# How replicas are formed in Frequency domain when a signal is sampled in Time Domain?

I know that sampling in one domian [time or frequency] gives raise to replicas in another domain[frequency / time]. How replicas are formed. What is this Time domain periodicity and frequency domain periodicity here in sampling.

-

Some people treat sampling in the time domain as the multiplication of a signal $x(t)$ by a periodic impulse train $\sum_n \delta(t-nT)$. Since the Fourier transform of a periodic impulse train is a periodic impulse train $\sum_k \delta\left(f-\frac{k}{T}\right)$ (I might be off by a factor of $T$ here, but the basic idea is correct), and multiplication in the time domain corresponds to convolution in the frequency domain, we get that the Fourier transform of the sampled signal is $$X(f)\star\sum_k \delta\left(f-\frac{k}{T}\right) = \sum_k X(f)\star\delta\left(f-\frac{k}{T}\right) = \sum_k X\left(f-\frac{k}{T}\right)$$ which is a periodic function of $f$ since $X(f)$ is replicated at intervals of $\frac{1}{T}$ Hz along the frequency axis. If $X(f)$ is bandlimited to $\left(-\frac{1}{2T},\frac{1}{2T}\right)$, the replicas do not overlap.

Other people resolutely refuse to consider impulse trains and come up with other, equally valid, means of arriving at the same result.

-