diff --git a/exo2/.SRCINFO b/exo2/.SRCINFO
new file mode 100644
index 0000000..e69de29
diff --git a/exo2/FArcEllipse.m b/exo2/FArcEllipse.m
new file mode 100644
index 0000000..fc3781b
--- /dev/null
+++ b/exo2/FArcEllipse.m
@@ -0,0 +1,34 @@
+classdef FArcEllipse < FEllipse
+ properties
+ angles
+ end
+ methods
+ function obj=FArcEllipse(a,b,angles,C)
+ obj = obj@FEllipse(a,b,C);
+
+ obj.angles = zeros(1,2);
+
+ % shift form -pi to pi
+ obj.angles(1) = rem(angles(1), 2*pi);
+ obj.angles(2) = rem(angles(2), 2*pi);
+
+ while obj.angles(2) < obj.angles(1)
+ obj.angles(2) = obj.angles(2) + 2 * pi;
+ end
+
+ % adjust boundaries
+ if obj.angles(2) < obj.angles(1)
+ obj.angles(2) = obj.angles(2) + 2 * pi;
+ end
+ end
+
+ function flag=inside(obj,x,y)
+ flag = inside@FEllipse(obj,x,y);
+ ang = atan2((y - obj.Cy) / obj.b, (x - obj.Cx) / obj.a);
+ while ang < obj.angles(1)
+ ang = ang + 2 * pi;
+ end
+ flag = flag & (ang < obj.angles(2));
+ end
+ end
+end
diff --git a/exo2/FBandPassFilter.m b/exo2/FBandPassFilter.m
new file mode 100644
index 0000000..7372927
--- /dev/null
+++ b/exo2/FBandPassFilter.m
@@ -0,0 +1,15 @@
+classdef FBandPassFilter < FFilter
+ methods
+ % band = [lower, upper] normalized as an image radius
+ % angles = [lower, upper] angular,region to check
+ function obj=FBandPassFilter(band, angles)
+ angles = [angles(2), angles(1) + pi; angles(2) + pi, angles(1)];
+ doms = {{FEllipse(band(2), band(2), [0,0]), false}, ...
+ {FEllipse(band(1), band(1), [0,0]), true}, ...
+ {FArcEllipse(1, 1, angles(1,:), [0,0]), true}, ...
+ {FArcEllipse(1, 1, angles(2,:), [0,0]), true}};
+
+ obj = obj@FFilter(doms);
+ end
+ end
+end
diff --git a/exo2/FDiscMap.m b/exo2/FDiscMap.m
new file mode 100644
index 0000000..fb28686
--- /dev/null
+++ b/exo2/FDiscMap.m
@@ -0,0 +1,31 @@
+% discretized map image
+classdef FDiscMap
+ properties
+ data % data set
+ Nx % size along X
+ Ny % size along Y
+ Cx % center point X
+ Cy % center point Y
+ end
+ methods
+ function obj=FDiscMap(data)
+ obj.data = data; % take referencies of the data
+ S = double(size(data));
+ obj.Nx = S(2);
+ obj.Ny = S(1);
+ obj.Cx = obj.Nx / 2.0 + 1.0; % offset 1
+ obj.Cy = obj.Ny / 2.0 + 1.0; % offset 1
+ end
+ function [i,j]=pixel(obj,x,y)
+ % first comp = Y
+ % second comp = X
+ i = int32(- (obj.Ny - 1) * y / 2 + obj.Cy);
+ j = int32((obj.Nx - 1) * x / 2 + obj.Cx);
+ end
+ % i <-> y, j <-> x
+ function [x,y]=point(obj,i,j)
+ x = (double(j) - obj.Cx) * 2 / (obj.Nx - 1);
+ y = -(double(i) - obj.Cy) * 2 / (obj.Ny - 1);
+ end
+ end
+end
diff --git a/exo2/FDomain.m b/exo2/FDomain.m
new file mode 100644
index 0000000..029d01a
--- /dev/null
+++ b/exo2/FDomain.m
@@ -0,0 +1,6 @@
+classdef (Abstract) FDomain
+ methods (Abstract)
+ % abstract method, check if point is inside the domain
+ inside(obj,x,y)
+ end
+end
diff --git a/exo2/FEllipse.m b/exo2/FEllipse.m
new file mode 100644
index 0000000..8f6481b
--- /dev/null
+++ b/exo2/FEllipse.m
@@ -0,0 +1,23 @@
+classdef FEllipse < FDomain
+ properties
+ a % horizontal semi-axis
+ b % vertical semi-axis
+ Cx % centre x
+ Cy % centre y
+ end
+ methods
+ % constructor
+ function obj=FEllipse(a,b,C)
+ obj.a = a;
+ obj.b = b;
+ obj.Cx = C(1); % copy array
