Fourier Frequency Identification

[AaronMartin]

I am a high school physics teacher and was thinking of demonstrating to students how frequencies of complex waveforms such as notes of instruments can be identified using a Fourier transform.

I haven't done Fourier transforms for a while and was quickly re-reading about them earlier this evening.

Essentially what I was after is given some data (collected using a data logger and microphone etc) or as an example a series of points generated from t=0 to t=10 in steps of 0.1 of 2 Sin[x + 1] + 2 Sin[3 x] + 2 Sin[4.6 x + 3]. Plotting this in Mathematica gives me a nice looking waveform.

If I then do a Fourier transform on the data produced using Fourier[data], Mathematica produces a Table of complex numbers.

Can someone please help me take these complex numbers and transform them into frequency information?

I attempted to do something like ListLinePlot[Abs[Fourier[data]], PlotRange -> All] but it produced a strange looking Plot which was nothing like I was expecting.

Thanks

Aaron

 

[Redbelly98]

I recommend using some simple trial functions, and seeing what you get for a spectrum. Something like sin(2πt), which has a frequency of 1 Hz (for t measured in seconds). See where the non-zero number(s) is/are in the spectrum; this corresponds to an amplitude of 1 at a frequency of 1 Hz.
Also try the function f(t)=1, to see how a 0-Hz (DC) signal is transformed. Keep things simple until you understand how the frequency and amplitude is related to the output. Then you can put more complicated signals in and know what the output signifies.

 

[oswaler]

Hello Aaron,

Fourier Frequency Identification is a powerful tool in understanding and analyzing complex waveforms. It allows us to break down a complex signal into its individual frequency components, making it easier to identify and study the various frequencies present in the waveform. I think it's great that you want to demonstrate this concept to your students, as it is a fundamental concept in physics and engineering.

To help you with your question, I would suggest looking into the concept of the Fourier Transform. This mathematical operation takes a signal in the time domain (such as your data collected from the microphone) and converts it into the frequency domain. This means that instead of having a plot of amplitude vs. time, you will have a plot of amplitude vs. frequency. This is what you were trying to achieve with your ListLinePlot[Abs[Fourier[data]], PlotRange -> All] command.

However, the reason why the plot did not look as expected is because the Fourier Transform produces a complex-valued output, as you mentioned. This is because the frequency components can have both a magnitude (amplitude) and a phase. To get a better understanding of the frequency components, it would be helpful to plot the magnitude and phase separately. You can do this by using the commands ListLinePlot[Abs[Fourier[data]], PlotRange -> All] and ListLinePlot[Arg[Fourier[data]], PlotRange -> All]. This will give you two plots, one showing the magnitude and the other showing the phase of the frequency components.

To convert the complex numbers into frequency information, you can use the command Abs[Fourier[data]]. This will give you the magnitude of each frequency component, which you can then plot against the corresponding frequency values. The resulting plot will show the frequency spectrum of your signal, with the different peaks representing the different frequencies present in your waveform.

I hope this helps you understand how to use the Fourier Transform to analyze complex waveforms. If you have any further questions, please don't hesitate to ask. Best of luck with your demonstration!