# Signal Processing – Discrete Fourier Transform and Incomplete Fourier Series

I'm working on a paper where I'm collecting sound pressure data from a chord's wave and trying to create a frequency spectrum to find the individual frequencies that make up the chord.

However, I can't process too many data points (software). To ensure a sampling rate higher than twice the highest note, I can't collect SPL data for the entire period of the chord's wave.

Would the DFT still work and pick up the individual frequencies on the frequency spectrum if not the entire chord's period is collected?

Thanks.

The DFT treats the data you collect as representing one entire period. It models a periodic function consisting of that data repeated over and over.

If you stop collecting before an entire period has elapsed, the data will model a periodic function consisting of that partial data repeated. It will not have the same shape as the real wave. It will have a higher fundamental frequency.

If you collect fewer data points spread out over the entire period, you will avoid that problem, but create another. As hyportnex said, frequency components that are high enough that you don't have two data points per cycle will be aliased. That is, they will look like a lower frequency. You will be unable to distinguish how much of the signal comes from the high frequency and the aliased low frequency.

• Thanks mmesser314. What if I have more than one period of data, but not a discrete number of periods? Say 1.5 periods? Would the DFT treat that as one entire period, or would it recognize it as more than one? Jan 16, 2023 at 16:06
• It will treat the data you give it as a period. Jan 16, 2023 at 17:09
• Hi mmesser314. Thanks for your response. I just had one more question: what if I gave the DFT say, exactly 2 periods of the same data. Would the DFT realize that there are two periods (assuming the data for each cycle is exactly the same) or still treat that as one period? Feb 4, 2023 at 23:08
• It always treats the data given to it as $1$ period. This would do no harm. If the data repeats after $1$ period, it also repeats after $2$ periods. That is to say if $T$ is the period of the data, then by definition $f(x) = f(x+T)$ for every $x$. So $f(x) = f(x+2T)$ is also true for every $x$, and $2T$ is also a period. Feb 4, 2023 at 23:43
• Thank you mmesser314. Feb 5, 2023 at 5:29

If you do not collect at least two samples per period for each and every tone that is in your signal then you will alias that particular oscillation (tone) into a lower frequency and you not only have now a wrong tone, but one that may interfere with a properly sampled tone and one that would have been otherwise correctly represented if not for being aliased (self-jammed) by something else.