Plotting a one-sided amplitude spectrum

Question

I have a continuous signal $x(t)$ such that

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

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 $3$Hz, $12$Hz and $15$Hz.

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

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?

Answer 1

I think you have to keep in mind 3 things:

1. The Fourier transform of the signal $\cos(2 \pi f_0 t)$ is given by $\frac{1}{2}(\delta(f+f_0) + \delta(f - f_0))$, where $\delta(f)$ is Dirac's delta. Therefore, one part of the spectrum lies at frequencies $> 0$ and one part at frequencies $< 0$. Note that negative frequencies are a consequence of using complex exponentials as basis functions in the definition of the Fourier transform. You can think of your signal (in this case a cosine, which is a real valued function), as being "crafted" out of two complex exponentials, one with a positive and one with a negative frequency, which, when added together, give you the cosine. In this case, the one-sided spectrum is enough, that is, the negative side of the spectrum can be recovered by simply mirroring the positive side. For general functions, this is not the case, however.
2. The corresponding amplitude levels are the ones that you mentioned, divided by one half.
3. If you have to plot the one-sided amplitude spectrum of the sampled signal, then you also have to plot the aliases. These are given by repeating the spectrum of the unsampled signal at integer multiples of the sampling frequency.

So in summary, lets assume that we sample with $f_s = 35 Hz$, then the one sided amplitude spectrum would look like this:

Signal spectrum: Dirac at: 3, 12, 15 Hz with coefficients 6, 3, and 1.5.

First alias around $35 Hz$:
Dirac at 20, 23, 32, 38, 47, 50 Hz with coefficients 1.5, 3, 6, 6, 3, 1.5

Other aliases accordingly.

Best regards!