SME
Signals & Systems · Problem type
Time shift and scaling
Use this for “x(t) peaks at t = 2 s. Where does y(t) = x(2t − 4) peak?”
Chart
(2,
1)
on
x
moves
to
(3,
1)
on
y:
t + 4
2
This is an example. Type your own values to start a new problem.
y(t)
=
x(
a
t
−
b (s)
)
a
×
t₁ (s)
−
b (s)
=
t₀ (s)
x peaks at t₀. y at time t reads x at a × t − b, so y peaks at the t₁ where a × t₁ − b = t₀: t₁ = (t₀ + b) ÷ a.
Formulas
a
×
t₁
−
b
=
t₀
2 × 3 − 4 = 2
Assumptions
•
y(t) = x(at − b): shift x by b, then scale time by a; a > 1 squeezes and 0 < a < 1 stretches.
•
A negative a also flips the signal in time, as x(−t) does.
•
The picture draws x as a pulse peaking at t₀; any feature of x moves the same way.
Step-by-step
WE KNOW
t₀ = +2 s, a = +2, b = +4 s
FIND
time on y (t₁)
Step 1 · Find time on y (t₁)
Use a × t₁ − b = t₀
y
at
t₁
reads
x
at
at₁
−
b
,
so
set
that
to
t₀
:
add
b
,
then
divide
by
a
.
t₁
=
t₀
+
b
a
t₁
=
2 + 4
2
t₁
=
6
2
t₁
=
+3
s
Check
Put the numbers back in. Both sides should match.
2 × 3 − 4 = 2 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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