Singularity Functions

1
Singularity Functions

• Singularity functions are either not finite or
  don't have finite derivatives everywhere
• The two singularity functions of interest here
  are
• (1) unit step function, u(t)
• and its construct the gate function
• (2) delta or unit impulse function, ?(t)
• and its construct the sampling function

2
Unit Step Function, u(t)

• The unit step function, u(t)
• Mathematical definition
• Graphical illustration

3
Extensions of the Unit Step Function

• A more general unit step function is u(t-a)
• The gate function can be constructed from u(t)
• a rectangular pulse that starts at t? and ends
  at t ? T
• like an on/off switch

u(t-?) - u(t- ?-T)
4
Delta or Unit Impulse Function, ?(t)

• The delta or unit impulse function, ?(t)
• Mathematical definition (non-pure version)
• Graphical illustration

5
Extensions of the Delta Function

• An important property of the unit impulse
  function is its sampling property
• Mathematical definition (non-pure version)

6
The Dirac delta (impulse) function can be
loosely thought of as a function on the real line
which is zero everywhere except at the origin
Impulse function is also constrained to
satisfy the identity