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I have a sequence $x[n]=\sin(\pi n+2)+\cos(2n/3 +1) $ I want to find its period and I have also attached my working I have two confusions

  1. Is there any effect of the constant term on period? constant"2"in bracket of sine and constant "1"in bracket of cos?

2)If period of sine term is a rational number,as shown in attached snapshot but period of cos term is irrational, what will be the overall period of composite signal x[n]?

enter image description here

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  • $\begingroup$ The sum is not a periodic signal benches the second term is not periodic; only the first term is periodic. $\endgroup$ – Dilip Sarwate Mar 3 at 15:42
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You need to work this from first principles. Strictly speaking, some $x[n]$ is periodic if, for some $N$, $x[n] = x[n + N]$ for all values of $n$. A looser definition would allow for "fractional $n$", i.e. someone might say that $x[n] = \cos n$ has a period over $N = 2\pi$ -- this would be correct and useful for some problems, and useless for others. If I were being strict, I'd say "quasi-periodic".

Since sine waves are periodic, periodicity of a sum of sines is easy to spot. Let $x[n] = \cos \theta_1 n + \phi_1 + \cos \theta_2 n + \phi_2$.

The first thing to note is that -- because the sine wave is periodic -- the $\phi_k$ don't matter. They're just phase shifts, but they don't change the period.

The second thing to note is that, in general, the first sine wave has a period of $2\pi\theta_1$, and the second has a period of $2\pi\theta_2$. So the overall $x[n]$ must repeat at the moment when both $\cos \theta_1 n + \phi_1$ and $\cos \theta_2 n + \phi_2$ repeat.

So does, and if so when does, it happen? (pause here and answer this yourself, if you can).

If you write out the problem as $x[n] = x[n + N]$ and solve for $N$, you'll be ahead, because you're out of wacky engineering and into high-school trigonometry. With DSP, you should always look for opportunities to reduce the problem to high-school math. They abound, and you can unconfuse yourself (at the cost of a lot of pencil lead) by doing so. Let

$$\cos \theta_1 n + \phi_1 + \cos \theta_2 n + \phi_2 = \cos \theta_1 (n + N) + \phi_1 + \cos \theta_2 (n + N) + \phi_2$$

Because this involves two distinct sine waves, you can separate the problem into two:

$$\cos \theta_1 n + \phi_1 = \cos \theta_1 (n + N) + \phi_1 $$ $$\cos \theta_2 n + \phi_2 = \cos \theta_2 (n + N) + \phi_2$$

By inspection, the first is periodic if $\theta_1 = 2 \pi \frac{p_1}{q_1}$ -- because unless $\theta_1$ is a rational factor of $\pi$ then the phase will never return to exactly the same as it was. By the same token, $\theta_2 = 2 \pi \frac{p_2}{q_2}$.

To find the perodicity of the overall function, just find $N$ where $\theta_1 (n + N) = \theta_2 (n + N) + 2 \pi k$. Substitute in $\theta_1 = 2 \pi \frac{p_1}{q_1}$ and $\theta_2 = 2 \pi \frac{p_2}{q_2}$, and do the math.

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    $\begingroup$ With DSP, you should always look for opportunities to reduce the problem to high-school math what..? ;-) $\endgroup$ – Fat32 Mar 3 at 16:24

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