Hello! I am having a difficult time understanding the "Nyquist Sampling Theorem." I was wondering if someone can help me understand this. Below is an example out of my book: Q: Suppose that an analog signal is given as x(t) = 5cos(2*pi*1000t), for t > 0 and is sampled at the rate of 8000 HZ a.) Sketch the spectrum for the original signal. A: They do the Euler Identity: 5cos(2*pi*1000t = (ej*2*pi*1000*t + e-j*2*pi*1000*t) / 2 5cos(2*pi*1000t)= 2.5*ej*2*pi*1000*t + 2.5*e-j*2*pi*1000*t The coefficient is c1 = 2.5 and c2 = 2.5 and they get the following graph: -------------------------------------------- According to whats in my book and my course shell, I am trying to do the example problem like it is in my course shell which is something like this: I know fs > 2* fmax x(t) = 5cos(2*pi*1000t) fs = (1 / Ts) x(nT) = 5cos((2*pi*1000*n) /fs) x(nT) = 5cos((2000*pi*n)/8000) x(nT) = 5cos((1/4)*pi*n) <-- This is where I get stuck. I don't recall the Euler Method and I don't understand how they get -1 and 1, and it shows 2.5. Am I doing this wrong and Eulor Identity is the only way to do this? Or is my procedure right the way I am doing it? Again if its correct, this is where I get stuck at. I have homework problems that covers similar materials like this and if someone can help me understand this, that would great.