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

Question

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?

Answer 1

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.

Answer 2

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 : $f + k \cdot f_s^*$.

Answer 3

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/