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- Laplace Transforms
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- Z-Transforms (ZT)
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- Signals and Systems - Resources
- Signals and Systems - Discussion
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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 (t) can be expressed as the sum of its even function e(t) and odd function o(t).
(t ) = e(t ) + 0(t )
where
e(t ) = ½[(t ) +(-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.
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