Sampling

The Sampling Theorem

The sampling theorem is a theoretical result of paramount importance for practical information transmission and processing. It states, that a band limited signal with no frequency components above a certain cut-off frequency is uniquely determined by its discrete values at equally spaced points, provided these samples are taken at a sampling rate equal to or greater than twice the cut-off frequency. The minimum sampling rate is known as the Nyquist rate.

The process of obtaining a set of samples from a continuous function of time x(t) is referred to as sampling. The samples can be considered to be obtained by passing x(t) through a sampler, which is a switch that closes and opens instanteously at the sampling instants kT. When the switch is closed, we obtain a sample x(kT). Otherwise the output of the sampler is zero. This ideal sampler is a ideal device, since in practice, it is impossible to obtain a switch that closes and opens instantaneously.
(See the textbook page 266ff.)

sampling of a continuous signal

Sampling of continuous signals

Introduction

Sampling is one of the most important topics in the domain of signals and systems. This applet explains the sampling of a continuous signal and the reconstruction of this signal from its samples. Its purpose is

To understand the process of sampling and how we can create a sampled signal.
To see the relationship between the spectrum of the continuous signal and the spectrum of the sampled signal .
To understand how to reconstruct the original signal.
To manipulate the spectrum of the sampled signal with lowpass filters and highpass filters.
To learn the sampling theorem and its application.

There are two main sections:

The first section treats the process of sampling: a selected signal is multiplied with an impuls train.
The second section treats the reconstruction of the original signal from the sampled signal: The box in the right side at the bottom show the spectrum of the sampled signal. With the selection of an appropriate filter you should find the original continuous signal.

Applet functions

This box shows the original continuous signal before sampling.

With this buttom you can select the input signal.

This box shows the impuls train. Select the sampling period T with a scrollbar and show it in a textfield.

This box shows the sampled signal (which results from the multiplication of the original signal with the impuls train).

This box shows the spectrum of the continuous signal.

Spectrum of the sampled signal. It is refeshed when you change the periode T of the impuls train, or the continuous signal.

Apply a filter to the sampled signal:

A lowpass filter: Select the bandwidth.
A bandpass filter: Select the center frequency and the bandwidth.
"filter ouput" shows the continuous signal which is the result of the filtering process.

Applet for Sampling