# What should the amplitude be when plotting 1-sided Amplitude Spectrum?

I have a continuous signal x(t) such that

$x(t)=12cos(6\pi t)+6cos(24\pi t)+3cos(30 \pi t)$

and is asked to sketch a 1-sided Amplitude Spectrum of the signal x(t) if sampled above the minimum sampling rate.

because $w=2\pi$, i worked out that the three signals are 3Hz, 12Hz and 15 Hz.

I'm just wondering, when I plot the Amplitude Spectrum should the Amplitude just be the corresponding coefficients? ie. 12 for 3Hz, 6 for 12Hz and 3 for 15Hz?

EDIT: Additionally, what's the difference between 1-sided Amplitude Spectrum and 2-sided Amplitude Spectrum? Does one offer any more benefit over the other?

• For real signals, $X(f) = X(-f)^\ast$, so no need to show both sides. Also, for power spectral density, $G_x(f)$ is non-negative, real, and symmetric. – Juancho Jun 20 '12 at 13:32

To complement @wrapperapps answer, you can express sinusoids as sum of exponentials: $\cos(2 \pi f_0 t) = (e^{j 2 \pi f_0 t} + e^{-j 2 \pi f_0 t})/2$.
Thus, the fourier transform yields $\delta(f+f_0)/2 + \delta(f-f_0)/2$: each frequency delta has amplitude $1/2$.