129 Note: To solve a system of ODEs, we need to change the function to return the right hand side of all the equations and the system and the pass the required number of initial conditions to the \typ{odeint} function.

   129 Note: To solve a system of ODEs, we need to change the function to

   130 return the right hand side of all the equations and the system and the

   131 pass the required number of initial conditions to the \typ{odeint}

   132 function.

   133 

   134 \section{FFT}

   135 \begin{itemize}

   136     \item We have a simple signal $y(t)$

   137     \item Find the FFT and plot it

   138 \end{itemize}

   139 \begin{lstlisting}

   140 In []: t = linspace(0, 2*pi, 500)

   141 In []: y = sin(4*pi*t)

   142 

   143 In []: f = fft(y)

   144 In []: freq = fftfreq(500, t[1] - t[0])

   145 

   146 In []: plot(freq[:250], abs(f)[:250])

   147 In []: grid()

   148 \end{lstlisting} 

   149 \begin{itemize}

   150     \item We now calculate the inverse Fourier transform.

   151     \item Then, verify the solution obtained. 

   152 \end{itemize}

   153 \begin{lstlisting}

   154 In []: y1 = ifft(f) # inverse FFT

   155 In []: allclose(y, y1)

   156 Out[]: True

   157 \end{lstlisting} 

   158 \begin{itemize}

   159 \item Let us add some noise to the signal

   160 \end{itemize}

   161 \begin{lstlisting}

   162 In []: yr = y + random(size=500)*0.2

   163 In []: yn = y + normal(size=500)*0.2

   164 

   165 In []: plot(t, yr)

   166 In []: figure()

   167 In []: plot(freq[:250],

   168   ...:      abs(fft(yn))[:250])

   169 \end{lstlisting}

   170 \begin{itemize}

   171     \item \typ{random}: produces uniform deviates in $[0, 1)$

   172     \item \typ{normal}: draws random samples from a Gaussian

   173         distribution

   174     \item Useful to create a random matrix of any shape

   175 \end{itemize}

   176 

   177 \begin{itemize}

   178 \item Now, we filter the noisy signal using a Wiener filter

   179 \end{itemize}

   180 \begin{lstlisting}

   181 In []: from scipy import signal

   182 In []: yc = signal.wiener(yn, 5)

   183 In []: clf()

   184 In []: plot(t, yc)

   185 In []: figure()

   186 In []: plot(freq[:250], 

   187   ...:      abs(fft(yc))[:250])

   188 \end{lstlisting}

   189