end{eqnarray*}
The result produced by this measure is expressed in {\it bits}\index{bits
of information}\,: if one elementary signal yields zero bit of information
($2^0$), then two well separated elementary signals will yield one bit of
information ($2^1$), four well separated elementary signals will yield two
bits of information ($2^2$), and so on. This can be observed by considering
the WVD of one, two and then four elementary atoms, and then by applying
the Renyi information on them. The file \index{\ttfamily renyi}{\ttfamily
renyi.m} computes this information measure\,:
\begin{verbatim}
     >> sig=atoms(128,[64,0.25,20,1]);
     >> [TFR,T,F]=tfrwv(sig);
     >> R1=renyi(TFR,T,F)       ------> -0.2075

     >> sig=atoms(128,[32,0.25,20,1;96,0.25,20,1]);
     >> [TFR,T,F]=tfrwv(sig);
     >> R2=renyi(TFR,T,F)       ------>  0.779

     >> sig=atoms(128,[32,0.15,20,1;96,0.15,20,1;...
                       32,0.35,20,1;96,0.35,20,1]);
     >> [TFR,T,F]=tfrwv(sig);
     >> R3=renyi(TFR,T,F)       ------>  1.8029
\end{verbatim}
We can see that if {\ttfamily R} is set to 0 for one elementary atom by
subtracting {\ttfamily R1}, we obtain a result close to 1 for two atoms
({\ttfamily R2-R1}=0.99) and close to 2 for four atoms ({\ttfamily
R3-R1}=2.01). If the components are less separated in the time-frequency
plane, the information measure will be affected by the overlapping of the
components or by the interference terms between them (see \cite{WIL91} for
more details on this analysis). In particular, it is possible to show that
the Renyi information measure provides a good indication of the time
separation at which the atoms are essentially resolved, with a better
precision than with the time-bandwidth product.

\section{Time-frequency analysis\,: help to decision}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

\subsection{General considerations}
%''''''''''''''''''''''''''''''''''
  The decision problem that one can have to solve when analyzing a
signal is threefold\,:
\begin{itemize}
\item detect if an observed signal contains a given information (i.e. say,
for a given false alarm probability, if {\it yes} or {\it no} the
information is present)\,;
\item estimate the parameters of a signal that we know to be present in
an observation\,;
\item classify a signal in one among different classes.
\end{itemize}
This problem, well known in theory in the general case, can be
reconsidered when dealing with non-stationary signals, emphasized by
the theory of time-frequency representations. Without going into
details, it has been shown that some of the known optimal strategies
of decision can be reformulated equivalently in the time-frequency
plane (like the matched-filter with the WVD for example). This result
is interesting for two reasons\,:
\begin{itemize}
\item on one hand, the time-frequency approach, compared to the
classical one (formulated in the time-domain in general), usually
provides a simpler interpretation of the decision test\,;
\item on the other hand, when the optimal solution for a given criterion
is not known in the decision theory, the time-frequency analysis can
be useful to formulate a sub-optimal solution based on the better
comprehension of the analyzed signal (for example, a time-frequency
detector can be easily modified to take into account variations of the
non-stationary signal to be detected, in order to improve the
robustness of the detector).
\end{itemize}
  The proposed solutions in the literature construct a decision test
(statistic)
\begin{itemize}
\item either as a general time-frequency correlation between a
time-frequency representation of the analyzed signal and some two
dimensional template, constructed using the {\it a priori} information
available on the signal,
\item or by applying a transform on the TF representation of the
analyzed signal, which brings to the fore some characteristic pattern
of the signal to be detected (or estimated or classified), and by
applying a test on this new space of decision. We consider in the
following an example of such approach, for the problem of the
detection and estimation of a linear frequency modulated signal
embedded in some white gaussian noise.
\end{itemize}

\subsection{An example\,: detection and estimation of linear FM signals}
%''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
  As we have seen in section \ref{WVD}, the WVD ideally concentrates the
linear chirp signals in the time-frequency plane. Thus, the problem of
detection and estimation of such a signal, which is not easily recognizable
in the time-domain, is reduced to the problem of detection and estimation
of a line in an image, which is a well known and easy-to-solve problem in
pattern recognition. This can be done by using the Hough transform,
dedicated to the detection of lines (\cite{BAR95}).

