The Dirac Impulse & Sifting Property
Interactive Analog Signal Processing (ECE 210)
Test Signal
f(t)
Cosine Wave
Triangle Wave
Exponential Decay (causal)
Unit Step
Pulse Width
ε
As width approaches 0, the height (1/ε) increases to maintain an area of 1. At 0, it becomes the ideal Dirac impulse.
0.50
Impulse Location
t
0
1.0
Integration Lower Limit
a
-2.0
Integration Upper Limit
b
3.0
Presets
Ideal Sifting
Outside Window
Pulse Approx.
Boundary Case
Enter Challenge Mode
Challenge:
Evaluate the integral exactly with
ε → 0
.
Check
f(t)
p
ε
(t-t
0
)
Window
[a,b]
Overlap
∫
a
b
f(t) p
ε
(t - t
0
) dt =
0.000
lim
ε → 0
∫
a
b
f(t) p
ε
(t - t
0
) dt = ∫
a
b
f(t) δ(t - t
0
) dt =
0.000
Endpoint convention: an impulse at
a
or
b
contributes one-half of
f(t
0
)
, matching the symmetric-pulse limit.