# 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?

## 3 Answers

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.

Sampling in one domain implies periodicity in the other. For example the Discrete Fourier Series (which the FFT is a special case off), requires both time and frequency domain signals to be discrete and periodic.

This really isn't a topic that can be exhaustively discussed on board like this. I would recommend spending some quality time with good text book such as this http://www.amazon.com/Digital-Signal-Processing-Alan-Oppenheim/dp/0132146355 or http://www.dspguide.com/

From my limited experience in signal processing, I know that some signals cannot be represented in a digital machine if they are in a band-limited form. The signal takes on a sequence of discrete sample values so that it takes the replicated form. So the question is how is that possible? It boils down to a simple yet very important equation : $$X(n)=sin(2πfnt_s)=Sin(2π(f+k \cdot f_s^*)nt_s)$$

While $$k$$ is an integer, this means that at any frequency $$x(n)$$ that we know represents the frequency $$f$$, can also represent sinewaves at other frequencies which is : $$f + k \cdot f_s^*$$.