I think I have understand how to mathematically and conceptually test for periodicity for sine and cosine waves. All continuous time sine waves are periodic.

\(f(t) = sin(2\pi f (t + T)\)

\(f(t) = sin(2\pi f t + 2\pi f T\)

\(f(t) = sin(2\pi f t) cos (2\pi f T) + sin(2\pi f T) cos (2\pi f t)\)

The sine term on the RHS will equal 0 as fT is always going to be one. This leaves the left hand side only.

\(f(t) = sin(2\pi f t) cos (2\pi f T)\)

The cosine term is equal to 1, therefore that proofs that the sinewave is periodic.

\(f(t) = sin(2\pi f t)\)

For the discrete version is not always periodic.

\(f(t) = sin(2\pi f (n + N)\)

\(f(t) = sin(2\pi f n + 2\pi f T\)

\(f(t) = sin(2\pi f n) cos (2\pi f N) + sin(2\pi f N) cos (2\pi f n)\)

For the discrete sinewave to be periodic f*N for the sine function on the RHS has to be an integer value i.e. there has to be at least samples for the entire band of vision 2pi.

I have a couple of questions:

1) For the function \(f(t) = sin(2\pi \frac{5}{8} t)\)

The check for periodicity is: \(\frac{2\pi \frac{5}{8}}{2\pi}\)

The function is periodic because the result is a rational number. the frequency is 1/8 Hertz and the periodic time is 8. What is the unit of T as there is no time. Also unless the numerator of the rational fraction is not a multiple of the denominator is right to just ignore it in the discrete domain.

2) If a function is not periodic is it true that its no good for DSP as taking the FT of it would be doing so with missing information which is an error.

Your advice and assistance will be greatly appreciated.