I am running through my old Digital Signal Processing textbook in preparation for a course in DSP, and in the following example from my textbook, I am not sure why: cos((5pi*n)/2 = cos(2*pin + pi(n/2)) = cos(pi*(n/2))

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Can someone explain why x1n = x2n? Is there some trigonometry that I am not getting?

  • 3
    \$\begingroup\$ it's because \$ \cos(2\pi n + \theta) = \cos(\theta) \$ for any real \$\theta\$ and any real integer \$n\$. the sine and cosine functions are periodic with period \$2 \pi\$. \$\endgroup\$ – robert bristow-johnson Aug 23 '16 at 22:01
  • \$\begingroup\$ The trick here is undersampling of higher frequency , you get the difference frequency which would be the same result as sampling that difference frequency. Thus either input frequency would result in identical outputs. This also assumes no "Shannon" LPF was used in this hypothetical question. This is intermodulation or "aliasing" \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Aug 23 '16 at 22:07
  • \$\begingroup\$ Because 0 degrees = 360 degrees. \$\endgroup\$ – Brian Drummond Aug 23 '16 at 23:58

They're exactly the same point, \$x=\pi/2\$ is the same point on the circle as \$x=5\pi/2\$, you've just done one extra rotation to get there


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