phase modulation

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 ω_c = 2πf, or
 if A, the carrier amplitude, is normalized to unity, or                                        (1-1)
 where θ is the same as ω_c.

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:

                                      (1-2)

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 2πf_c for ω_c,

                               (1-3)

Equation (1-3) is a general and very useful equation. Using this equation enables the following types of modulation.

1. If you periodically change some function of A with an input signal, the result is amplitude modulation.
2. If you periodically change some function of ω(where ω = f_c) with an input signal, the result is frequency modulation.
3. 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(ω_c). Next, using the expanded form v(t) = A sin(ω_c(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 (ω_c(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 ω_c, and

the instantaneous radial frequency can be stated as:

                                        (1-4)

Remember that you can visualize both the terms ω_c and ω_c(t) + φ(t) as rotating vectors: ω_c rotates with a fixed velocity and ω_c(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, f_i as shown in the following equation:

                                (1-5)

Here f_i continually runs ahead or behind the reference carrier inherent in ω_c or, more properly, A sin(ω_c). Equation (1-5) shows that a phasor representing the motion of the composite angle ω_c(t) + φ(t) continually runs ahead or behind the phasor of the non-modulated reference carrier angle ω_c. 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.

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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notation
in eqn. 1-3 and elsewhere: omega, incorrectly specified as a function of t, i.e., (t), should actually be multiplied by t; phi is written correctly, as a function of t.
- manfred.caranci@verizon.net - Jan 11, 2011

Is figure 19A correct?
It seems as though the red and blue sine waves in Figure 19A do not correspond to the red and blue arrows in Figure 19B. Isn't it the case that when the red arrow in 19B is at zero, the blue arrow will be positive? Thanks!
- May 18, 2010