Sampling
Sampling is the process of converting a continuous time signal (consisting of infinite number of points) to a discrete time signal (finite points). This process enables the conversion of analog signals to digital signals that can be stored digitally on a storage device.
Sampling can be thought of taking the values of a continuous signal at discrete time instances (specified by the time period T). This can be achieved by multiplying the given continuous time signal by a train of dirac delta functions separated by the time period T. This can be mathematically represented as follows:
$ X_{S}(t) = S_{\tau}(t). X(t) $ [Eq. 1]
where $ S_{\tau}(t) = P_{T}(t) = \sum_{K=-\infty}^{\infty} \delta (t - KT) $ [Eq. 2]
Since the direc delta function is not practically possible in the time domain, we sample the continuous time signal with a repeated rect function instead (with an area equal to 1). Thus Eq. 2 becomes:
$ S_{\tau}(t) = rep_{T}\left(\frac{1}{\tau} rect\left(\frac{t}{\tau}\right)\right) $ [Eq. 3]
The analysis of $ X_{S}(t) $ in the frequency domain reveal important aspects of sampling. This has been done below:
$ X_{S}(t) = S_{\tau}(t). x(t) \longrightarrow X_{S}(f) = S_{\tau}(f) * X(f) $ [Eq. 4]
Taking the Fourier transform of Eq. 3 gives:
$ S_{\tau}(t) = rep_{T}\left(\frac{1}{\tau} rect\left(\frac{t}{\tau}\right)\right) \longrightarrow S(f) = \frac{1}{T} comb_{\frac{1}{T}}\left(sinc(\tau f)\right) $ [Eq. 5]
where $ comb_{T}\left(X(f)\right) = \sum_{K=-\infty}^{\infty} X(f) . \delta (f - KT) $ [Eq. 6]
Substituting the value of $ comb_{\frac{1}{T}}\left(sinc(\tau f)\right) $ in Eq. 5, we get:
$ S_{\tau}(f) = \frac{1}{T} \sum_{K=-\infty}^{\infty} sinc(\tau f)) . \delta (f - \frac{K}{T}) = \frac{1}{T} \sum_{K=-\infty}^{\infty} sinc(\tau \frac{K}{T}) . \delta (f - \frac{K}{T}) $ [Eq. 7]
Substituting in Eq. 4:
$ X_{S}(f) = S_{\tau}(f) * X(f) \Rightarrow X_{S}(f) = \left[\frac{1}{T} \sum_{K=-\infty}^{\infty} sinc(\tau \frac{K}{T}) . \delta (f - \frac{K}{T}) \right] * X(f) $
$ X_{S}(f) = \frac{1}{T} \sum_{K=-\infty}^{\infty} sinc(\tau \frac{K}{T}) . X(f - \frac{K}{T}) $ [Eq. 8]
The $ sinc(\tau \frac{K}{T}) $ term in Eq. 8 appears because we use the rect function (Eq. 3) instead of the ideal direc delta function (Eq. 2) in the sampling process. If, however, the ideal direc delta function is used (or in other words $ \tau \rightarrow 0 $), Eq. 8 becomes:
$ X_{S}(f) = \frac{1}{T} \sum_{K=-\infty}^{\infty} X(f - \frac{K}{T}) $ [Eq. 9]
This shows that the sampling process in time produces identical (scaled) copies of the signal in the frequency domain repeated every $ \frac{1}{T} $. Because of this fact, the signal can be reconstructed back by passing it through a low-pass filter with a cut-off frequency of $ \frac{1}{2T} $ and a gain of $ T $.
