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Answer:
a) periodic (N = 1)
b) not periodic
c) not periodic
d) periodic (N = 8)
e) periodic (N = 16)
Explanation:
For function to be a periodic: f(n) = f(n+N)
![a) x[n]=sin(\frac{8\pi}{2}n+1)\\\\sin(\frac{8\pi}{2}n+1)=sin(4\pi n+1)](https://tex.z-dn.net/?f=a%29%20x%5Bn%5D%3Dsin%28%5Cfrac%7B8%5Cpi%7D%7B2%7Dn%2B1%29%5C%5C%5C%5Csin%28%5Cfrac%7B8%5Cpi%7D%7B2%7Dn%2B1%29%3Dsin%284%5Cpi%20n%2B1%29)
It is periodic with fundamental period N = 1
![b) x[n]=cos(\frac{n}{8} -\pi)\\\\\frac{1}{8} N=2\pi k](https://tex.z-dn.net/?f=b%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7Bn%7D%7B8%7D%20-%5Cpi%29%5C%5C%5C%5C%5Cfrac%7B1%7D%7B8%7D%20N%3D2%5Cpi%20k)
N must be integer. So it is nor periodic
![c) x[n]=cos(\frac{\pi}{8} n^2)\\\\cos(\frac{\pi}{8} (n+N)^2)=cos(\frac{\pi}{8} (n^2+N^2+2nN)\\\\N^2 = 16 \:\:or\:\:2nN=16](https://tex.z-dn.net/?f=c%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20n%5E2%29%5C%5C%5C%5Ccos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20%28n%2BN%29%5E2%29%3Dcos%28%5Cfrac%7B%5Cpi%7D%7B8%7D%20%28n%5E2%2BN%5E2%2B2nN%29%5C%5C%5C%5CN%5E2%20%3D%2016%20%5C%3A%5C%3Aor%5C%3A%5C%3A2nN%3D16)
Since N is dependent to n. So it is not periodic.
![d) x[n]=cos(\frac{\pi }{2} n) cos(\frac{\pi }{4} n)\\\\x[n] = \frac{1}{2} cos(\frac{3\pi }{4} n) + \frac{1}{2} cos(\frac{\pi }{4} n)\\\\N_1=8\:\:and\:\:N_2=8\\](https://tex.z-dn.net/?f=d%29%20x%5Bn%5D%3Dcos%28%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20n%29%20cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20n%29%5C%5C%5C%5Cx%5Bn%5D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20cos%28%5Cfrac%7B3%5Cpi%20%7D%7B4%7D%20n%29%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20n%29%5C%5C%5C%5CN_1%3D8%5C%3A%5C%3Aand%5C%3A%5C%3AN_2%3D8%5C%5C)
So it is periodic with fundamental period N = 8.
![e) x[n]=2cos(\frac{\pi }{4} n)+sin(\frac{\pi }{8} n)-2cos(\frac{\pi }{2} n+\frac{\pi }{6} )\\\\N_1=8\:\:and\:\:N_2=16\:\:and\:\:N_3=4](https://tex.z-dn.net/?f=e%29%20x%5Bn%5D%3D2cos%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%20%20n%29%2Bsin%28%5Cfrac%7B%5Cpi%20%7D%7B8%7D%20n%29-2cos%28%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20n%2B%5Cfrac%7B%5Cpi%20%7D%7B6%7D%20%29%5C%5C%5C%5CN_1%3D8%5C%3A%5C%3Aand%5C%3A%5C%3AN_2%3D16%5C%3A%5C%3Aand%5C%3A%5C%3AN_3%3D4)
So it is periodic with N = 16.
Answer:
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements.
I think the answer is B) 1 a.m. and 5 a.m. I very sorry if it is wrong plz let me know if it is wrong bro thankx
Fossil fuels when burned let out toxins and pollution in the air. Because of that scientists and engineers work together to find ways to not be so dependent on fossil fuels so we are not polluting our air.
