Plotting line spectra of a CT signal

A line spectrum shows the frequency content of a periodic continuous-time signal as vertical lines at the discrete frequencies present (the harmonics). There are two plots: the amplitude (magnitude) spectrum and the phase spectrum.

Procedure

1. Express the signal as a sum of cosines (or a Fourier series): x(t) = A₀ + Σ Aₖ cos(2πfₖt + φₖ). Convert sines using sin θ = cos(θ - 90°), and make all amplitudes positive by adjusting the phase by ±180°.
2. Identify each frequency fₖ, its amplitude Aₖ and its phase φₖ.
3. Plot amplitude Aₖ against frequency as a vertical line at each fₖ (single-sided), with DC at f = 0.
4. Plot phase φₖ against frequency in the same way.
5. For a two-sided spectrum, use the complex form: each cosine gives lines of height Aₖ/2 at +fₖ and -fₖ, with phases +φₖ and -φₖ; the DC value stays A₀.

Example

x(t) = 4 + 6 cos(2π·10t + 30°) + 2 sin(2π·30t)

Rewrite 2 sin(2π·30t) = 2 cos(2π·30t - 90°).

Amplitude (single-sided)        Phase
 6 |     |                  30 |     *
 4 | |   |                   0 +-----+-----+--> f
 2 | |   |      |                       10    30
   +-+---+------+--> f    -90 |            *
   0   10     30

The two-sided spectrum has lines of height 3 at ±10 Hz and 1 at ±30 Hz, with DC 4. The spacing of the lines is the fundamental frequency (here 10 Hz); a periodic signal always gives a discrete (line) spectrum.