+ obj.Cy = C(2);
+ end
+
+ % inside method override
+ function flag=inside(obj,x,y)
+ % ellipse equation
+ flag = ((x - obj.Cx) / obj.a)^2 + ((y - obj.Cy) / obj.b)^2 < 1;
+ end
+ end
+end
diff --git a/exo2/FFilter.m b/exo2/FFilter.m
new file mode 100644
index 0000000..f1100ae
--- /dev/null
+++ b/exo2/FFilter.m
@@ -0,0 +1,60 @@
+% collect domains to filter, then apply
+classdef FFilter
+ properties
+ domains {} % cell of domains to verify
+ % each element of the cell is composed by
+ % an array [dom, flag] where:
+ % dom = domain of filtering
+ % flag = invert if false (apply domain filtering for all points outside the domain instead of inside)
+ end
+ methods
+ function obj=FFilter(domains)
+ K = length(domains);
+ obj.domains = cell(K);
+ % verify domain integrity
+ for k=1:K
+ if length(domains{k}) == 2
+ obj.domains{k} = {domains{k}{1}, domains{k}{2}};
+ else
+ warning('Invalid input, skipping it')
+ obj.domains = obj.domains{1:end-1};
+ end
+ end
+ end
+ % add a specific domain with the corresponding flag
+ function add(obj,dom,flag)
+ obj.domains = {obj.domains, [dom, flag]};
+ end
+
+ % img = a discmap object
+ function out=apply(obj, img)
+ [Ni, Nj] = size(img); % get image boundaries
+ map = FDiscMap(img); % create a discrete map of the image
+ out = zeros(Ni,Nj);
+ minval = min(min(img));
+ for i=1:Ni
+ for j=1:Nj
+ [x,y] = map.point(i,j); % get corresponding point
+ if obj.mustfilter(x,y)
+ out(i,j) = minval; % apply filter, minval supports log scaling
+ else
+ out(i,j) = img(i,j); % don't apply
+ end
+ end
+ end
+ end
+ % find whether the point is inside at least one of the specific domains
+ function match=mustfilter(obj,x,y)
+ match = false;
+ K = length(obj.domains);
+ k = 1;
+ while k <= K && ~match
+ dom = obj.domains{k}{1};
+ flag = obj.domains{k}{2};
+ % oscure if
+ match = match | ~xor(dom.inside(x,y), flag);
+ k = k + 1;
+ end
+ end
+ end
+end
diff --git a/exo2/FSquare.m b/exo2/FSquare.m
new file mode 100644
index 0000000..cbdb8b0
--- /dev/null
+++ b/exo2/FSquare.m
@@ -0,0 +1,24 @@
+classdef FSquare < FDomain
+ properties
+ Bx {0.0} % bottom-left x
+ By {0.0} % bottom-left y
+ Tx {1.0} % top-right x
+ Ty {1.0} % top-right y
+ end
+ methods
+ % constructor
+ function obj=FSquare(B, T)
+ if T(1) < B(1) | T(2) < B(2)
+ warning('Square construction: Top corner is less than bottom corner')
+ end
+ obj.Bx = B(1);
+ obj.By = B(2);
+ obj.Tx = T(1);
+ obj.Ty = T(2);
+ end
+ % inside method override
+ function flag=inside(obj,x,y)
+ flag = (obj.x < Tx) & (obj.x > Bx) & (obj.y < Ty) & (obj.y > By);
+ end
+ end
+end
diff --git a/exo2/Lecture3.pdf b/exo2/Lecture3.pdf
new file mode 100644
index 0000000..7133472
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diff --git a/exo2/analyseimg.m b/exo2/analyseimg.m
new file mode 100644
index 0000000..0b2a775
--- /dev/null
+++ b/exo2/analyseimg.m
@@ -0,0 +1,37 @@
+filename = "stm.png";
+
+%filter = @(L) region_filter(region_filter(L, [0, 0.5 - 0.005], [1, 0.5 + 0.005]), [0.5 - 0.005, 0], [0.5 + 0.005, 1]);
+%filter = @(L) ellipt_filter(L, 0.01, 0.01, [0.5, 0.5]);
+%filter = @(L) ellipt_ifilter(L, 0.05, 0.05, [0.5, 0.5]);
+%filter = @(L) ellipt_ifilter(region_ifilter(L, [0.5 - 0.01, 0], [0.5 + 0.01, 1]), 0.05, 0.05, [0.5, 0.5]);
