SME
Signals & Systems · Problem type
Energy and power of a signal
Use this for “Find the energy of x(t) = 2e⁻⁴ᵗu(t) and the average power of 2 cos(10t + 30°).”
Chart
Shaded: ∫ from 0 to 0.75 of x²(t) dt = 0.4988
This is an example. Type your own values to start a new problem.
E (J)
=
(
A (V)
)
2
÷
(2
×
a (s⁻¹)
)
P (W)
=
(
A (V)
)
2
÷
2
Formulas
E
=
A
2
2 ×
a
0.5
=
2
2
2 × 4
P
=
A
2
2
2
=
2
2
2
Assumptions
•
Energy and power are normalized: the signal is a voltage across 1 Ω, so the instantaneous power is x(t)².
•
For Ae−atu(t) with a > 0, E = ∫ A²e−2at dt from 0 to ∞ = A² ÷ (2a); its average power is 0.
•
For A cos(ωt + φ), cos² averages ½ over a period, so P = A² ÷ 2 for any ω and φ; its energy is infinite.
Step-by-step
WE KNOW
A = 2 V, a = 4 s⁻¹
FIND
pulse energy (E), sinusoid power (P)
Step 1 · Find pulse energy (E)
Use E = A² ÷ (2 × a)
x
2
=
A
2
e
−2at
starts
at
A
2
and
falls
with
rate
2
a
,
so
the
area
under
it
is
A
2
2
a
.
E
=
2
2
2 × 4
E
=
2
2
8
E
=
4
8
E
=
0.5
J
Step 2 · Find sinusoid power (P)
Use P = A² ÷ 2
x
2
=
A
2
cos²(ωt
+
φ)
swings
between
0
and
A
2
,
averaging
half
of
A
2
.
P
=
2
2
2
P
=
4
2
P
=
2
W
Check
Put the numbers back in. Both sides should match.
0.5
=
2
2
2 × 4
✓
2
=
2
2
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Continuous and discrete signals
Main lesson
Time shift and scaling
Use this for “x(t) peaks at t = 2 s. Where does y(t) = x(2t − 4) peak?”
Period of a discrete cosine
Use this for “Find the fundamental period of x[n] = cos(3πn ÷ 4).”
Linear and time-invariant systems
Use this for “Is the system y(t) = t·x(t) linear? Is it time-invariant?”