Explaining DFT Formula

[Vishnu M]

Could you please unpack the terms in the DFT equation for me please and explain how they interact with one-another (perhaps via. the sine wave or unit circle as analogies)

i struggled to conceptualise what k was, and why the exponential term is as it is and what that means

[Simon]

I’ll start you off with this and you try to complete it – I’m going to use square brackets to index everything to emphasise that we are dealing with discrete (= digital) signals only, and to match the notation in the sp-m1-4-discrete-fourier-transform notebook.

x[n] is the waveform. n is indexing time. To plot x[n], we place n on the horizontal axis and label it “time (samples)” then plot the value of x[n] at each value of n. We have drawn the waveform.

X[k] is the Discrete Fourier Transform of x[n]. k is indexing BLANK. To plot X[k], we place BLANK on the horizontal axis and label it “BLANK” then plot the value of X[k] at each value of k. We have drawn the BLANK.

Can you fill in some of the BLANKs?

[Vishnu M]

I believe that:

k is indexing the component frequencies we are running DFT on to get their coefficients of correlation with the complex waveform we are trying to deconstruct. We iterate from k to K where K is the nyquist frequency.

To plot X[k] we place frequencies from k to K on the horizontal axis and label it frequencies, and then we plot the magnitude of that frequency given X[k] on the vertical axis, and we have drawn our frequency spectrum…..

I hope i have got it?

[Simon]

Yes, you’ve got it. k is indexing frequency – sometimes we say it is the DFT “bin”.

When plotting X[k] the axis will run from 0 to K (not from k to K). We could label that axis with the integers 0 to K (some of the notebooks do that), or we could convert k to Hz and label the axis with frequency.

How do we convert the integer k into a frequency in Hz ?

[Vishnu M]

is k the number of cycles of the circle we do in a second, and so hertz would just be k…? sorry i’m finding it quite confusing

[Simon]

Getting closer… some more clues:

The lowest value of k (apart from 0), is 1. That’s the lowest frequency DFT “bin” and corresponds to the lowest frequency basis function.

The lowest frequency basis function makes just one cycle within the analysis frame. So its frequency in Hertz must be 1 / “duration of the analysis frame in seconds”

We can calculate the duration of the analysis window in seconds if we know how many samples are in the frame and what the sampling frequency is: there is a simple relationship between these three values.

Now we know the frequency of the first bin k=1 we can work out the frequency of the bin k=2 and so on all the way up to K.

[Author]

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