+
+rgbimage = imread(filename);
+grayimg = rgb2gray(rgbimage);
+
+fftimg = fftshift(fft2(grayimg));
+
+%filtfftimg = filter(fftimg);
+filt = FBandPassFilter([0.04, 0.12], [pi/2 - 0.3, pi/2 + 0.3]);
+filtfftimg = filt.apply(fftimg);
+
+filtimg = ifft2(fftshift(filtfftimg));
+
+figure
+imshow(grayimg)
+
+figure
+imshow(contrast(log(abs(fftimg))))
+
+figure
+imshow(contrast(filt.apply(log(abs(fftimg)))))
+
+figure
+imshow(contrast(abs(filtimg)))
+
+function out=contrast(L)
+ % renormalize
+ minval = min(min(L));
+ maxval = max(max(L));
+
+ out = uint8((L - minval) * 255 / (maxval - minval));
+end
diff --git a/exo2/angular_filter.m b/exo2/angular_filter.m
new file mode 100644
index 0000000..c69b25d
--- /dev/null
+++ b/exo2/angular_filter.m
@@ -0,0 +1,3 @@
+% angles = angles [a1, a2] bounded from 0 to 2 * pi, C = center
+function out=angular_filter(L, angles, C)
+
diff --git a/exo2/ellipt_filter.m b/exo2/ellipt_filter.m
new file mode 100644
index 0000000..f577910
--- /dev/null
+++ b/exo2/ellipt_filter.m
@@ -0,0 +1,20 @@
+function out=ellipt_filter(L, a, b, C)
+ [Ni, Nj] = size(L);
+ out = L(:,:);
+ Na = int32((Nj-1) * a); % x
+ Nb = int32((Ni-1) * b); % y
+ C = [int32((Nj-1) * C(1)), int32((Ni-1) * C(2))]; % center
+ % ellipse parametrisation
+ ellipt = @(x) int32(Nb * sqrt(1.0 - (double(x - C(2))/double(Na))^2));
+ xmin = C(2) - Na + 1;
+ xmax = C(2) + Na + 1;
+ % loop through x
+ for i=xmin:xmax
+ range = ellipt(double(i-1));
+ ymin = C(1) - range + 1;
+ ymax = C(1) + range + 1;
+ for j=ymin:ymax
+ out(j,i) = 0;
+ end
+ end
+end
diff --git a/exo2/ellipt_ifilter.m b/exo2/ellipt_ifilter.m
new file mode 100644
index 0000000..6e795ef
--- /dev/null
+++ b/exo2/ellipt_ifilter.m
@@ -0,0 +1,20 @@
+function out=ellipt_ifilter(L, a, b, C)
+ [Ni, Nj] = size(L);
+ out = zeros(Ni, Nj);
+ Na = int32((Nj-1) * a); % x
+ Nb = int32((Ni-1) * b); % y
+ C = [int32((Nj-1) * C(1)), int32((Ni-1) * C(2))]; % center
+ % ellipse parametrisation
+ ellipt = @(x) int32(Nb * sqrt(1.0 - (double(x - C(2))/double(Na))^2));
+ xmin = C(2) - Na + 1;
+ xmax = C(2) + Na + 1;
+ % loop through x
+ for i=xmin:xmax
+ range = ellipt(double(i-1));
+ ymin = C(1) - range + 1;
+ ymax = C(1) + range + 1;
+ for j=ymin:ymax
+ out(j,i) = L(j,i);
+ end
+ end
+end
diff --git a/exo2/exercises4_5.pdf b/exo2/exercises4_5.pdf
index 81eabdc..00caaff 100644
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diff --git a/exo2/fourier.aux b/exo2/fourier.aux
new file mode 100644
index 0000000..fd2ede0
--- /dev/null
+++ b/exo2/fourier.aux
@@ -0,0 +1,33 @@
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diff --git a/exo2/fourier.pdf b/exo2/fourier.pdf
new file mode 100644
index 0000000..f48d355
Binary files /dev/null and b/exo2/fourier.pdf differ
diff --git a/exo2/fourier.tex b/exo2/fourier.tex
index a26335e..4ab5676 100644
--- a/exo2/fourier.tex
+++ b/exo2/fourier.tex
@@ -1,266 +1,421 @@
\documentclass{article}
% Change "article" to "report" to get rid of page number on title page
\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{setspace}
\usepackage{fancyhdr}
\usepackage{lastpage}
\usepackage{extramarks}
\usepackage{chngpage}
\usepackage{soul}
