SME
Signals & Systems · Problem type
Step response of a difference equation
Use this for “Find y[3] and the final value when y[n] = 0.5y[n − 1] + x[n] and x[n] is the unit step.”
Diagram
This is an example. Type your own values to start a new problem.
y[n]
=
α
y[n
−
1]
+
x[n]
y[
n
]
=
(1
−
α
+
1
)
÷
(1
−
α
)
=
y[n]
y[∞]
=
1
÷
(1
−
α
)
=
y[∞]
Each sample adds the next power of α: y[0] = 1, y[1] = 1 + α, so y[n] = 1 + α + … + αⁿ, a geometric sum.
Formulas
y[
n
]
=
1 −
α
n
+ 1
1 −
α
1.875
=
1 −
0.5
3 + 1
1 − 0.5
y[∞]
=
1
1 −
α
2
=
1
1 − 0.5
Assumptions
•
The input is the unit step u[n], and the output starts from rest: y[−1] = 0.
•
H(z) = z ÷ (z − α) has one pole at z = α; |α| < 1 puts it inside the unit circle, so y settles.
Step-by-step
WE KNOW
α = 0.5, n = 3
FIND
output at n (y[n]), final value (y[∞])
Step 1 · Find output at n (y[n])
Use y[n] = (1 − αⁿ⁺¹) ÷ (1 − α)
The
geometric
sum
1
+
α
+
…
+
αⁿ
in
one
line.
y[
n
]
=
1 −
0.5
3 + 1
1 − 0.5
y[
n
]
=
1 −
0.5
4
1 − 0.5
y[
n
]
=
1 − 0.0625
0.5
y[
n
]
=
0.9375
0.5
y[
n
]
=
1.875
(y[0]
to
y[3]:
1,
1.5,
1.75,
1.875)
Step 2 · Find final value (y[∞])
Use y[∞] = 1 ÷ (1 − α)
The
final
value
is
H(1)
=
1
1 −
α
,
the
whole
sum
of
the
powers
of
α
.
y[∞]
=
1
1 − 0.5
y[∞]
=
1
0.5
y[∞] = 2
Check
Put the numbers back in. Both sides should match.
1.875
=
1 −
0.5
3 + 1
1 − 0.5
✓
2
=
1
1 − 0.5
✓
Topic check
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