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Question Number 202771 by ali009 last updated on 03/Jan/24

determine whether this signal is periodic or not if it′s periodic specify it′s faundemwntal period y(n)=cos((n/8))cos(((nπ)/8))

$${determine}\:{whether}\:{this}\:{signal}\:{is}\:{periodic} \\ $$$${or}\:{not}\:{if}\:{it}'{s}\:{periodic}\:{specify}\:{it}'{s}\: \\ $$$${faundemwntal}\:{period} \\ $$$${y}\left({n}\right)={cos}\left(\frac{{n}}{\mathrm{8}}\right){cos}\left(\frac{{n}\pi}{\mathrm{8}}\right)\: \\ $$

Answered by Frix last updated on 03/Jan/24

It′s not periodic. To see what happens: π≈3 cos (n/8) cos ((3n)/8) ⇒ period is 8π π≈((22)/7) cos (n/8) cos ((11n)/(28)) ⇒ period is 112π π≈((355)/(113)) cos (n/8) cos ((355n)/(904)) ⇒ period is 904π ... cos (n/8) cos ((nπ)/8) ⇒ period is ∞

$$\mathrm{It}'\mathrm{s}\:\mathrm{not}\:\mathrm{periodic}.\:\mathrm{To}\:\mathrm{see}\:\mathrm{what}\:\mathrm{happens}: \\ $$$$\pi\approx\mathrm{3} \\ $$$$\mathrm{cos}\:\frac{{n}}{\mathrm{8}}\:\mathrm{cos}\:\frac{\mathrm{3}{n}}{\mathrm{8}}\:\Rightarrow\:\mathrm{period}\:\mathrm{is}\:\mathrm{8}\pi \\ $$$$\pi\approx\frac{\mathrm{22}}{\mathrm{7}} \\ $$$$\mathrm{cos}\:\frac{{n}}{\mathrm{8}}\:\mathrm{cos}\:\frac{\mathrm{11}{n}}{\mathrm{28}}\:\Rightarrow\:\mathrm{period}\:\mathrm{is}\:\mathrm{112}\pi \\ $$$$\pi\approx\frac{\mathrm{355}}{\mathrm{113}} \\ $$$$\mathrm{cos}\:\frac{{n}}{\mathrm{8}}\:\mathrm{cos}\:\frac{\mathrm{355}{n}}{\mathrm{904}}\:\Rightarrow\:\mathrm{period}\:\mathrm{is}\:\mathrm{904}\pi \\ $$$$... \\ $$$$\mathrm{cos}\:\frac{{n}}{\mathrm{8}}\:\mathrm{cos}\:\frac{{n}\pi}{\mathrm{8}}\:\Rightarrow\:\mathrm{period}\:\mathrm{is}\:\infty \\ $$

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