Signal Processing/Signals

Introduction:

Before we dive into the more rigorous mathematical and physical definition of signals, let us try to understand what is meant by signals. A signal is any entity that brings about a reaction or a change. When we speak, a signal is created in the form of the pressure pulses travelling through the air. When this signal reaches someone's ears, they process it and can hear what is said. Thus the signal causes some effect. Even an image that we see is a signal, that causes perception of vision and conveys some information.

Signals, If we have to define them, we can call them as a function of time. We can think of a signal as a deliberate variation in some property of the medium used to convey the data. We have variety of signals, Light, Sound, Electronic, Electromagnetic are a few of them. A few of the examples of Signals would be;

1. An electrical voltage travelling along copper wires between your telephone and the local exchange.

2. Pulses of light (though we might not be able to see them) in a fibre-optic cable

3. The radio emissions that are picked up by a mobile telephone or radio receiver.

A signal can be mathematically expressed as $$s(t)$$, meaning it is the function of time. All these can provide the necessary variations to represent the data. In the first example we can relate the changes in voltage to changes in electrical energy. With the other examples – light and radio waves – we need to think in terms of waves of energy, usually referred to as electromagnetic radiation. Electromagnetic radiation is caused by changes in electrical and magnetic fields. Electromagnetic radiation can support signals even when there is no physical medium (such as a cable) involved/ We would be dealing with the EM Wave part later On.

Signals need not always be functions of time. In our example of image, the signal is a function of its position. A pixel in any image is specified by its x- and y-coordinates. However, most of the signals we encounter are functions of time, as they change with respect to time.

Signals can be classified according to various properties. Some of these classifications are:

1. Periodic and Non-periodic signals: A periodic signal is one which repeats after a given time. The time after which it repeats is called the period of the signal. A periodic signal can be represented as

$$ s(t+T) = s(t) $$

where T is the period of the signal.

The most common and most important example of periodic signals is the sine wave. This can be verified as follows

sin(t+2π) = sin(t)

Thus the period of a sine wave is 2π.



On the other hand, a non-periodic signal is any signal that does not repeat itself after any period of time, however large that period may be. Most of the signals we come across in real life are non-periodic. For example, speech is a non-periodic signal.

It should be bourne in mind that a true periodic signal is not possible in reality. A periodic signal demands that the signal should repeat itself after every T (time period) time. This means that the signal should exist for eternity and should have started only when time started. However, this is not possible, and thus we do not come across periodic signals in practice. When we talk of periodic signals in signal processing, or in communication in general, we usually take a reference point in time, and say time started from then (for our purpose, time may as well have assumed to start from when our experiment starts). Also, the signal is assumed to be periodic only till the experiment ends.

2. Gaussian signals

These signals are characterized by their gaussian distribution. The statistics characteristics of a gaussian signal are the followings:

kurtosis=3 skewness=0

If you need more detail about what these terms mean you can go to the Statistics Ground Zero page or to the Statistics and Distribution page.

If you choose to generate a random signal from a Power Spectral Density, you will get a gaussian signal unless you specify your machine otherwise.

3. Non-gaussian signals

Non-gaussian signals are unstationary signals.