# 5.2: 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

$x(t)=A\cos(ωt+φ)$

illustrated in Figure, 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 $$T=\frac {2π} ω$$ between negative- to-positive zero crossings (or positive peaks or negative peaks), and the phase $$φ$$ characterizes the time $$τ=\frac {−φ} ω$$ when the signal reaches its first peak. With $$τ$$ so defined, the signal $$x(t)$$ may also be written as

$x(t)=A\cosω(t−τ)$

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 peak was achieved. With the substitution $$ω=\frac {2π} T$$ we obtain a third way of writing $$x(t)$$:

$x(t)=A\cos\frac {2π} {T} (t−τ)$

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
T
in seconds (sec)
φ
τ
in seconds (sec)

Exercise $$\PageIndex{1}$$

Show that $$x(t)=A\cos2πT(t−τ)$$ is “periodic with period T," meaning that $$x(t+mT)=x(t)$$ for all integer $$m$$.

Exercise $$\PageIndex{2}$$

The inverse of the period T is called the “temporal frequency” of the cosinusoidal signal and is given the symbol $$f$$; the units of $$f=\frac 1 T$$ are (seconds)−1 or hertz (Hz). Write $$x(t)$$ in terms of $$f$$. How is $$f$$ related to $$ω$$? Explain why $$f$$ gives the number of cycles of $$x(t)$$ per second.

Exercise $$\PageIndex{3}$$

Sketch the function $$x(t)=110\cos[2π(60)t−\frac π 8]$$ versus $$t$$. Repeat for $$x(t)=5\cos[2π(16×10^6)t+\frac π 4]$$ and $$x(t)=2cos[\frac {2π} {10^{−3}}(t−\frac {10^{−3}} 8)]$$. For each function, determine $$A,ω,T,f,φ$$, and $$τ$$. Label your sketches carefully.

The signal $$x(t)=A\cos(ωt+φ)$$ can be represented as the real part of a complex number:

$x(t)=\mathrm{Re}[Ae^{j(ωt+φ)}]=\mathrm{Re}[Ae^{jφ}e^{jωt}]$

We call $$Ae^{jφ}e^{jωt}$$ the complex representation of $$x(t)$$ and write

$x(t)↔Ae^{jφ}e^{jωt}$

meaning that the signal $$x(t)$$ may be reconstructed by taking the real part of $$Ae^{jφ}e^{jωt}$$. In this representation, we call $$Ae^{jφ}$$ the phasor or complex amplitude representation of $$x(t)$$ and write

$x(t)↔Ae^{jφ}$

meaning that the signal $$x(t)$$ may be reconstructed from $$Ae^{jφ}$$ by multiplying with $$e^{jωt}$$ and taking the real part. In communication theory, we call $$Ae^{jφ}$$ the baseband representation of the signal $$x(t)$$.

Exercise $$\PageIndex{4}$$

For each of the signals in Problem 3, give the corresponding phasor representation $$Ae^{jφ}$$.

## Geometric Interpretation

Let's call

$Ae^{jφ}e^{jωt}$

the complex representation of the real signal $$A\cos(ωt+φ)$$. At $$t=0$$, the complex representation produces the phasor

$Ae^{jφ}$

This phasor is illustrated in the Figure. In the figure, $$φ$$ is approximately $$\frac {−π} {10}$$ 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 $$e^{jωt_1}$$ just rotates the phasor $$Ae^{jφ}$$ through an angle of $$ωt_1$$! See Figure. Therefore, as we run t from 0, indefinitely, we rotate the phasor $$Ae^{jφ}$$ indefinitely, turning out the circular trajectory of the Figure. When $$t=\frac {2π} ω$$ then $$e^{jωt} = e^{j2π} = 1$$. Therefore, every $$(2πω)$$ seconds, the phasor revisits any given position on the circle of radius A. We sometimes call $$Ae^{jφ}e^{jωt}$$ a rotating phasor whose rotation rate is the frequency $$ω$$:

$\frac {d} {dt} ωt=ω$

This rotation rate is also the frequency of the cosinusoidal signal $$A\cos(ωt+φ)$$.

In summary, $$Ae^{jφ}e^{jωt}$$ is the complex, or rotating phasor, representation of the signal $$A\cos(ωt+φ)$$. In this representation, $$e^{jωt}$$ rotates the phasor $$Ae^{jφ}$$ through angles $$ωt$$ at the rate $$ω$$. The real part of the complex representation is the desired signal $$A\cos(ωt+φ)$$. This real part is read off the rotating phasor diagram as illustrated in the Figure. In the figure, the angle $$φ$$ is about $$\frac {−2π} {10}$$. As we become more facile with phasor representations, we will write $$x(t)=\mathrm{Re}[Xe^{jωt}]$$ and call $$Xe^{jωt}$$ the complex representation and $$X$$ the phasor representation. The phasor $$X$$ is, of course, just the phasor $$Ae^{jφ}$$.