4.1 Phasors: phasor representation of signals

There are two key ideas behind the phasor representation of a signal:

1. a real, time-varying signal may be represented by a complex, time-varying signal; and
2. a complex, time-varying signal may be represented as the product of a complex number that is independent of time and a complex signal that is dependent on time.

Let's be concrete. The signal

illustrated in [link] , is a cosinusoidal signal with amplitude A , frequency ω , and phase φ . The amplitude A characterizes the peak-to-peak swing of $2A$ , the angular frequency ω characterizes the period $T=\frac{2\pi }{\omega }$ between negative- to-positive zero crossings (or positive peaks or negative peaks), and the phase φ characterizes the time $\tau =\frac{-\phi }{\omega }$ when the signal reaches its first peak. With τ so defined, the signal $x\left(t\right)$ may also be written as

When τ is positive, then τ is a “time delay” that describes the time (greater than zero) when the first peak is achieved. When τ is negative, then τ is a “time advance” that describes the time (less than zero) when the last peakwas achieved. With the substitution $\omega =\frac{2\pi }{T}$ we obtain a third way of writing $x\left(t\right)$ :

In this form the signal is easy to plot. Simply draw a cosinusoidal wave with amplitude A and period T ; then strike the origin $(t=0)$ so that the signal reaches its peak at τ . In summary, the parameters that determine a cosinusoidal signal have the following units:

A , arbitrary (e.g., volts or meters/sec, depending upon the application)

ω , in radians/sec (rad/sec)

T , in seconds (sec)

φ , in radians (rad)

τ , in seconds (sec)

The signal $x\left(t\right)=Acos(\omega t+\phi )$ can be represented as the real part of a complex number:

We call $A{e}^{j\phi }{e}^{j\omega t}$ the complex representation of $x\left(t\right)$ and write

meaning that the signal $x\left(t\right)$ may be reconstructed by taking the real part of $A{e}^{j\phi }{e}^{j\omega t}$ . In this representation, we call $A{e}^{j\phi }$ the phasor or complex amplitude representation of $x\left(t\right)$ and write

meaning that the signal $x\left(t\right)$ may be reconstructed from $A{e}^{j\phi }$ by multiplying with $e^{j\omega t}$ and taking the real part. In communication theory, we call $A{e}^{j\phi }$ the baseband representation of the signal $x\left(t\right)$ .

Geometric Interpretation. Let's call

the complex representation of the real signal $Acos(\omega t+\phi )$ . At $t=0$ , the complex representation produces the phasor

This phasor is illustrated in [link] . In the figure, φ is approximately $\frac{-\pi }{10}$ If we let $t$ increase to time $t_1$ , then the complex representation produces the phasor

We know from our study of complex numbers that $e^{j\omega t_1}$ just rotates the phasor $A{e}^{j\phi }$ through an angle of $\omega t_1$ ! See [link] . Therefore, as we run t from 0, indefinitely, we rotate the phasor $A{e}^{j\phi }$ indefinitely, turning out the circular trajectory of [link] . When $t=\frac{2\pi }{\omega }$ then $e^{j\omega t}=e^{j2\pi }=1$ . Therefore, every $\left(\frac{2\pi }{\omega }\right)$ seconds, the phasor revisits any given position on the circle of radius A . We sometimes call $A{e}^{j\phi }{e}^{j\omega t}$ a rotating phasor whose rotation rate is the frequency $\omega$ :