periodic

[logistic_guy]

Determine which of the following sinusoids are periodic and compute their fundamental period.

\(\displaystyle \bold{(a)} \cos 0.01\pi n\)

\(\displaystyle \bold{(b)} \cos \pi\frac{30n}{105}\)

\(\displaystyle \bold{(c)} \cos 3\pi n\)

\(\displaystyle \bold{(d)} \sin 3n\)

\(\displaystyle \bold{(e)} \sin \pi\frac{62n}{10}\)

 

[logistic_guy]

\(\displaystyle \textcolor{red}{\text{Definition}}\).

A discrete-time cosine \(\displaystyle x[n] = \cos \omega n\) or a discrete-time sine \(\displaystyle x[n] = \sin \omega n\) is periodic if and only if:

\(\displaystyle \frac{\omega}{2\pi} = \frac{r}{N_p}\)

where \(\displaystyle r, N_p\) are integers with no common factors in lowest term. Then, the fundamental period is \(\displaystyle N_p\).


\(\displaystyle \bold{(a)}\)

We have \(\displaystyle \omega = 0.01\pi = \frac{\pi}{100}\), then

\(\displaystyle \frac{\omega}{2\pi} = \frac{\frac{\pi}{100}}{2\pi} = \frac{1}{200}\)

Then, it is \(\displaystyle \textcolor{blue}{\text{periodic}}\) with \(\displaystyle N_p = \textcolor{blue}{200}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)} \cos \pi\frac{30n}{105}\)

\(\displaystyle \omega = \frac{30\pi}{105} = \frac{6\pi}{21} = \frac{2\pi}{7}\)

\(\displaystyle \frac{\omega}{2\pi} = \frac{\frac{2\pi}{7}}{2\pi} = \frac{1}{7}\)

Then, it is \(\displaystyle \textcolor{blue}{\text{periodic}}\) with \(\displaystyle N_p = \textcolor{blue}{7}\)

 

[logistic_guy]

\(\displaystyle \bold{(c)} \cos 3\pi n\)

\(\displaystyle \omega = 2\pi f = 3\pi\)

\(\displaystyle f = \frac{3\pi}{2\pi} = \frac{3}{2} = \frac{r}{N_p}\)

Therefore, according to the definition:

The discrete-time signal is \(\displaystyle \textcolor{blue}{\text{periodic}}\) with \(\displaystyle N_p = \textcolor{blue}{2}\)

 

[logistic_guy]

\(\displaystyle \bold{(d)} \sin 3n\)

\(\displaystyle \omega = 2\pi f\)

\(\displaystyle f = \frac{\omega}{2\pi} = \frac{3}{2\pi} \neq \frac{r}{N_p}\)

where \(\displaystyle r, N_p \in \mathbb{Z}\).

Then, the dsicrete-time signal \(\displaystyle \textcolor{blue}{\text{non-periodic}}\) as \(\displaystyle \frac{3}{2\pi}\) is \(\displaystyle \textcolor{red}{\text{irrational}}\)

 

[logistic_guy]

\(\displaystyle \bold{(e)} \sin \pi\frac{62n}{10}\)

\(\displaystyle \omega = 2\pi f\)

\(\displaystyle f = \frac{\omega}{2\pi} = \frac{\frac{\pi 62}{10}}{2\pi} = \frac{31}{10} = \frac{r}{N_p}\)

The discrete-time signal is \(\displaystyle \textcolor{blue}{\text{periodic}}\) with \(\displaystyle N_p = \textcolor{blue}{10}\)