first functional FFT analysis of the data

module3/exo3/exercice_python_fr.org

@@ -145,18 +145,22 @@ On possède des données toutes les semaines.
 
   # Conversion of the data to numpy array
   co2 = np.array(concentration)
+
+  #Remove mean value
+  co2 = co2 - np.mean(co2)
   # Concentration FFT
   concentrationFFT = np.fft.fft(co2)
   # We take the norm of the FFT
-  concentrationFFTNorm = np.absolute(concentrationFFT)
+  concentrationFFTNorm  = np.absolute(concentrationFFT)
+  concentrationFFTAngle = np.angle(concentrationFFT)
 
   # Frequency axis
   N  = co2.shape[0] # Number of samples
   Te = 1            # Sampling interval
   t  = np.arange(N)  # time reference
 
-  freq = np.arange(N)/(N*Te)
-  #freq = np.fft.fftfreq(N)
+  #freq = np.arange(N)/(N*Te)
+  freq = np.fft.fftfreq(N)*N*(1/N*Te)
 
 
   plt.figure(figsize=(10,5))
@@ -169,3 +173,106 @@ On possède des données toutes les semaines.
 
 #+RESULTS:
 [[file:dataFFT.png]]
+
+** FFT zoom
+
+On effectue un zoom sur la partie positive du graphique pour tenter de
+caractériser les différérentes fréquences d'oscillations du signal.
+
+#+begin_src python :results file :session :var matplot_lib_filename="fftZoom.png" :exports both
+  startIndx = 10
+  fftzoomIndx = int(N/6)
+
+  plt.figure(figsize=(10,5))
+  plt.plot(freq[startIndx:fftzoomIndx],concentrationFFTNorm[startIndx:fftzoomIndx])
+  plt.tight_layout()
+
+  plt.savefig(matplot_lib_filename)
+  matplot_lib_filename
+#+end_src
+
+#+RESULTS:
+[[file:fftZoom.png]]
+
+On remarque que les "petites" oscillations ont une amplitude
+d'environ 1400. On tentera de filtrer donc le signal au dessus de ce
+seuil.
+** Filtrage des grandes oscillations 
+
+On élimine toutes les valeurs au dessus du seuil établit précédemment.
+
+#+begin_src python :results file :session :var matplot_lib_filename="smallOsillations.png" :exports both
+  # Extraction of the small oscillations
+  smallAmplitude = 1400
+  smallNorm = np.copy(concentrationFFTNorm)
+  smallNorm[smallNorm > smallAmplitude] = 0
+
+  plt.figure(figsize=(10,5))
+  plt.plot(freq,smallNorm)
+  #plt.plot(freq[startIndx:fftzoomIndx],small[startIndx:fftzoomIndx])
+  plt.tight_layout()
+
+  plt.savefig(matplot_lib_filename)
+  matplot_lib_filename
+#+end_src
+
+#+RESULTS:
+[[file:smallOsillations.png]]
+
+** Reconstruction du signal :  petites oscillations 
+
+#+begin_src python :results file :session :var matplot_lib_filename="smallOsillationsTime.png" :exports both
+  smallFreqSignal = smallNorm*np.exp(1j*concentrationFFTAngle)
+  smallSignal = np.fft.ifft(smallFreqSignal)
+  smallSignal = np.real(smallSignal)
+
+  plt.figure(figsize=(10,5))
+  plt.plot(t,smallSignal)
+  plt.tight_layout()
+
+  plt.savefig(matplot_lib_filename)
+  matplot_lib_filename
+#+end_src
+
+#+RESULTS:
+[[file:smallOsillationsTime.png]]
+
+** Filtrage des petites oscillations 
+
+On élimine toutes les valeurs au dessus du seuil établit précédemment.
+
+#+begin_src python :results file :session :var matplot_lib_filename="bigOsillations.png" :exports both
+  # Extraction of the big oscillations
+  bigAmplitude = 1400
+  bigNorm = np.copy(concentrationFFTNorm)
+  bigNorm[bigNorm < bigAmplitude] = 0
+
+  plt.figure(figsize=(10,5))
+  plt.plot(freq,bigNorm)
+  #plt.plot(freq[startIndx:fftzoomIndx],big[startIndx:fftzoomIndx])
+  plt.tight_layout()
+
+  plt.savefig(matplot_lib_filename)
+  matplot_lib_filename
+#+end_src
+
+#+RESULTS:
+[[file:bigOsillations.png]]
+
+** Reconstruction du signal : grandes oscillations 
+
+#+begin_src python :results file :session :var matplot_lib_filename="bigOsillationsTime.png" :exports both
+  bigFreqSignal = bigNorm*np.exp(1j*concentrationFFTAngle)
+  bigSignal = np.fft.ifft(bigFreqSignal)
+  bigSignal = np.real(bigSignal)
+
+  plt.figure(figsize=(10,5))
+  plt.plot(t,bigSignal)
+  plt.tight_layout()
+
+  plt.savefig(matplot_lib_filename)
+  matplot_lib_filename
+#+end_src
+
+#+RESULTS:
+[[file:bigOsillationsTime.png]]