ex-6: went on writing on the FFT

notes/sections/6.md

@@ -15,7 +15,7 @@ where:
 
 - $E$ is the electric field amplitude, default set $E = \SI{1e4}{V/m}$;
 - $a$ is the radius of the slit aperture, default set $a = \SI{0.01}{m}$;
-- $\theta$ is the angle specified in @fig:fenditure;
+- $\theta$ is the angle specified in @fig:slit;
 - $J_1$ is the Bessel function of first order;
 - $k$ is the wavenumber, default set $k = \SI{1e-4}{m^{-1}}$;
 - $L$ default set $L = \SI{1}{m}$.
@@ -46,7 +46,7 @@ where:
   \node [cyclamen] at (5.5,-0.4) {$\theta$};
   \node [rotate=-90] at (10.2,0) {screen};
 \end{tikzpicture}
-\caption{Fraunhofer diffraction.}
+\caption{Fraunhofer diffraction.}\label{fig:slit}
 }
 \end{figure}
 
@@ -101,11 +101,8 @@ same bin width of the previous one, but a smaller number of them ($m < n$), was
 filled with $m$ points according to a Gaussian distribution with mean $\mu$,
 corresponding to the central bin, and variance $\sigma$.  
 Then, the original histogram was convolved with the kernel in order to obtain
-the smeared signal. The result is shown in @fig:convolved.
-
-![Same sample of @fig:original convolved with the
-  kernel.](images/6_convolved.pdf){#fig:convolved}
-
+the smeared signal. Some results in terms of various $\sigma$ are shown in
+@fig:convolved.  
 The convolution was implemented as follow. Consider the definition of
 convolution of two functions $f(x)$ and $g(x)$:
 
@@ -124,8 +121,9 @@ $$
 $$
 
 where $j$ runs over the bins of the kernel.  
-For a better understanding, see @fig:dot_conv. Thus, the third histogram was
-obtained with $n + m - 1$ bins, a number greater than the initial one.
+For a better understanding, see @fig:dot_conv. As can be seen, the third
+histogram was obtained with $n + m - 1$ bins, a number greater than the initial
+one.
 
 \begin{figure}
 \hypertarget{fig:dot_conv}{%
@@ -194,6 +192,104 @@ obtained with $n + m - 1$ bins, a number greater than the initial one.
 }
 \end{figure}
 
-\textcolor{red}{Missing various $\sigma$ comparison.}
 
-## Unfolding
+## Unfolding with FFT
+
+Two different unfolding routines were implemented, one of which exploiting the
+Fast Fourier Transform.  
+This method is based on the property of the Fourier transform according to
+which, given two functions $f(x)$ and $g(x)$:
+
+$$
+  \hat{F}[f*g] = \hat{F}[f] \cdot \hat{F}[g]
+$$
+
+where $\hat{F}[\quad]$ stands for the Fourier transform of its argument.  
+Thus, the implementation of this tecnique lies in the computation of the
+Fourier trasform of the signal and the kernel, the product of their transforms
+and the anti-transformation of the result. Being the histogram a discrete set of
+data, the Discrete Fourier Transform (DFT) was emploied.
+
+In order to accomplish this procedure, every histogram was transformed into a
+vector. The kernel vector was 0-padded in order to make its length the same as
+the one of the signal, making it feasable to implement the dot product between
+the vectors.
+
+FFT are efficient algorithms for calculating the DFT. Namely, given a set of $n$
+values {$z_i$}, each one is transformed into:
+
+$$
+  x_j = \sum_{k=0}^{n-1} z_k \exp \left( - \frac{2 \pi i j k}{n} \right)
+$$
+
+The evaluation of the DFT is a matrix-vector multiplication $W \vec{z}$. A
+general matrix-vector multiplication takes $O(n^2)$ operations. FFT algorithms,
+instad, use a divide-and-conquer strategy to factorize the matrix into smaller
+sub-matrices. If $n$ can be factorized into a product of integers $n_1$, $n_2
+\ldots n_m$, then the DFT can be computed in $O(n \sum n_i) < O(n^2)$
+operations, hence the name.  
+The inverse Fourier transform is thereby defined as:
+
+$$
+  z_j = \frac{1}{n}
+        \sum_{k=0}^{n-1} x_k \exp \left( \frac{2 \pi i j k}{n} \right)
+$$
+
+In GSL, `gsl_fft_complex_forward()` and `gsl_fft_complex_inverse()` are 
+functions which allow to compute the foreward and inverse transform,
+respectively.
+
+In this special case, the sequence which must be transformed is made of real
+numbers, but the Fourier transform is not real: it is a complex sequence wich
+satisfies:
+
+$$
+  z_k = z^*_{n-k}
+$$
+
+Where $z^*$ is the conjugate of $z$. A sequence with this symmetry is called
+'half-complex'. This structure requires a particular storage layouts for the
+forward transform (from real to half-complex) and inverse transform (from
+half-complex to real). As a consequence, the routines are divided into two sets:
+`gsl_fft_real` and `gsl_fft_halfcomplex`. The symmetry of the half-complex
+sequence implies that only half of the complex numbers in the output need to be
+stored. This works for all lengths: when the length is even, the middle value
+where $k = n/2$ is real. Thus, only $n$ real numbers are required to store the
+half-complex sequence (real and imaginary parts).  
+The ratio between the kernel and the sigmal transformations where therefore
+made between complex numbers.
+If the time-step of the DFT is $\Delta$, then the frequency-domain includes
+both positive and negative frequencies, ranging from $-1 / (2 \Delta)$ to
+$+1 / (2 \Delta$). The positive frequencies are stored from the beginning of
+the array up to the middle, and the negative frequencies are stored backwards
+from the end of the array. \textcolor{red}{Useful?}
+
+
+---
+
+<div id="fig:convolved">
+![Convolved. $\sigma = 0.05 \Delta \theta$](images/noise-0.05.pdf){width=7cm}
+![Deconvolved. $\sigma = 0.05 \Delta \theta$](images/deco-0.05.pdf){width=7cm}
+
+![Convolved. $\sigma = 0.1 \Delta \theta$](images/noise-0.1.pdf){width=7cm}
+![Deconvolved. $\sigma = 0.1 \Delta \theta$](images/deco-0.1.pdf){width=7cm}
+
+![Convolved. $\sigma = 0.5 \Delta \theta$](images/noise-0.5.pdf){width=7cm}
+![Deconvolved. $\sigma = 0.5 \Delta \theta$](images/deco-0.5.pdf){width=7cm}
+
+![Convolved. $\sigma = 1 \Delta \theta$](images/noise-1.pdf){width=7cm}
+![Deconvolved. $\sigma = 1 \Delta \theta$](images/deco-1.pdf){width=7cm}
+
+Signal convolved with kernel on the left and deconvolved signal on the right.
+Increasing values of $\sigma$ from top to bottom. $\Delta \theta$ is the bin
+width.
+</div>
+
+As can be seen from the plots on the left in @fig:convolved, increasig the
+value of $\sigma$ implies a stronger smoothing of the curve, while the
+deconvolution process seems not to be affected by $\sigma$ amplitude changes:
+it always gives the same outcome.
+
+It was also implemented the possibility to add a Poisson noise to the
+distribution to check weather the deconvolution is affected or not by this
+kind of noise.