SME
Signals & Systems · Problem type
Period of a discrete cosine
Use this for “Find the fundamental period of x[n] = cos(3πn ÷ 4).”
Diagram
k
N
=
3
8
in
lowest
terms
Period: 8 samples
This is an example. Type your own values to start a new problem.
Ω₀ (rad/sample)
=
2π
×
k
÷
N
N₀ (samples)
=
N
÷
gcd(
k
,
N
)
Write the cosine’s frequency as 2π × k ÷ N: 3π ÷ 4 = 2π × 3 ÷ 8, so k = 3 and N = 8. The period is N once k ÷ N is in lowest terms.
Formulas
Ω₀
=
2π
×
k
N
3π/4
=
2π
×
3
8
N₀
=
N
gcd(
k
,
N
)
8
=
8
gcd(3, 8)
Assumptions
•
Write the frequency as Ω₀ = 2πk ÷ N with whole numbers k and N: k full cycles every N samples.
•
Put k ÷ N in lowest terms: the period N₀ is what is left of N, N ÷ gcd(k, N).
•
When Ω₀ ÷ 2π is irrational (Ω₀ = 1 rad, say), cos(Ω₀n) never repeats: it has no period.
•
Ω₀ and Ω₀ + 2π give the same samples, so k can be taken below N.
Step-by-step
WE KNOW
k = 3, N = 8
FIND
frequency (Ω₀), period (N₀)
Step 1 · Find frequency (Ω₀)
Use Ω₀ = 2π × k ÷ N
k
full
turns
of
2π
spread
over
N
samples.
Ω₀
=
2π
×
3
8
Ω₀
=
3π/4
rad/sample
(≈
2.36
rad/sample)
Step 2 · Find period (N₀)
Use N₀ = N ÷ gcd(k, N)
Cancel
the
common
factor
of
k
and
N
:
the
samples
first
repeat
after
what
is
left
of
N
.
N₀
=
8
gcd(3, 8)
N₀
=
8
samples
Check
Put the numbers back in. Both sides should match.
3π/4
=
2π
×
3
8
✓
8
=
8
gcd(3, 8)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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