\subsubsection{The Hough transform for lines}
Consider the polar parameterization of a line
\[x\ \cos\theta+y\ \sin\theta=\rho\]
(this parameterization is much more adapted to this problem than the
Cartesian one). For each point $(x,y)$ of an image $I$, the Hough transform
associates a sinusoid in the plane $(\rho,\theta)$, whose points have an
amplitude equal to the intensity of the pixel $(x,y)$. So to all the points
in $I$, the Hough transform associates a pencil of sinusoids which
intersect themselves in the plane $(\rho,\theta)$. In other words, the HT
performs integrations along lines on the image $I$, and the value of each
integral is affected to the point $(\rho,\theta)$ corresponding to the
parameters of this line. Therefore, if on the image $I$ some pixels with
high intensities are concentrated along a straight line, we will observe in
the domain $(\rho,\theta)$ a peak whose coordinates are directly related to
the parameters of the lines.

  This method can be easily applied to other parametric curves, like
hyperbola for example. This transform is computed in the file
\index{\ttfamily htl}{\ttfamily htl.m}.

\subsubsection{The Wigner-Hough transform}
\index{Wigner-Hough transform}
When applying the Hough transform to the Wigner-Ville distribution of the
signal
\begin{eqnarray*}
x(t)=e^{j2\pi(\nu_0 t+\beta/2 t^2)} + n(t)
\end{eqnarray*}
observed during an observation time $T$ ($n(t)$ is a noise assumed white
and gaussian), we obtain a new transform called the {\it Wigner-Hough
transform} (WHT), whose expression is
\begin{eqnarray}
\label{WHT}
WH_x(\nu_0,\beta) &=& \int_T W_x(t,\nu_0+\beta t)\ dt\\
 &=& \int_{-\infty}^{+\infty}\int_T x(t+\tau/2)\ x^*(t-\tau/2)\
	 e^{-j2\pi(\nu_0+\beta t)\tau}\ dt\ d\tau\nonumber
\end{eqnarray}
  The comparison of the WHT to a threshold is the proposed detection test,
and the estimates of the unknown parameters $\nu_0$ and $\beta$ are given
by the coordinates of the detected peak in the space of the parameters
$(\nu_0,\beta)$. Thanks to the unitarity property of the WVD (Moyal's
formula), it is possible to show that this detection test is {\it
asymptotically} the {\it optimal detector} (i.e. optimal when $T$ tends to
infinity). Besides, the {\it estimators} are {\it asymptotically efficient}
(i.e. they asymptotically reach the Cramer-Rao lower bounds).  Compared to
the classical decision test usually used in this case, the generalized
likelihood ratio test (GLRT), this method presents the following advantages
in the case of multicomponent signals\,:
\begin{itemize}
\item it is free from the estimation of the initial phase and amplitude
of each component, which usually do not bring any information, and
\item its complexity do not increase with the number of components $N_c$,
unlike the GLRT whose complexity increases linearly with $N_c$.
\end{itemize}

  Here is an illustration of this decision test\,: first, we consider
a linear chirp signal embedded in a white gaussian noise, with a 1\,dB
signal-to-noise ratio\,:
\begin{verbatim}
     >> N=64; sig=sigmerge(fmlin(N,0,0.3),noisecg(N),1);
\end{verbatim}
Now, if we analyze it with the WVD followed by the Hough transform (see
fig. \ref{Ex1fig3} and \ref{Ex1fig4}),
\begin{verbatim}
     >> tfr=tfrwv(sig); contour(tfr,5); grid
     >> htl(tfr,N,N,1);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig3.eps}}
\caption{\label{Ex1fig3}WVD of a noisy chirp signal (SNR=1 dB) : while the
chirp is hardly readable in the time-representation, the line still clearly
appear in the WVD}
\end{figure}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=8cm
\centerline{\epsfbox{figure/ex1fig4.eps}}
\caption{\label{Ex1fig4}Wigner-Hough transform of the previous noisy
chirp\,: the peak corresponds to the chirp signal (and the side-lobes to
the noise), and its coordinates give estimators of the chirp
parameters. The detection test consists in comparing this peak to a
threshold (threshold fixed by the chosen criterion)}
\end{figure}
we obtain, in the parameters' space $(\rho,\theta)$, a peak representing
the chirp signal, significantly more energetic than the other peaks
corresponding to the noise. The decision test is then very simple\,: it
consists in applying a threshold on this representation, positioned
according to a detection criterion\,; if the peak is higher than the
threshold, then the chirp is said to be present, and the coordinates of
that peak $(\hat{\rho},\hat{\theta})$ provide estimates of the chirp
parameters (the change from $(\hat{\rho},\hat{\theta})$ to
$(\hat{\nu_0},\hat{\beta})$ corresponds to the change from polar to
Cartesian coordinates).

