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Singularity Signals

  1. Step signal,
  2. Ramp signal, and
  3. Impulse signal,
Description: Description: 1337.png
(a) Unit step; (b) Step function of magnitude K
Description: Description: 1343.png
(c) Shifted unit step function;
Description: Description: 1349.png
(d) Gate function

Step Signal

This function is also known as Heaviside unit function. It is defined as given below,
f(t) = u(t) = 1 for t > 0
= 0 for t < 0
and is undefined at t = 0.
A step function of magnitude K is defined as
f(t) = Ku(t) = K for t > 0
= 0 for t < 0
and in undefined at t = 0.
A shifted or delayed unit step function is defined as
f(t) = u(t – T) = 1 for t > T
= 0 for t < T
and is undefined at t = T.
Another function, called gate function, can be obtained from step function as follows.
Therefore, g(t) = Ku(t – a) – Ku(t – b)

Ramp Signal

A unit ramp function is defined as
f(t) = r(t) = t for t ≥ 0
= 0 for t < 0
A ramp function of any slope K is defined as
f(t) = Kr(t) = Kt for t ≥ 0
= 0 for t < 0
A shifted unit ramp function is defined as
f(t) = r(t – T) = t for t ≥ T
= 0 for t < T

Impulse Signal

This function is also known as Dirac Delta function, denoted by d(t). This is a function of a real variable t, such that the function is zero everywhere except at the instant t = 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity.
Description: Description: 1355.png
(a) Unit ramp function;

Description: Description: 1361.png
(b) Ramp function;

Description: Description: 1368.png
(c) Shifted unit ramp function
Consider a gate function as shown in figure.
The function is compressed along the time-axis and stretched a long the y-axis, keeping the area under the pulse as unity. As a  0, the value of Description: Description: 2467.png   and the resulting function is known as impulse.
It is defined as δ(t) = 0 for t ≠ 0
Also, δ(t) = Description: Description: 2508.png
(a) Generation of impulse function from gate function (b) Impulse Signal

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