lecture5

CEE 572: Lecture 5 Impulse Response Functions for Rational Systems Jeff Scruggs Fall 2023

Dirac Delta Functions I Consider a family of functions Δ( t , T ) , as shown below T T t 2 2 1 T − Δ ( t , T )

Dirac Delta Functions I Consider a family of functions Δ( t , T ) , as shown below T T t 2 2 1 T − Δ ( t , T ) I Define the Dirac delta function δ ( t ) as the limit δ ( t ) = lim T → 0 Δ( t , T ) Also called a Dirac impulse or a unit impulse function

Dirac Delta Functions I Consider a family of functions Δ( t , T ) , as shown below T T t 2 2 1 T − Δ ( t , T ) I Define the Dirac delta function δ ( t ) as the limit δ ( t ) = lim T → 0 Δ( t , T ) Also called a Dirac impulse or a unit impulse function I Properties of δ ( t ) : I δ ( t ) = ∞ : t = 0 0 : t 6 = 0 I R ∞ -∞ δ ( t ) dt = R 0 + 0 - δ ( t ) dt = 1

Impulse Response Functions I Dirac delta functions are usually depicted as arrows: t 0 δ ( t )