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

- a real, time-varying signal may be represented by a complex, time-varying signal; and
- 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 Figure 1, is a cosinusoidal signal with amplitude *A*, frequency
*ω*, and phase *φ*. The amplitude *A* characterizes the peak-to-peak swing of *ω* characterizes the period *φ* characterizes the time *τ* so defined, the signal

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

In this form the signal is easy to plot. Simply draw a cosinusoidal wave
with amplitude *A* and period *T*; then strike the origin *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)

Show that

The inverse of the period

Sketch the function

The signal *represented* as the real part of a complex number:

We call

meaning that the signal *phasor* or complex amplitude representation of

meaning that the signal *baseband* representation of the signal

For each of the signals in Problem 3.3, give the corresponding
phasor representation

**Geometric Interpretation.** Let's call

the complex representation of the real signal *phasor*

This phasor is illustrated in Figure 2. In the figure, *φ* is approximately

We know from our study of complex numbers that *t* from 0,
indefinitely, we rotate the phasor *A*. We sometimes call *rotating phasor* whose rotation rate
is the frequency

This rotation rate is also the frequency of the cosinusoidal signal

In summary, *A**φ* is about

Sketch the imaginary part of

(MATLAB) Modify Demo 2.1 in "The Function e^{x} and e^{jθ}" so that *t* to get signals like those of Figure 1. You
should observe something like Figure 4 using the subplot features discussed
in An Introduction to MATLAB. (In the figure,

**Positive and Negative Frequencies.** There is an alternative phasor representation for the signal ^{x} s^{jθ}", namely, cos

In this formula, the term *ω*. The term *ω*. The physically meaningful frequency for a cosine is *ω*, a positive number like *A*

Sketch the two-phasor representation of

**Adding Phasors.** The sum of two signals with common frequencies
but different amplitudes and phases is

The rotating phasor representation for this sum is

The new phasor is

Write the phasor

**Differentiating and Integrating Phasors.** The derivative of the signal

This finding is very important. It says that the derivative of

These two phasor representations are entirely equivalent. The first says that the phasor

The integral of

This finding shows that the integral of

The phasor

**An Aside: The Harmonic Oscillator.** The signal

has the solution

Try it:

The constants

Show how to compute

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