Data filtering with Fourier

Feature image for  filtering with DFT Tips & Tricks - showing the filtered and the original input signals

A recent support inquiry inspired the topic of this newsletter. The user wished to use Fourier transform to filter a signal using only K frequency components with the highest amplitudes.

NumXL DFT wizard supports low-pass filtering by reconstructing the signal using the lower K components, thus removing the higher frequency noise components, and generating a smoother signal. 

In this figure, we show the 'Options' Tab of the NumXL Fourier Transform Wizard Dialog, and we highlighted the section defining the parameters of the low-pass filtering using the first N-components to reconstruct the input signal.

 

Unfortunately, we can’t be sure that the first K components are the ones with the highest amplitudes, so we can’t use the wizard for our objective. To proceed with our goal, first, we generate the Fourier spectrum:

In the figure, we show the Fourier transform (i.e., DFT/FFT) amplitude of our input data using the first 110 components

 

Next, we identify the N-components (e.g., N=11) with the highest amplitudes, and derive a new DFT spectrum to include only those components, and set the rest to zero:

In this figure, we show a modified plot of the Fourier transform (i.e., DFT/FFT) amplitude, as we removed all components (i.e., set to zero) of lower amplitude values

 

Now, using the modified DFT spectrum and the IDFT function, we can reconstruct the filtered signal.

In this figure, we show both the original input signal and the output filtered (using the DFT components of the highest amplitudes) signal

 

Conclusion:

In this issue, we have demonstrated a few steps to implement a simple filter in the frequency-domain with Fourier transform. You can use this technique, and apply, with little-to-no modification, and build a wide range of more sophisticated filter functions.

Note that we have not touched on the phase part of the Fourier transform, but instead left it unchanged. Shall you wish to implement a filter function that affects the Phase, then you’d account for the change in the Phase and the amplitude of each component in the modified DFT spectrum.

 

For more information on Fourier transform functions, click here!

 

 

Example spreadsheet used for this newsletter:

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