\usepackage[usenames,dvipsnames]{color}
\usepackage{graphicx,float,wrapfig}
\usepackage{ifthen}
\usepackage{listings}
\usepackage{courier}
+\usepackage{mathtools}
+\usepackage{gensymb}
+% definition of the absolute value
+\DeclarePairedDelimiter\abs{\lvert}{\rvert}
+
+\usepackage{scalerel,stackengine}
+
+\stackMath
+\newcommand\reallywidehat[1]{%
+\savestack{\tmpbox}{\stretchto{%
+ \scaleto{%
+ \scalerel*[\widthof{\ensuremath{#1}}]{\kern-.6pt\bigwedge\kern-.6pt}%
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+ }{\textheight}%
+}{0.5ex}}%
+\stackon[1pt]{#1}{\tmpbox}%
+}
% !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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% We put the blank space above in order to make sure this
% \marginpar gets correctly placed.
}%
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+ {
+ \begin{minipage}{1.05\linewidth}
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+ #2
+ \end{minipage}
+ \hspace{1cm}
+ \begin{minipage}{\linewidth}
+ #3
+ \end{minipage}
+ \end{minipage}
+ }
+
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\label{#1}
\end{figure}}
% Includes a MATLAB script.
% The first parameter is the label, which also is the name of the script
% without the .m.
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+%\newcommand{\matlabscript}[2]
+% {\begin{itemize}\item[]\lstinputlisting[caption=#2,label=#1]{#1.m}\end{itemize}}
\newcommand{\matlabscript}[2]
- {\begin{itemize}\item[]\lstinputlisting[caption=#2,label=#1]{#1.m}\end{itemize}}
+ {\lstinputlisting[caption=#2,label=#1]{#1.m}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Make title
%\title{\vspace{2in}\textmd{\textbf{\hmwkClass:\ \hmwkTitle\ifthenelse{\equal{\hmwkSubTitle}{}}{}{\\\hmwkSubTitle}}}\\\normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate}\\\vspace{0.1in}\large{\textit{\hmwkClassInstructor\ \hmwkClassTime}}\vspace{3in}}
\title{\vspace{2in}\textmd{\textbf{\hmwkClass:\ \hmwkTitle\ifthenelse{\equal{\hmwkSubTitle}{}}{}{\\\hmwkSubTitle}}}\\\normalsize\vspace{0.1in}\small{Due\ on\ \hmwkDueDate}\\\vspace{0.1in}\large{\textit{ \hmwkClassTime}}\vspace{3in}}
\date{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{spacing}{1.1}
\maketitle
% Uncomment the \tableofcontents and \newpage lines to get a Contents page
% Uncomment the \setcounter line as well if you do NOT want subsections
% listed in Contents
%\setcounter{tocdepth}{1}
\newpage
\tableofcontents
\newpage
-% When problems are long, it may be desirable to put a \newpage or a
-% \clearpage before each homeworkProblem environment
+\section{Introduction}
-\newpage
+%Interpolating and analysing signals is a problem
-\begin{homeworkProblem}
+\section{Theoretical basis}
+
+\subsection{The fourier transform}
+
+\paragraph{Definition in $L^2$}
+Let $f \in L^2(\mathbb{R})$, then it's \textit{Fourier transform} $\hat{f}$ is defined as
+
+\begin{align} \label{defnt}
+ \hat{f}(k) := \int_{-\infty}^{\infty} f(x) \cdot e^{-2 \pi k x} dx & &
+ f(x) := \int_{-\infty}^{\infty} \hat{f}(k) \cdot e^{2 \pi x k} dk
+\end{align}
+
+where the second definition is called the \textit{inverse Fourier transform} and
+the bijection is a consequence of $L^2(\mathbb{R})$ [\cite{stubbe}].
