Classification of signals
Signals are represented in various forms, but the most important method of representation is how a particular signal is classified.
Based on different features, there are five methods by which signals are classified.
1. Continuous-time and discrete-time signals
In this method, signals are classified based on how they are defined as a function of time. A signal x(t) is said to be a continuous-time signal if it is defined for all time t.
It arises naturally when a physical such as an acoustic wave or a light wave is converted into an electrical signal. This conversion is effected by a transducer. For example a microphone, and a photocell.
On the other hand, a discrete-time signal is defined only at discrete instants of time. Therefore, the independent variable has discrete values only that are uniformly spaced. It is oftentimes derived from a continuous-time signal by sampling at a uniform rate.
So, if T denotes the sampling period and n denotes an integer that can assume positive and negative values, then sampling a continuous-time signal x(t) at time t=nT, gives a sample with the value x(nT5). For better understanding it can be presented as
x[n]=x(nT5), where n=0,+-1, +-2, and so on.
Discrete time signal is represented as a sequence of numbers…x[-2], x[-1], x[0], x[1], x[2],…, which has continuous values. This sequence of number is called time series.
2. Even and odd signals
A continuous signal is said to be an even signal if
x(-t)=x(t) for all t.And it is said to be an odd signal if
x(-t)=-x(t) for all t.
3. Deterministic and random signals
Signals that have are certain concerning its value at any time is known to be the deterministic signal. Deterministic signals may be modeled as completely specified functions of time.
On the other hand, a random signal is one about which there is uncertainty before it occurs. These signals may be viewed as belonging to an ensemble or a group of signals. The ensemble of signals is known as a random process. An example of a random signal is the noise generated by the amplifier of a radio or television.
Its amplitude fluctuates from being positive to negative in a completely random way. Another example can be the signal received in a radio communication system.
4. Energy signal and power signal
In electrical systems, a signal may be represented as voltage or current. Let us assume a voltage v(t) developed across a resistor R, producing current i(t). The instantaneous power dissipated in the resistor will be given by
p(t)=v^2(t)/R. It equals p(t)=Ri^2(t).
The instantaneous power in both cases is proportional to the square of the amplitude of the signal.
Moving forward, for the resistance R of 1 ohm, the equations i and ii will take on the mathematical form.
Accordingly, in signal analysis, power is defined in terms of a 1-ohm resistor. Therefore, regardless of whether a given signal represents a voltage or a current, we may express the instantaneous power p(t) as,
p(t)=x^2(t).
A signal is referred to as an energy signal if it satisfies the following condition:
0<E<infinity.
And, a signal is referred to as a power signal if it satisfies the following condition:
0<P<infinity.
Basic operations on signals
One of the most important topics of signals and systems is to understand how systems are used to process and manipulate signals.
To understand that, we need to go through a few basic operations, and also identify two classes of operations.
1. Operations performed on dependent variables
- Amplitude scaling
Let x(t) denote a continuous-time signal. And y(t) represents the result of amplitude scaling applied on x(t), will be defined as:
y(t)=cx(t).
- Addition
Let x1(t) and x2(t) denote a pair of continuous time signals. Then the result y(t) is obtained by:
y(t)=x1(t) + x2(t).
- Multiplication
Let x1(t) and x2(t) denote a pair of continuous-time signals. Then the result y(t) is obtained by:
y(t)=x1(t)x2(t)
- Differentiation
Let x(t) denote a continuous-time signals. Then the result y(t) is obtained by:
y(t)=dx(t)/dt.
2. Operations on an independent variable
- Time scaling
Let x(t) denote a continuous-time signals. Then the result y(t) is obtained by time scaling an independent variable, with time t, by a factor ‘a’ is defined by:
y(t)=x(at).
If a>1, the signal y(t) is compressed version of x(t).
If 0<a<1, the signal y(t) is expanded version of x(t).
Reference