  In the case of a multi-component signal, the problem of interference
terms appear. However, due to the oscillating structure of these terms, the
integration (\ref{WHT}) operated by the Hough transform on the WVD will
attenuate them. This can be observed on the following example\,: we
superpose two chirp signals with different initial frequencies and sweep
rates (see fig. \ref{Ex1fig5} and \ref{Ex1fig6})\,:
\begin{verbatim}
     >> sig=sigmerge(fmlin(N,0,0.4),fmlin(N,0.3,0.5),1);
     >> tfr=tfrwv(sig); contour(tfr,5); grid
     >> htl(tfr,N,N,1);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig5.eps}}
\caption{\label{Ex1fig5}WVD of two simultaneous chirp signals :
interference terms appear between the two components}
\end{figure}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=8cm
\centerline{\epsfbox{figure/ex1fig6.eps}}
\caption{\label{Ex1fig6}Wigner-Hough transform of the two-component chirp
signal : two main peaks are present, characterizing the two chirp
components, while the cross terms present in the WVD only introduce small
side-lobes in the Wigner-Hough transform}
\end{figure}
We can see that the components are well separated in the parameter
space, in spite of the use of a nonlinearity in the WHT. Again, the
coordinates of the two peaks provide estimates of the different
parameters.

\section{Analysis of local singularities}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\index{singularity}\index{Holder exponent}\index{regularity} If the
time-frequency representations are useful to bring to the fore the
progression with time of the frequency of a signal, the time-scale
representations are more adapted to the analysis of irregular structures
and singularities, or of signals presenting self-similarities (such as
fractional Brownian motion, \cite{GON92}). We give in the following such an
example with the analysis of {\it local singularities}, thanks to the
scalogram and the Unterberger distribution.

  The local regularity of a signal can be characterized by its {\it Holder}
(or {\it Lipschitz} or {\it scaling}) {\it exponent}\,: for a signal $x(t)$
which is {\it uniformly Holder} $H$, there exists a constant $C$ such that
\begin{eqnarray*}
|x(s)-x(t)|\ \ \leq\ \ C\ |s-t|^H,\ \ 0<H<1.
\end{eqnarray*}
$H$ then represents the {\it exponent of regularity}\index{exponent of
regularity} of the signal. If we consider the wavelet transform
$T_x(t,a;\Psi)$ of this signal, with an analyzing wavelet $\Psi$ such that
$t\ \Psi(t)$ is absolutely integrable, then one can show that
\begin{eqnarray*}
|T_x(t,a;\Psi)|&\leq& C\ |a|^{H+1/2}\ \int_{-\infty}^{+\infty} |t|^H\
|\Psi(t)|\ dt\\
 &=& O(|a|^{H+1/2})\ \ \forall\ t,
\end{eqnarray*}
or, in terms of scalogram and behavior when $a$ tends to 0,
\begin{eqnarray*}
E\left[|T_x(t,a;\Psi)|^2\right] \ \sim\ |a|^{2H+1},\ \ a\rightarrow 0.
\end{eqnarray*}
where $E[.]$ refers to the expectation. This means that the regularity
of the signal can be recovered from the behavior of its scalogram at
small scales, and it is possible to show that the reciprocal is true.

  Since they are time-dependent in nature, the wavelet-based
techniques also allow an estimation of the local regularity of a
signal. In some sense, time-scale methods offer in this respect a
framework similar to the one provided by time-frequency analysis for
tracking the time evolution of spectral features. Indeed, if we now
have, at a given time $t_0$,
\begin{eqnarray}
\label{regularity}
|x(t_0+\tau)-x(t_0)|\ \leq\ C\ |\tau|^{H(t_0)},\ \ 0<H(t_0)<1,
\end{eqnarray}
then we can establish the inequality
\begin{eqnarray*}
|T_x(t,a;\Psi)|&\leq& C\ |a|^{H(t_0)+1/2}\ \int_{-\infty}^{+\infty}
|t|^{H(t_0)}\ |\Psi(t)|\ dt \\
 &&+ C\ |t-t_0|^{H(t_0)}\ \int_{-\infty}^{+\infty} |\Psi(t)|\ dt\\
 &=& O(|a|^{H(t_0)+1/2} + |t-t_0|^{H(t_0)}).
\end{eqnarray*}
We then obtain an image of the signal's regularity at the small scales
of its wavelet transform (or scalogram), but accompanied with a time
localization. The reciprocal is also true, which means that an
appropriate decrease of the wavelet (scalogram) coefficients in a
cone-shaped region of the time-frequency plane allows one to estimate
the local regularity of a signal.

