phase modulation

Modulation of the phase of the carrier.

To properly develop a backdrop for phase modulation, you should first understand modulation and its carrier. The forms of the simple, general equation for v(t) are:

or
if A, the carrier’s amplitude is normalized to unity and
where q is the same as w_c [E 31]

These are the simple equations for an unmodulated carrier. They describe an unmodulated sinusoidal wave propagating at a rate of radians-per-second. That wave can be defined as 2 * * f (were f is some designated frequency). You can consider any of these forms as the prototype or reference equation since you can use each one as the foundation for the following two, more complicated, carrier equations.

The second general equation adds the idea of a phase displacement, that is, phase shift as:

[E 32]

The addition of the term produces an equation for a wave that is displaced by a certain fixed angle from the above mentioned reference wave. This shift in phase can be illustrated using a trigonometric waveform or a rotating vector waveform as shown in Figure 19.

Figure 19 A shows two sine waves that are identical in both frequency and amplitude. The only difference is that the blue wave would cross the center line before the red wave and is thus phase-shifted (is leading) a fixed amount from the red wave. B in turn, shows rotating vectors (phasors) representing these two sine waves. Since the vectors rotate in the conventional counterclockwise direction, the blue vector is also leading the red vector.

The third equation changes both the frequency and phase terms. Now, both (=2f), the frequency term, and , the phase term, are shown as instantaneous functions of time. Thus, the entire equation is a function of time as:

or [E 33]

This third equation is a very general and very useful equation. For example, using this equation:

1. If you can devise a method to periodically change some function of "A" with an input signal, the result is amplitude modulation.

2. If you can devise a method to periodically change some function of "" (which is really changing the value of "f_c"), with an input signal), the result is frequency modulation.
3. If you can devise a method to periodically change some function of "" with an input signal, the result is phase modulation.

We can expand on the ideas in the preceding statements further. To start, recall that the equation for the reference, non-modulated, carrier is . Next, using the expanded form for a modulated carrier, if the termis signal modulated, it produces a carrier that is periodically changing in phase. You can visualize this change as either instantaneously moving the position of the reference carrier wave or the position of the reference phasor. Another way of stating it is that the angle [] of v(t) is phase modulated around the reference wave c.

Using the equations ; the equation , where "" is the same as c; and the equation , the instantaneous radial frequency can be stated as:


(Remember that you can visualize both the terms and as rotating vectors. is rotating with a fixed velocity and is rotating with a variable (modulated) velocity. Classical mechanics shows that velocity is the derivative of length or displacement and acceleration is, in turn, the derivative of velocity. Thus, a changing (modulated or accelerated) rotation is the derivative of a fixed rotation -- that is, you can consider a rotating (carrier) vector with a modulated relative phase as the derivative of a rotating (carrier) vector with a fixed relative phase.)

Likewise, you can calculate the instantaneous frequency as:


Here f_i , the instantaneous carrier frequency, is continually running ahead or behind the reference carrier inherent in or, more properly, . E 35 shows that a phasor representing the motion of the composite anglewill continually run ahead or behind the phasor of the non-modulated reference carrier angle . Whereas this phase modulation is accomplished when the value of the phase angle is changed as a direct function of the modulating signal, frequency modulation will be accomplished only when the value of the phase angle is changed with the derivative of the modulating signal.

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Refer to the RF & Communications Resources page for additional information about RF terminology, fundamentals, and National Instruments RF products.

Information Contributed By: Bob Libbey, Retired RCA Engineer and Adjunct Professor, New Jersey Institute of Technology.

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