digital periodic sinewaves

[jagsee1972]

Hello friends,
Newbee to DSP. I have a questions to which I think i have the answer, I would just like confirmation.
In the continuous time domain sine waves and cosine waves are always periodic, however in the discrete time domain they may or may not be. In order for the discrete function to be periodic:

cos(ω(n+N) + Φ) where ω=2Πf

cos(2Πfn + Φ) then ωN = 2Π r where r is an integer multiple

This means that periodicity is obtained if and only if:

2ΠfN=2Π r

f=r/N

Doesn't this simply mean that there must be an integer multiple of the samples that make up up the cycle. For example if there needs to be 12 samples for a cycle and only 9 are taken that signal will not be periodic, however if there are 12, 24, 36 then it would be.
Although I can't see why you would take more samples then 12 in this example.

Thank you in advance.

 

[FvM]

It's not clear what you are actually asking for?

 

[jagsee1972]

My word that was a quick reply. In order for discrete time cosine wave to be periodic the discrete samples taken must be equal or an integer multiple of the cycle. I have seen the equation in a book but just wanted to know if i have interrupted it correctly.

 

[dashxdr]

I think I may have some useful insights, but I'm not sure if it will answer your question.

Sine waves have a period of 2pi, which is an irrational number. Computers can only approximate irrational numbers. Whenever you go to sample your sine wave, you're taking an approximation in the value (because computers don't have infinite precision) and you're taking an approximation in the moment in time of that value. Or perhaps there's only one approximation. That is, you could have a rational time, then you'd require an irrational value for sine. Or you could decide you want a rational value for sine, and then you'd need an irrational value for time.

If you have a waveform of N samples, you can only fill it with complete cycles of a sine wave. That means

waveform = sin(2*PI*i*k/N) where 0<=i<N and k is 1, 2, 3, ...

Your question seems to ask whether non-integer values of k make any sense. I don't believe they do for any fixed size waveform table.