# Singularity Signals

1. Step signal,
2. Ramp signal, and
3. Impulse signal,
(a) Unit step; (b) Step function of magnitude K

(c) Shifted unit step function;

(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.

(a) Unit ramp function;

(b) Ramp function;

(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    and the resulting function is known as impulse.

It is defined as δ(t) = 0 for t ≠ 0
and
Also, δ(t) =

(a) Generation of impulse function from gate function (b) Impulse Signal