+
+\paragraph{Properties}
+The following properties can be straigh-forward demonstrated:
+
+\begin{itemize}
+ \item Linearity: $\forall f,g \in L^2, a \in \mathbb{R}: \reallywidehat{f + a \cdot g} = \hat{f} + a \cdot \hat{g}$
+ \item Scaling: $\forall a \in \mathbb{R}: \reallywidehat{f(a \cdot x)} = \frac{1}{\abs*{a}} \hat{f}(\frac{k}{a})$
+ \item Translation: $\forall x_0 \in \mathbb{R}: \reallywidehat{f(x - x_0)} = e^{-2 \pi i x_0 k} \hat{f}(k)$
+ \item Modulation: $\forall k_0 \in \mathbb{R}: \reallywidehat{e^{2 \pi i x k_0} f(x)} = \hat{f}(k - k_0)$
+ \item Parseval identity: $\forall f,g \in L^2: \langle f, g \rangle = \langle \hat{f}, \hat{g} \rangle$ where $\langle \cdot , \cdot \rangle$ is the canonical $L^2$ scalar product (see [\cite{stubbe}])
+\end{itemize}
+
+\paragraph{Convolution definition and theorem}
+Let $f, g \in L^2(\mathbb{R})$, then the \textit{convolution} of $f$ with respect to $g$ is defined as follow:
+
+\begin{equation} \label{convolution}
+ (f \circledast g)(x) := \int_{-\infty}^{\infty} f(y)g(x - y) dy
+\end{equation}
+
+Having listed enough properties of the \textit{Fourier transform}, it's now possible to relate these two concepts by stating the \textit{convolution} theorem:
+\newline
+
+\begin{equation} \label{conv_thm}
+ \reallywidehat{f \circledast g} = \hat{f} \cdot \hat{g}
+\end{equation}
+
+This result is very useful in its practical applications, because it allows to convert
+a convolution into a product operation.
-First Problem
+\section{Discrete Fourier transform}
- \begin{homeworkSection}{(1)}
- Answer to q.1
-\matlabscript{simple}{A script which does the calculation for q.1.}
- \end{homeworkSection}
+The passage from the continous to a discrete expression of the Fourier transform is
+necessary in order to process digital signals. The interest on doing such an elaboration
+is that an analogic circuit which computes the continous
+Fourier transform on a generic signal doesn't forcely exist.
- \begin{homeworkSection}{(2)}
- Answer to q.2
- \end{homeworkSection}
+\subsection{Discretization}
-\begin{homeworkSection}{(3)}
- Answer to q.3
- \end{homeworkSection}
+In order to pass to a discrete representation of the Fourier transform, some conventions need to be set:
+\begin{itemize}
+ \item The signal is evaluated on a limited set of time arguments.
+ \item The first time of the sequence is $0$
+ \item The function is supposed to be constant between one sample and another.
+ \item The function is zero in all other points.
+\end{itemize}
-\begin{homeworkSection}{(4)}
- Answer to q.4
- \end{homeworkSection}
+Let $\{f_n\}_{0 < n < N}$ be the signal samples derived from a function $f(x)$, where
+$N$ is the total number of samples and $T$ the sampling period.
