Signal and Noise

Learn It!

Goal

\(x(t)=3 cos(6 \pi t) \)

If \(x(t)=3 cos(6 \pi t) \) is sampled every \(0.5\) seconds starting at \(t=0\). What are the first 4 sampled values?

Method 1

Plug and Chug

t=0.0

t=0.5

t=1.0

t=1.5

t=0.5

t=1.0

t=1.5

We know that the sample points are each equally spaced, so we know which values of \(t\) we are sampling at

\(x(0.0)=3 cos(6 \pi \cdot 0.0) =3 cos(0)=3\)

\(x(0.5)=3 cos(6 \pi \cdot 0.5) =3 cos(3 \pi)=-3\)

\(x(1.0)=3 cos(6 \pi \cdot 1.0) =3 cos(6 \pi)=3\)

\(x(1.5)=3 cos(6 \pi \cdot 1.5) =3 cos(9 \pi)=-3\)

\(x(0.5)=3 cos(6 \pi \cdot 0.5) =3 cos(3 \pi)=-3\)

\(x(1.0)=3 cos(6 \pi \cdot 1.0) =3 cos(6 \pi)=3\)

\(x(1.5)=3 cos(6 \pi \cdot 1.5) =3 cos(9 \pi)=-3\)

Just plug in the four time points into the function we are sampling. That's all there is to it!

The cosine of an even multiple of \(\pi\) ( such as 0, \(2\pi\), \(4\pi\) )is +1, the cosine of an odd multiple of \(\pi\) ( such as \(\pi\), \(3\pi\)) is -1

The cosine of an even multiple of \(\pi\) ( such as 0, \(2\pi\), \(4\pi\) )is +1, the cosine of an odd multiple of \(\pi\) ( such as \(\pi\), \(3\pi\)) is -1

+

?

Put your calculator in radian mode!

Always be sure to put your calculator in radian mode on these kinds of problems.
You will get the answer wrong despite working the problem correctly if you are in degree mode.

★

(3,-3,3,-3)

The correct answer.