  If we further impose to condition (\ref{regularity}) that the signal
presents an asymptotic spectral decrease,
\begin{eqnarray*}
X(\nu) \ \sim\ |\nu|^{-(1+2H(t_0))}\ e^{j2\pi \nu t_0}\ \ \mbox{ for }\ \
|\nu|\rightarrow\infty,
\end{eqnarray*}
then we have the following approximation for the active Unterberger
distribution\,:
\begin{eqnarray*}
U_x(t,a) \ \sim\ |a|^{2(1+H(t_0))}\ \delta(t-t_0),\ \ a\rightarrow 0.
\end{eqnarray*}
Thus, the Unterberger distribution follows a law along scales which
gives access to the strength of the singularity ($H$), and along time to
the localization of this singularity.

  The file \index{\ttfamily holder}{\ttfamily holder.m} estimates the
Holder exponent of any signal from an affine time-frequency representation
of it.\\

o {\it Example}\\ For instance, we consider a 64-points Lipschitz
  singularity (see \index{\ttfamily anasing}{\ttfamily anasing.m}) of
  strength $H=0$, centered at $t_0=32$,
\begin{verbatim}
     >> sig=anasing(64);
\end{verbatim}
and we analyze it with the scalogram (Morlet wavelet with half-length = 4, see
fig. \ref{Ex1fig7}),
\begin{verbatim}
     >> [tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig7.eps}}
\caption{\label{Ex1fig7}Scalogram of a Lipschitz singularity at time
$t=32$, of strength $H=0$}
\end{figure}
The time-localization of the singularity can be clearly estimated from
the scalogram distribution at small scales :
\begin{verbatim}
     >> H=holber(tfr,f,1,256,32)    ------>  H=-0.0381
\end{verbatim}

If we now consider a singularity of strength H=-0.5 (see
fig. \ref{Ex1fig8}),
\begin{verbatim}
     >> sig=anasing(64,32,-0.5);
     >> [tfr,t,f]=tfrscalo(sig,1:64,4,0.01,0.5,256,1);
\end{verbatim}
\begin{figure}[htb]
\epsfxsize=10cm
\epsfysize=10cm
\centerline{\epsfbox{figure/ex1fig8.eps}}
\caption{\label{Ex1fig8}Scalogram of a Lipschitz singularity at time
$t=32$, of strength $H=-0.5$}
\end{figure}
we notice the different behavior of the scalogram along scales, whose
decrease is characteristic of the strength $H$. The estimation of the
Holder exponent at $t=32$ gives :
\begin{verbatim}
     >> H=holber(tfr,f,1,256,32)    ------>  H=-0.5107
\end{verbatim}
which is close to 0.5.

The same conclusions can be observed from the active Unterberger
distribution.


\section{Introduction}
Visual tasks, such as object classification and detection, have been successfully approached through the supervised learning paradigm. However, since manually labelled data is costly and not scalable, unsupervised learning is gaining momentum.\newline
Recently a new unsupervised learning paradigm is raising: \emph{self-supervised} learning. The idea is to exploit different labeling that is freely available besides or within visual data, and to use them as intrinsic reward signals to learn general-purpose features.
For example, \cite{context_prediction} uses the relative spatial co-location of patches in images as a label. In \cite{unsupervised_models_recognition} they use object correspondence obtained through tracking in videos, and \cite{learning_by_moving} uses ego-motion information obtained by a mobile agent such as the Google car \cite{landmark_identification}. The features obtained with these approaches have been successfully transferred to classification and detections tasks with encouraging performances when compared to the supervised task.\newline
A fundamental difference between \cite{context_prediction} and \cite{unsupervised_models_recognition} is that the former method uses single images, while the latter use multiple images related through a temporal or viewpoint transformation. While it is true that biological agents typically make use of multiple images and other information, it is also true that single snapshot may carry more information than the one that has been extracted so far. In \cite{Noroozi_2016} Noorozi and Favaro worked on a novel supervised task the \emph{Jigsaw puzzle reassembly} problem, which builds features that yield high performance when transferred to detection and classification task. The experimental evaluations show them that the learned features capture semantically relevant content, and their method outperforms state of the art methods in several transfer learning benchmarks.\newline
In this report we would like to implement the method found in \cite{Noroozi_2016} and to apply the fine-tuning task to solve the classification task of Food Images \cite{food_images} competition yielded by Kaggle \cite{kaggle}.