Introduction to Signals and Systems

Definition of Signal

In a communication system, the word ‘signal’ is commonly used. Therefore we must know its exact meaning.

• Mathematically, signal is described as a function of one or more independent variables.
• Basically it is a physical quantity. It varies with some dependent or independent variables.
• So the term signal is defined as “A physical quantity which contains some information and which is function of one or more independent variables.”

Classification of Signals:

There are various types of signals. Every signal has its own characteristics. The processing of signals mainly depends on the characteristic of that particular signal. So classification of signal is necessary. Broadly the signals are classified as below:

• Continuous and discrete time signals
• Continuous valued and discrete valued signals
• Periodic and non-periodic signals
• Even and odd signals
• Energy and power signals
• Deterministic and random signals
• Multichannel and multidimensional signals

Continuous and discrete time signals:

Continuous signal:

A signal of continuous amplitude is called continuous signal or analog signal. Continuous signal has some value at every instant of time.

Examples:

Sine wave, cosine wave, triangular wave etc. similarly some electrical signals derived from physical quantities like temperature, pressure, sound etc. are also an examples of continuous signals.

Examples of continuous signals

Mathematical expression:

Mathematically a continuous signal can be expressed as,

x(t)=A sin(wt+?)

Here A= amplitude of signal

w = angular frequency=2pf

?= phase shift

Characteristics:

• For every fix value of t, x(t) is periodic in nature.
• If the frequency (1/t) is increased then the rate of oscillation also changes.

Discrete time signal:

In this case the value of signal is specified only at specific time. So signal represented at “discrete interval of time” is called as discrete time of signal.

The discrete time signal is generated from continuous time signal by using the sampling operation. This process is shown in figure below.

Example of Discrete time signal

• Consider a continuous analog signal as shown in figure a). This signal is continuous in nature from –infinity to +infinity.
• The sampling pulses are shown in figure b). These are train of pulses. Here the samples are taken with Ts as sampling time.
• Figure c) shows the discrete time signal
• For signal shown in figure a), the expression is x(t)=A cos (wt)
• And for signal shown in fig c) , the expression is x(t)= A cos (wn)

Characteristics:

• Discrete time sinusoidal signals are identical when their frequencies are separated by integer multiple of 2p.
• If the frequency of discrete time sinusoidal is rational number, then such signal is periodic in nature.
• For the discrete time sinusoidal, the highest oscillation is obtained when angular frequency w=+p or –p.

Continuous valued or discrete valued signals:

Continuous valued signals:

If the variation in the amplitude of signal is continuous then, it is called continuous valued signal. Such signals may be continuous or discrete in nature. Following figure shows the examples of continuous valued signals.

Examples of continuous signals

Example of Discrete time signal

Discrete valued signals:

If the variation in the amplitude of signal is not continuous but the signal has certain discrete amplitude levels then such signal is called as discrete valued signal. Such signal may be again continuous or discrete in nature as shown in figure below.

Discrete amplitude signal continuous in nature

Discrete amplitude signal discrete in nature

Periodic and Non-periodic signals:

Periodic signal:

A signal which repeats itself after a fixed time period or interval is called as periodic signal. The periodicity of continuous time signal can be defined mathematically as,

x(t)=x(t+T0)

This is called as condition of periodicity. Here T0 is called as fundamental period. That means after this period signal repeats itself.

For the discrete time signal, the condition of periodicity is,

x(n)=x(n+N)

Here number ‘N’ is the period of signal. The smallest value of N for which the condition of periodicity exists is called fundamental period.

Following figure shows the examples of periodic signals:

Periodic signals

Non-periodic signals:

A signal which does not repeat itself after a fixed time period or does not repeat at all is called as non-periodic or aperiodic signal.

In other words we can say that, the period of non-periodic signal is infinity.

Following figure shows the non-periodic signal:

Non-periodic signal

Even and Odd Signals:

Even signals:

An even signal is also called as symmetrical signal. A continuous time signal x(t) is said to be even or symmetrical if it satisfies the following condition:

Condition for symmetry: x(t)=x(-t)……..for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. That means x(-t) is the signal which is reflected about vertical axis.

Condition for symmetry: x(n)=x(-n)…….for discrete time signal.

Following figure shows the even signal:

Even signals

Odd signal:

A continuous time signal x(t) is said to be odd signal if it satisfies following condition:

Condition for odd signal: x(-t)=-x(t)…….for continuous time signal.

Here x(-t) indicates that the signal is present for negative time period. While –x(t) indicates that the amplitude of the signal is negative. Thus odd signal is not symmetric about vertical axis.

Condition for odd signal: x(-n)=-x(n)……for discrete time signal.

Following figure shows the odd signal:

Odd signals

Note: amplitude of odd signal at origin is always zero.