Three of the following signals represent valid discretetime

Three of the following signals represent valid discrete-time Fourier transforms, and them of them do not. Which three are valid, and which three are invalid? Justify your answer.

Three of the following signals represent valid discrete-time Fourier transforms, and them of them do not. Which three are valid, and which three are invalid? Justify your answer. (a) (b) X(eju) sin (w/2) (c) (d) X (e jw) sin(2w) (e)

Solution

For discrete time fourier transform to be valid it must be periodic.

As seen signal (b), (c), (d) are pure sinusoidal form, hence they are periodic and hence valid DTFT function

Whereas signal (a), (e) are summation of sinusoidal and polynomial. so they are not periodic. Hence Invalid DTFT

Signal (f) is impulse function which also non-periodic hence invalid DTFT