Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b.

The question is: Determine the fundamental period of

$x[n] = cos(0.6n\pi + 0.3\pi)$

Since x[n], is in square brackets, this means discrete time, not continuos time, thus the answer must be an integer.

The test is: $cos(0.6n\pi + 0.3%pi)$ = $cos(0.6(n+Nk)\pi + 0.3\pi)$

Where N is a positive integer (and the value we seek) and k is any integer.

Where N is the answer.

Step 1: Inverse cos() both sides

$0.6n\pi + 0.3\pi$ = $0.6(n+Nk)\pi + 0.3\pi$

step 2: Simplify, remove common term $0.3\pi$ from both sides And expand the () on the LHS

$0.6n\pi$ = $0.6n\pi + 0.6Nk\pi$

step 3: Remove $0.6\pi$, it is a common common terms:

$0$ = $0.6Nk\pi$

Step 4: The 0 on the LHS is really problematic, but we can also write it as $2\pi$ because $0$ equals $2\pi$

Thus $2\pi$ = $0.6Nk\pi$

step 5: Remove $\pi$ from both sides

$2$ = $0.6Nk$

$Nk$= $2/0.6$, or $3.33333$ repeating

One of these two (N or k) must be a non-integer.

Or is this problem misleading, in that this signal is periodic in the continuous domain, and aperiodic in the discrete domain.

share|cite|improve this question
$\large{\tt LaTeX}\ ?$. – Felix Marin Sep 29 '13 at 0:25
yea, I just figured out how to do that here. – duane Sep 29 '13 at 1:17
Also, if $\cos x = \cos y$, then not only $ x = y + 2\pi k$ possible, but $x = -y + 2\pi m$ is possible too – Evgeny Oct 2 '13 at 4:08

I don't know why you make it so complicated. For any sinusoid of the form $\cos( \omega t + a )$ the fundamental period is $T= 2\pi /\omega$ . And any integer multiple of this is also a period, i.e $ k \, 2 \pi / \omega$

For $cos(0.6 \pi n+0.3 \pi)$ we have $T=k \, 2/0.6= k \, 10/3$. This is not an integer for $k=1$, but it is for $k=3$ Then the fundamental discrete period is $10$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.