+Because $\forall x \in \mathbb{R} \setminus [0, (N-1)T]: f(x) = 0$ the fourier integration domain
+restricts to $[0, (N-1)T]$, then applying a discretisation on $x$, given by $x_n = nT, 0 < n < N$,
+the integral reduces itself into a sum:
+
+\begin{equation} \label{dft_deriv}
+ \hat{f}(k) = \sum_{n=0}^{N-1} \int_{nT}^{(n+1)T} f_n \cdot e^{-2\pi i nT k} dx = T \sum_{n=0}^{N-1} f_n \cdot e^{-2\pi i nT k}
+\end{equation}
+
+Then taking the indicisation of the $k$-space as $k_m = \frac{m}{NT}, 0 < m < N$, the
+output results a $N$-sized set of values $\{\hat{f}_m\}_{0 < m < N}$:
+
+\begin{equation} \label{dft_fake}
+ \hat{f}_m = T \sum_{n=0}^{N-1} f_n \cdot e^{-2\pi i \frac{nm}{N}}
+\end{equation}
+
+Notice that this expression of the discrete Fourier transform (or DFT) still depends on $T$ and since it's a scaling term,
+it's presence is arbitrary and it could be removed.
+In order to guaratee the application bijectivity, it's important evaluate the
+inverse discrete fourier transform by taking the adjoint of the sum given in equation (\ref{dft_fake})
+on the $\{\hat{f}_m\}_{0 < m < N}$ set and setting it to $\{f_n\}_{0 < n < N}$.
+
+\begin{equation} \label{inverse_dft}
+ f_n = \frac{T}{NT} \sum_{m=0}^{N-1} \hat{f}_m \cdot e^{2\pi i \frac{nm}{N}} = \frac{1}{N} \sum_{m=0}^{N-1} \hat{f}_m \cdot e^{2\pi i \frac{nm}{N}}
+\end{equation}
+
+The previous expression shows that the inverse DFT doesn't depend on the sampling period $T$, then
+there's no reason to keep that dependency for the DFT. This consideration allows to redefine the DFT as follow:
+
+\begin{equation} \label{dft}
+ \hat{f}_m = \sum_{n=0}^{N-1} f_n \cdot e^{-2\pi i \frac{nm}{N}}
+\end{equation}
+
+\section{\textit{Matlab} implementation}
+
+\newpage
+
+\begin{homeworkProblem}
+
+% When problems are long, it may be desirable to put a \newpage or a
+% \clearpage before each homeworkProblem environment
+
+ \begin{homeworkSection}{(1) Straigh forward implementation}
+
+ \unlimwrap[0.55\linewidth]{
+ Using the formulas given in the equations (\ref{inverse_dft}) and (\ref{dft}),
+ the resulting \textit{matlab} code for the DFT is straight forward as it
+ shows to listing (\ref{mydft}).
+ \textit{Matlab} storage system of vectors and matrices is very flexible but
+ it's not clear how a matrix is treated when the initialisation is handled externally.
+ So, in order to determine the size of the output, the best approach is to
+ adapt it to the input size. That's what the lines (3) and (5) do: they take the input size format
+ and expand it into the output initialization.
+ }{
+ \matlabscript{mydft}{A \textit{matlab} implementation of the DFT}
+ }
+
+ \end{homeworkSection}
+
+ \begin{homeworkSection}{(2) Testing}
+ There are two essential aspects to check whether the algorithm works:
+ \begin{enumerate}
+ \item No bugs, all behaviours are expected
+ \item Lowest possible time complexity
+ \end{enumerate}
+
+ Both tasks can be executed using a script which compares the code execution of (\ref{mydft}) with
+ the \textit{Matlab} built-in \textit{fft}.
+ At first glance, the complexity is supposed to be $\mathcal{O}(N^2)$ where $N$ is the size of the input array,
+ which is justified by the fact that the code loops $N$ times (index $j$) inside another
+ $N$ sized loop (index $i$).
+ On the other hand the \textit{Matlab} built-in function \textit{assert} helps in order to verify
+ that the output of \textit{mydft} and \textit{fft} correspond.
+
+ \end{homeworkSection}
\end{homeworkProblem}
\newpage
\begin{homeworkProblem}
-Second problem....
