From Continuous Time to DiscreteTime

Most signals in real life, e.g., heartbeat, brain waves, sound, music, wireless communication waveforms, etc., are continuous-time signals. To process such a signal using DSP techniques, it must be converted into a sequence of numbers. This can be done through the process of periodic sampling with the help of a device known as an analog-to-digital converter (ADC).

Time domain

Consider an analog signal $x(t)$ as a function of time shown in the figure below. Then drawn in the figure is the spectrum of this signal.

For digital signal processing, this signal needs to be converted into a series of samples, i.e., into a discrete-time version $x[n]$. This can be accomplished by sampling $x(t)$ at regular intervals of $T_s$ seconds. This process is mathematically represented as:

$$\begin{equation*} x[n] = x(t)\bigg| _{t=nT_S} \end{equation*}$$

The time interval denoted by $T_s$ ...