Fourier Analysis of a sound signal using Mathematica

[neoromeo]

Homework Statement

I am trying to construct a Mathematica notebook that will be able to import sound in the form of a .wmv file and then create the frequency spectrum for a given time interval.

Homework Equations

I managed to complete this part, though I am trying to figure out:
a) How could I make a filter to cut off certain frequencies and then,
b) using the Inverse Fourier transform to rebuild the signal.

The Attempt at a Solution

The following code does the import and frequency analysis job.

Clear All

file = "C:\\SOUND\\adriana.wav";

data = Flatten@Import[file, "Data"];

Import[file, "Options"]

snd = Import[file, "Sound"]

Length[data]

SetOptions[ListLinePlot, ImageSize -> {500, 150}, AspectRatio -> 0.25,
PlotRange -> All];

SetOptions[ListPlot, ImageSize -> {500, 150}, AspectRatio -> 0.25,
PlotRange -> All];

ListLinePlot[data[[1 ;; 4820100 ;; 100]]]

ListLinePlot[Abs[Fourier[data[[500000 ;; 500000 + 88200]]]]]

ListLinePlot[Abs[Fourier[data[[200000 ;; 600000]]]][[1 ;; 600]]]


Thanks in advance :)

 

[diazona]

a) How could I make a filter to cut off certain frequencies and then,

You have the Fourier transform of the input data - how would you manipulate that to cut off certain frequencies? (If you don't know the Mathematica functions for it, at least conceptually: how could you do this?)

b) using the Inverse Fourier transform to rebuild the signal.

Look at the Mathematica documentation for Fourier and it should point you to the proper function.

 

[neoromeo]

a) I guess I 'll have to make a loop that will check for the certain intensities on the frequency region I want to cut off and make everything zero. Another way could be the convolution of the original data with some other function, but it's getting tricky when it comes to real data.

b) Actually I managed to do the Inverse Fourier, though it's really distorted compared to the original, even for small time fragments... I have to clear the spectrum first and then try again.

 

[diazona]

Mathematica is generally better at working with functions than loops (or arrays, lists, etc.). So you should investigate using an envelope function. How would you construct the function? (Simple example: suppose you wanted to cut off all frequencies above 1000 Hz, what function would you use?)

Note that convolutions have a special relationship with Fourier transforms that make them very easy to compute.

 

[paranormal]

Great job on successfully constructing a Mathematica notebook that can import sound and create a frequency spectrum! As for your questions about creating a filter and using the Inverse Fourier transform, here are some suggestions:

a) To create a filter, you can use the built-in function "BandpassFilter" in Mathematica. This function allows you to specify a range of frequencies to keep in your signal and filter out the rest. For example, if you want to keep frequencies between 100 Hz and 500 Hz, you can use the following code:

filteredData = BandpassFilter[data, {100, 500}, SampleRate -> 44100];

This will create a new dataset called "filteredData" that only contains frequencies between 100 Hz and 500 Hz.

b) To use the Inverse Fourier transform, you can use the built-in function "InverseFourier" in Mathematica. This function takes in the frequency domain data and converts it back to the time domain. For example, if you want to rebuild the signal for the filtered data above, you can use the following code:

rebuiltSignal = Abs[InverseFourier[filteredData]];

This will create a new dataset called "rebuiltSignal" that contains the time domain data for the filtered frequencies.

I hope this helps! Keep up the great work on your Mathematica notebook.