# Homework Help: Nyquist Sampling Theorem - I just need help understand this.

1. Mar 17, 2012

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:

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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.