+Second problem: Spectral and cepstral analysis
+
+ \begin{homeworkSection}{(1) Straigh forward implementation of the inverse DFT}
+ \matlabscript{mydftinverse}{A \textit{matlab} implementation of the inverse DFT}
+
+ \end{homeworkSection}
\end{homeworkProblem}
\end{spacing}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/exo2/fourier.toc b/exo2/fourier.toc
new file mode 100644
index 0000000..7c98df9
--- /dev/null
+++ b/exo2/fourier.toc
@@ -0,0 +1,16 @@
+\contentsline {section}{Introduction}{3}%
+\contentsline {section}{Theoretical basis}{3}%
+\contentsline {subsection}{The fourier transform}{3}%
+\contentsline {paragraph}{Definition in $L^2$}{3}%
+\contentsline {paragraph}{Properties}{3}%
+\contentsline {paragraph}{Convolution definition and theorem}{3}%
+\contentsline {section}{Discrete Fourier transform}{3}%
+\contentsline {subsection}{Discretization}{3}%
+\contentsline {section}{\textit {Matlab} implementation}{4}%
+\contentsline {section}{Problem 1}{5}%
+\contentsline {subsection}{(1) Straigh forward implementation}{5}%
+\contentsline {subsection}{(2) Testing}{5}%
+\contentsline {subsection}{(3)}{5}%
+\contentsline {subsection}{(4)}{5}%
+\contentsline {section}{Problem 2}{6}%
+\contentsline {subsection}{(1) Straigh forward implementation of the inverse DFT}{6}%
diff --git a/exo2/mydft.m b/exo2/mydft.m
index 72da226..63ebc2c 100644
--- a/exo2/mydft.m
+++ b/exo2/mydft.m
@@ -1,14 +1,15 @@
function result=mydft(input_array)
% convert to cell
s = mat2cell(size(input_array), 1, 2);
% expand into an argument list
result = zeros(s{:});
N = length(result);
-
+
for m=1:N
result(m) = 0;
for n=1:N
- result(m) = result(m) + input_array(n) * exp(-2 * pi * 1i * (n-1) * (m-1) / N);
+ w = exp(-2 * pi * 1i * (n-1) * (m-1) / N);
+ result(m) = result(m) + input_array(n) * w;
end
end
end
diff --git a/exo2/mydftinverse.m b/exo2/mydftinverse.m
index 6cb2253..6833345 100644
--- a/exo2/mydftinverse.m
+++ b/exo2/mydftinverse.m
@@ -1,15 +1,16 @@
function result=mydftinverse(input_array)
% convert to cell
s = mat2cell(size(input_array), 1, 2);
% expand into an argument list
result = zeros(s{:});
N = length(result);
for m=1:N
result(m) = 0;
for n=1:N
- result(m) = result(m) + input_array(n) * exp(2 * pi * 1i * (n-1) * (m-1) / N);
+ w = exp(2 * pi * 1i * (n-1) * (m-1) / N);
+ result(m) = result(m) + input_array(n) * w;
end
result(m) = result(m) / N;
end
end
diff --git a/exo2/output/testmydft.out b/exo2/output/testmydft.out
new file mode 100644
index 0000000..98b731c
--- /dev/null
+++ b/exo2/output/testmydft.out
@@ -0,0 +1,18 @@
+Ns times
+1024 0.007726
+2048 0.002314
+4096 0.002574
+8192 0.003421
+16384 0.010608
+32768 0.016271
+65536 0.039539
+131072 0.036993
+262144 0.048733
+524288 0.070934
+1.04858e+06 0.141809
+2.09715e+06 0.211301
+4.1943e+06 0.322412
+8.38861e+06 0.618665
+1.67772e+07 1.16364
+3.35544e+07 2.40053
+6.71089e+07 7.27462
diff --git a/exo2/region_filter.m b/exo2/region_filter.m
new file mode 100644
index 0000000..5dfbcbe
--- /dev/null
+++ b/exo2/region_filter.m
@@ -0,0 +1,12 @@
+% t = top boundary point, b = bottom boundary point
+function out=region_filter(L, t, b)
+ [Ni, Nj] = size(L);
