# Singularity Signals

- Step signal,
- Ramp signal, and
- 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*< 0and 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*< 0A ramp function of any slope

*K*is defined as*f*(

*t*) =

*Kr*(

*t*) =

*Kt*for

*t*≥ 0

= 0 for

*t*< 0A 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*≠ 0and

Also,

*δ*(*t*) =**(a) Generation of impulse function from gate function (b) Impulse Signal**