Why is the Signal from a Discrete Fourier Transform considered periodic?

[Natalie Johnson]

https://en.wikipedia.org/wiki/Discrete_Fourier_transform

Why is the signal obtained from a DFT periodic?

The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the phrase
"X[k] (the frequency coefficients form the DFT) can be interpreted as Fourier coefficients of the periodic continuation of the signal x[n]."
Where in the equation are the infinite amount of frequency coefficients constructed?

My Guess - The N point time signal is decomposed into a set number of sinusoids (frequencies present in the time domain signal), each of these sinusoids is N points in length. Since all these sinusoids sum together to obtain the time domain signal and sinusoids are periodic, the time signal composed from them can be considered periodic... Probably wrong

Please advise

 

[BvU]

Why is the signal obtained from a DFT periodic?

See 4.4 in your own link !

Where in the equation are the infinite amount of frequency coefficients constructed?

Nowhere. But a periodic discrete signal can manage with a finite set.

Interesting topics: cutoff frequency, aliasing, folding, etc.; MIT text
Advice: try to fully understand the normal Fourier transform -- then it's a small step to DFT

 

[Natalie Johnson]

See 4.4 in your own link !
Nowhere. But a periodic discrete signal can manage with a finite set.

Interesting topics: cutoff frequency, aliasing, folding, etc.; MIT text
Advice: try to fully understand the normal Fourier transform -- then it's a small step to DFT

Is the mathematical definition in that section comparable to what I have written in my guess explanation? Because section 4.4 only shows the frequency spectrum is periodic...

Also, that mathematical explanation in section 4.4, it has k+N ... but the values of k only go from 0 to N/2 (0 to nyquist frequency), so is this including negative frequencies?

The book I have been reading is 'DSP for Engineers and Scientists' and its pretty good but it didn't explain why its periodic and went straight into time domain aliasing

 

[BvU]

The DFT works with a set of samples from a time-domain signal, not with a time-domain signal. The sampling process can't distinguish signals with a frequency higher than the sampling frequency divided by 2 from signals with a higher frequency (the aliasing effect).

The set of samples has a finite size and the DFT assumes periodic extension -- that establishes the lowest frequency in the DFT that can be distinguished. The DFT can be reversed to regenerate the original set of samples, not necessarily the original time-domain signal (unless the frequencies in that original time-domain signal are limited in range from the aforementioned lowest to sampling frequency divided by 2).

I can't really follow your 'guess'

But you can follow my advice: check out the convolution theorem and study the FT of a comb function.​

 

[BvU]

(my post #4 missed a response to some later edits in your post #3)

Because section 4.4 only shows the frequency spectrum is periodic..

Good point (my oversight ). The periodicity of ##x_n## is already mentioned under 2. Motivation and is obvious from ##x_{n+N} = x_n##.

Also, that mathematical explanation in section 4.4, it has k+N ... but the values of k only go from 0 to N/2

I clearly see 0 and N-1 as limits for ##n##, not ##k## !? Where do you see k go from 0 to N/2 ? In some other context, perhaps ?

DSP for Engineers and Scientists

My google can't find a book with that title ?

 

[Natalie Johnson]

Its from dsp guide.com. Its got massive amazon reviews on the usa site and from chapter 8 onwards its really relevant to all this. The author gives the pdf for free.

The frequency domain goes from 0 to N/2 but the author says with neg frequencies its double this and is then periodic.

I think the limits are for both k and n