# Samples of the Fourier transform of a continuous-time signal

Assume that I manually compute the Fourier transform, $$X_a(f)$$, of a continuous-time aperiodic signal, $$x_a(t)$$, and then take equally-spaced samples of this Fourier transform, $$X(k) \equiv X_a(k F_0)$$. Is there a discrete-time periodic signal, $$x(n)$$, whose Fourier series coefficients satisfy the relation $$c_k = X(k)$$, where $$c_k$$ is defined as $$c_k = \frac{1}{N}\sum_0^{N-1} x(n) e^{-j2\pi kn/N}$$?

A continuous-time signal, $$x_a(t)$$, has no notion of sampling time, but when we take samples, $$X(k)$$, of its Fourier transform, $$X_a(f)$$, for computer analysis, we are discretizing the signal in frequency-domain. I am trying to understand if this frequency-domain discretization would give back a discrete-time signal, $$x(n)$$, (through inverse Fourier transform/series) that has some relation to the original continuous-time signal.