Real signal as sum of phasors

Let us start with a general description of a signal $x (t)$ , which consists of a DC component, with value $A_{0}$ , and a sum of $N$ sinusoidal components, each with a different frequency $f_{k}$ , amplitude $A_{k}$ and phase $ϕ_{k}$ , as

$x (t) = A_{0} + \sum_{k = 1}^{N} A_{k} \cos (2 π f_{k} t + ϕ_{k}) .$

Now by using the Euler equations we can write this equation as the following sum of rotating phasors:

$\begin{aligned} x (t) & = A_{0} + \sum_{k = 1}^{N} {(\frac{A_{k}}{2} e^{j ϕ_{k}}) e^{j 2 π f_{k} t} + (\frac{A_{k}}{2} e^{- j ϕ_{k}}) e^{- j 2 π f_{k} t}} \\ = X_{0} + \sum_{k = 1}^{N} {\frac{X_{k}}{2} e^{j 2 π f_{k} t} + \frac{X_{k}^{*}}{2} e^{- j 2 π f_{k} t}}, \end{aligned}$

where $X_{0} = A_{0}$ and $X_{k} = A_{k} e^{j ϕ_{k}}$ . From this description we can derive the following general property:

For real signals the values of the complex weights of the phasors with negative frequencies are the complex conjugated versions of the complex weights of the phasors with positive frequencies.

Last updated on Apr 4, 2020