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I am just begining learning about DFT and I am bit unsure on what is happening mathematically during DFT.

I've sampled a signal , a sine wave $sin(1000\cdot 2\pi t)$. And performed DFT to calculate the $C_k$-coefficients, by plotting the $C_k$ coefficients with different $k$ values.

The frequency resolution is $8000/205$ where $205$ is the amount of datapoint/samples and $8000$ sample frequency.

I can by using the plot read, where $|C_k|$ is highest and then using my frequency resolution, determine the frequency.

I see that the amount of samples have a lot to say about determining the frequency. If I use a lower amount of samples, or does not use all the samples, will the estimated frequency either be lower, or higher. Why does is do that?

ex.

k := Round[1/((8000/100)/1000)]
freqres := 8000/100 // N
estimated freq = 468.293

k := Round[1/((8000/205)/1000)]
freqres := 8000/205// N
estimated freq = 992

k := Round[1/((8000/500)/1000)]
freqres := 8000/500 
estimated freq = 992
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