Q. Explain Sampling and Pulse Modulation?

In most analog circuits, signals are processed in their entirety. However, in many modern electric systems, especially those that convert waveforms for processing by digital circuits, such as digital computers, only sample values of signals are utilized for processing. Sampling makes it possible to convert an analog signal to discrete form, thereby permitting the use of discrete processing methods. Also, it is possible to sample an electric signal, transmit only the sample values, and use them to interpolate or reconstruct the entire waveform at the destination. Sampling of signals and signal reconstruction from samples have widespread applications in communications and signal processing.

One of the most important results in the analysis of signals is the sampling theorem, which is formally presented later. Many modern signal-processing techniques and the whole family of digital communication methods are based on the validity of this theorem and the insight it provides. The idea leading to the sampling theorem is rather simple and quite intuitive. Let us consider a relatively smooth signal x₁(t), which varies slowly and has its main frequency content at low frequencies, as well as a rapidly changing signal x₂(t) due to the presence of high-frequency components. Suppose we are to approximate these signals with samples taken at regular intervals, so that linear interpolation of the sampled values can be used to obtain an approximation of the original signals. It is obvious that the sampling interval for the signal x₁(t) can be much larger than the sampling interval necessary to reconstruct signal x₂(t) with comparable distortion. This is simply a direct consequence of the smoothness of the signal x₁(t) compared to x₂(t). Therefore, the sampling interval for the signals of smaller bandwidths can be made larger, or the sampling frequency can be made smaller. The sampling theorem is, in fact, a statement of this intuitive reasoning.