Fourier transform of a periodic/aperiodic signal

Question

Generally speaking, I know that periodic signals (continuous-time domain signals) with period 2pi/wo have a spectrum with equidistance Delta-impulses of distance w0.

My question is that, if we have a spectrum with equidistance Delta-impulses of distance w0, does it necessarily means that the time domain signal is periodic? Or for example, if our time domain signal itself is not periodic, like a sin(wt) when w is not a rational ratio, will the spectrum be periodic?!

Also, does a one-sided spectrum with equidistance Delta-impulses of distance w0 correspond to a periodic signal in time domain?

Answer 1

Equidistant Dirac impulses in the spectrum imply a periodic time domain signal.

As pointed out in a comment, in continuous time, the signal $x(t)=\sin(\omega_0t)$ is always periodic, regardless of the value of $\omega_0$.

Your question about the spectrum being periodic is unclear to me.

A one-sided spectrum with equidistant Dirac impulses also implies a periodic time-domain signal, but that signal is necessarily complex-valued because its spectrum has no Hermitian symmetry, as required for real-valued signals.

Answer 2

@Niousha. Regarding your first paragraph, in theory a real-valued continuous-time domain sinusoidal signal, of infinite time duration, will have only one positive-frequency impulse in the frequency domain. (I say "in theory" because such a signal does not exist in Nature. Such a signal is strictly an abstract concept, like a perfect circle or one of Euclid's lines having infinite length and zero thickness, that only exists in the minds of signal processing people.)

For discrete-time signals (sequences), the answer to your questions in your 2nd & 3rd paragraphs is "No." For example generate a 23-sample sequence defined by x = sin(2*pi*3*n/23) where n = 0,1,2,3,...22. Such a sinusoidal sequence has only one non-zero positive-frequency spectral component but it is not periodic.