Phase modulation (PM) is the process of modulating the phase of the carrier.
To properly develop a backdrop for PM, you should first understand modulation and its carrier. The forms of the general equation for v(t) are:
where ωc = 2πf, or
if A, the carrier amplitude, is normalized to unity, or (1-1)
where θ is the same as ωc.
The equations for an unmodulated carrier 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 reference equation because you can use each one as the foundation for the following more complicated carrier equations (1-2 and 1-3).
Add a phase displacement—that is, a phase shift—to obtain the first more-complicated equation:
Adding the φ term displaces the wave by a certain fixed angle φ from the wave referenced in (1-1). This phase shift can be illustrated using a trigonometric waveform or a rotating vector waveform as shown in Figure 1.
Figure 1. Phase-Shifted Sine Waves
Figure 1 shows (in A) two sine waves that are identical in both frequency and amplitude. The only difference is that the blue wave crosses the center line before the red wave and is thus phase-shifted (is leading) a fixed amount from the red wave. In B, rotating vectors (phasors) represent these two sine waves. Because the vectors rotate in the counterclockwise direction, the blue vector leads the red vector.
The second more-complicated equation changes both the frequency and phase terms. Now, both the frequency term, and the phase term are shown as instantaneous functions of time. Thus, the entire equation is a function of time, as follows:
or, substituting 2πfc for ωc,
Equation (1-3) is a general and very useful equation. Using this equation enables the following types of modulation.
- If you periodically change some function of A with an input signal, the result is amplitude modulation.
- If you periodically change some function of ω(where ω = fc) with an input signal, the result is frequency modulation.
- If you periodically change some function of φ(t) with an input signal, the result is phase modulation.
We can further expand on the ideas in the preceding statements. Recall that the equation for the reference, non-modulated, carrier is v(t) = sin(ωc). Next, using the expanded form v(t) = A sin(ωc(t) + φ(t)) for a modulated carrier, if the term φ(t) is signal-modulated, it produces a carrier that periodically changes in phase. You can visualize this change as either instantaneously moving the reference carrier wave position or the reference phasor position. Another way of stating it is that the angle (ωc(t) + φ(t)) of v(t) is phase-modulated around the reference wave c.
Using the following three equations
where θ is the same as ωc, and
the instantaneous radial frequency can be stated as:
Remember that you can visualize both the terms ωc and ωc(t) + φ(t) as rotating vectors: ωc rotates with a fixed velocity and ωc(t) + φ(t) rotates with a variable (modulated) velocity. Classical mechanics shows that velocity is the derivative of length or displacement, and acceleration is 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 changing phase as the derivative of a rotating (carrier) vector with a fixed relative phase.
Likewise, you can calculate the instantaneous frequency, fi as shown in the following equation:
Here fi continually runs ahead or behind the reference carrier inherent in ωc or, more properly, A sin(ωc). Equation (1-5) shows that a phasor representing the motion of the composite angle ωc(t) + φ(t) continually runs ahead or behind the phasor of the non-modulated reference carrier angle ωc. Whereas this phase modulation occurs 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 phase angle value changes with the derivative of the modulating signal.
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Information Contributed By: Bob Libbey, Retired RCA Engineer and Adjunct Professor, New Jersey Institute of Technology.