+ out = L(:,:);
+ abst = [int32((Nj-1) * t(1)) + 1, int32((Ni-1) * t(2)) + 1];
+ absb = [int32((Nj-1) * b(1)) + 1, int32((Ni-1) * b(2)) + 1];
+ for i=abst(1):absb(1)
+ for j=abst(2):absb(2)
+ out(j,i) = 0;
+ end
+ end
+end
diff --git a/exo2/region_ifilter.m b/exo2/region_ifilter.m
new file mode 100644
index 0000000..6cd32bc
--- /dev/null
+++ b/exo2/region_ifilter.m
@@ -0,0 +1,12 @@
+% t = top boundary point, b = bottom boundary point
+function out=region_ifilter(L, t, b)
+ [Ni, Nj] = size(L);
+ out = zeros(Ni,Nj);
+ abst = [int32((Nj-1) * t(1)) + 1, int32((Ni-1) * t(2)) + 1];
+ absb = [int32((Nj-1) * b(1)) + 1, int32((Ni-1) * b(2)) + 1];
+ for i=abst(1):absb(1)
+ for j=abst(2):absb(2)
+ out(j,i) = L(j,i);
+ end
+ end
+end
diff --git a/exo2/stm.png b/exo2/stm.png
new file mode 100644
index 0000000..1ff37a5
Binary files /dev/null and b/exo2/stm.png differ
diff --git a/exo2/test.m b/exo2/test.m
index ae27750..23bdce6 100644
--- a/exo2/test.m
+++ b/exo2/test.m
@@ -1,32 +1,52 @@
% This script tests functor.m for correctness
-functor = @(x) myfft(x);
+functor = @(x) mydft(x);
fprintf('Test 1: Gaussian ...');
sample = exp(-linspace(-4,4,256).^2);
assert(all(abs(functor(sample) - fft(sample))<1e-10));
fprintf('\tpassed\n');
fprintf('Test 1.1: Gaussian with a different Matlab dimension ...');
sample = exp(-linspace(-4,4,256).^2)';
assert(all(abs(functor(sample) - fft(sample))<1e-10));
fprintf('\tpassed\n');
fprintf('Test 1.2: Gaussian complex ...');
sample = 1i*exp(-linspace(-4,4,256).^2);
assert(all(abs(functor(sample) - fft(sample))<1e-10));
fprintf('\tpassed\n');
fprintf('Test 2: sawtooth ...');
sample = linspace(-1,1,256);
assert(all(abs(functor(sample) - fft(sample))<1e-10));
fprintf('\tpassed\n');
fprintf('Test 3: sin and sin2 ...');
sample = sin(linspace(-pi,pi,256));
assert(all(abs(functor(sample) - fft(sample))<1e-10));
sample = sin(2*linspace(-pi,pi,256));
assert(all(abs(functor(sample) - fft(sample))<1e-10));
fprintf('\tpassed\n');
+Ns = 10:26;
+times = zeros(length(Ns),1);
+
+% Time complexity test
+for k=Ns
+ fprintf('\tEvaluating step %d \n', k);
+ sample = exp(-linspace(-4,4,2^k).^2)';
+ tic;
+ fft(sample);
+ times(k-Ns(1)+1) = toc;
+end
+
+Ns = 2.^Ns
+
+figure
+grid on
+xlabel('N')
+ylabel('Elapsed time')
+loglog(Ns, times)
+
fprintf('All tests passed!\n');
diff --git a/exo2/texput.log b/exo2/texput.log
new file mode 100644
index 0000000..81e7944
--- /dev/null
+++ b/exo2/texput.log
@@ -0,0 +1,21 @@
+This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2020.3.10) 18 MAR 2020 09:04
+entering extended mode
+ restricted \write18 enabled.
+ %&-line parsing enabled.
+**fourier.t
+
+! Emergency stop.
+<*> fourier.t
+
+End of file on the terminal!
+
+
+Here is how much of TeX's memory you used:
+ 4 strings out of 492167
+ 115 string characters out of 6131558
+ 59016 words of memory out of 5000000
+ 4468 multiletter control sequences out of 15000+600000
+ 3640 words of font info for 14 fonts, out of 8000000 for 9000
+ 1141 hyphenation exceptions out of 8191
+ 0i,0n,0p,12b,6s stack positions out of 5000i,500n,10000p,200000b,80000s
+! ==> Fatal error occurred, no output PDF file produced!