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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.

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$\large{\tt LaTeX}\ ?$. –  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
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1 Answer 1

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 $10$

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