Lowest Frequency Questions

[Ian Jiang]

Hi, Catherin

I have some questions on window & series of basis waves.
When we sample time points from a signal with a window, the samples could be resolved as a weighted sum of series of basis sin waves.
It is stated that the lowest frequency of a windowed signal could be acquired by 1/window duration.
Since the window size is fixed during sampling, does it imply that, as we don’t know anything happening outside this particular window, we treat this duration as a full cycle (even if that might not be the case should we zoom out)?
More on that, say we have an entire signal which is just a simple sin wave with a frequency of 10Hz. And we have a window that happens to contain two complete cycles of it (0.2s), then as 1/window duration suggests, we should assume that we don’t know what’s outside the window and give its lowest frequency by 50Hz instead of a global truth of 10Hz?
My last question is since there’s only one unique combination of basis functions to produce our signal, does it mean that we are giving zero coefficients to any basis sin wave that has a frequency lower than the window’s lowest frequency we just computed?

Thank you!

[Catherine Lai]

Hi Ian,

You’re right in saying that when we take a window, we ignore everything else that is happening outside the window. So if there are frequencies present in the signal that have a period longer than the window length, we won’t be able to capture them.

This isn’t quite shown in the example you give (though it may be just a typo!). If the window length=0.2 s, then 1/window length = 5 , so the minimum frequency you could capture with that window length is 5 Hz and multiples of that up to half the sampling rate. So in this case you would be able to capture a 10 Hz signal.

If instead the window length = 0.02 s, the frequencies would be 1/0.02 = 50 Hz, and multiples of that (100 Hz, 150 Hz, etc). So in that case, you wouldn’t accurately capture the 10 Hz wave. In this case, since the input frequency 10 Hz falls between the DFT analysis frequencies, you would get leakage: you would see the largest non-zero magnitude at 50 Hz, but also other frequencies in the DFT output will have non-zero magnitudes.

For your last question, the basis functions are exactly the sinusoids (cosine functions to be specific) with frequencies matching the DFT output frequencies. So, if you have an input size of N=10 samples, you will have N=10 basis functions. All (infinitely many) other potential sinsusoids are ignored as they are outside the basis set. So, you can kind of think of them as having zero coefficients, but it’s better to think of the ones that don’t match the DFT analysis frequencies as not being a part of that specific basis function set (determined by the input size and the sampling rate). In that case you only have to deal with N functions rather than an infinite number!

cheers,
Catherine

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