Signals Classification

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Signals are classified into the following categories:

• Continuous Time and Discrete Time Signals

• Deterministic and Non-deterministic Signals

• Even and Odd Signals

• Periodic and Aperiodic Signals

• Energy and Power Signals

• Real and Imaginary Signals

Continuous Time and Discrete Time Signals

A signal is said to be continuous when it is defined for all instants of time.

A signal is said to be discrete when it is defined at only discrete instants of time/

Deterministic and Non-deterministic Signals

A signal is said to be deterministic if there is no uncertainty with respect to its value at any instant of time. Or, signals which can be defined exactly by a mathematical formula are known as deterministic signals.

A signal is said to be non-deterministic if there is uncertainty with respect to its value at some instant of time. Non-deterministic signals are random in nature hence they are called random signals. Random signals cannot be described by a mathematical equation. They are modelled in probabilistic terms.

Even and Odd Signals

A signal is said to be even when it satisfies the condition x(t) = x(-t)

Example 1: t2, t4 cost etc.

Let x(t) = t2

x(-t) = (-t)2 = t2 = x(t)

$\therefore, $ t2 is even function

Example 2: As shown in the following diagram, rectangle function x(t) = x(-t) so it is also even function.

A signal is said to be odd when it satisfies the condition x(t) = -x(-t)

Example: t, t3 ... And sin t

Let x(t) = sin t

x(-t) = sin(-t) = -sin t = -x(t)

$\therefore, $ sin t is odd function.

Any function ƒ(t) can be expressed as the sum of its even function ƒ_e(t) and odd function ƒ_o(t).

ƒ(t ) = ƒ_e(t ) + ƒ₀(t )

where

ƒ_e(t ) = ½[ƒ(t ) +ƒ(-t )]

Periodic and Aperiodic Signals

A signal is said to be periodic if it satisfies the condition x(t) = x(t + T) or x(n) = x(n + N).

Where

T = fundamental time period,

1/T = f = fundamental frequency.

The above signal will repeat for every time interval T₀ hence it is periodic with period T₀.

Energy and Power Signals

A signal is said to be energy signal when it has finite energy.

$$\text{Energy}\, E = \int_{-\infty}^{\infty} x^2\,(t)dt$$

A signal is said to be power signal when it has finite power.

$$\text{Power}\, P = \lim_{T \to \infty}\,{1\over2T}\,\int_{-T}^{T}\,x^2(t)dt$$

NOTE:A signal cannot be both, energy and power simultaneously. Also, a signal may be neither energy nor power signal.

Power of energy signal = 0

Energy of power signal = ∞

Real and Imaginary Signals

A signal is said to be real when it satisfies the condition x(t) = x*(t)

A signal is said to be odd when it satisfies the condition x(t) = -x*(t)

Example:

If x(t)= 3 then x*(t)=3*=3 here x(t) is a real signal.

If x(t)= 3j then x*(t)=3j* = -3j = -x(t) hence x(t) is a odd signal.

Note: For a real signal, imaginary part should be zero. Similarly for an imaginary signal, real part should be zero.