How do the Frequency, Sample Rate and Duration affect the DFT of a Sinusoid?
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- čas přidán 14. 09. 2020
- Uses an example to show how the Frequency (f), Sample Rate (1/T), and Sample Length (L) affect the Discrete Fourier Transform (DFT) of a Sinusoid.
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We had a discrete cosine time signal with 1 period/cycle = 6 samples. Could you please show us how to sketch the spectrum, if 12 samples are taken to calculate the DFT.
This is exactly the same as what is shown at the 3:07 min mark of the video, except that in the video we have 100 samples per period, and we started showing a time domain vector that was one period long, and then showed one that was 10 periods long. in your case you are asking about 6 samples per period, and 2 periods long - which is just a scaled version of what is shown in the video. Hopefully you can work it out from that.
@@iain_explains Thank you so much. Your explanations are very clear. Could you make a video for explaining how to sketch the results if the time signal was weighted with a Hamming window, Hanning or Blackman window function prior to DFT.
@@sarithasukumaran5940 Thanks for the suggestion. I'll add it to my "to do" list. You could try it for yourself if you like. I made the Matlab code for this video available on my website iaincollings.com - you'll find this video listed under the heading "Discrete Fourier Transform", and click on the PDF link, then you can apply your own filters.
Dear Professor
For first example:
Freq: 8hz, SampTime = 2.4sec, SampRate = 100sps
What is samples for period (N) = ?
at which M value we observe spike.
I got N = 25.
Please explain
Good question. I should have gone back and explained the first scenario I showed. Samples per period = 100/8 = 12.5 and using the formulas shown in the video, 1/12.5 = (m-1)/(Length of the vector) and the Length of the vector is 2.4 x 100 so therefore m=20.2 so the peak of the FFT will be in the 20th element (which you can confirm by running the code for yourself which you can find at iaincollings.com on the Signals and Systems page, under the DFT heading.) You might also find this video helpful: "How does the Discrete Fourier Transform DFT relate to Real Frequencies?" czcams.com/video/pIFz84oj9cA/video.html
@@iain_explains Thank you Professor for your explanation. Can N be a fraction here (12.5) ? I believe N (samples per period) should be an integer always right in descrete time?
True, technically I shouldn't have said m=20.2, since m is an integer, yes. What I should have said was that the 20th element of the vector corresponds to a frequency of 2pi(20-1)/(2.4x100), which is the closest (of all the discrete frequencies in that vector) to the actual signal's frequency, which is 2pi/12.5 , so you can expect that the spike will be centred on m=20, but it will be spread out into neighbouring elements of the vector too, because that's what happens whenever your total sample length does not exactly equal an integer multiple of the signal's wavelength ... but I thought it would be OK to give the shorter (less precise) answer ... but I can see now that it was a bit confusing. Hopefully this more complete answer helps.
@@iain_explains Sorry to bother again professor, But thank you for the great explanation. My query is regarding "N" value, but not "m".
N is the samples per period right. Here N values comes 12.5 . But N should be an integer right. ? in that case may we consider N to be (12.5 *2) = 25 ? ...
for the same signal ==> Freq: 8hz, SampTime = 2.4sec, SampRate = 100sps.
x(t) = x(T+ N). Considering N to be integer, N = 25. please correct me if i am wrong
No, N is the length of the vector in the time domain. It is an integer. It's the number of samples of the signal that have been recorded/stored.