Angle Modulation

Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 4: Continuous-Time Analysis: The Fourier Transform. Signal Processing and Linear Systems (2nd ed., pp. 401-414). Oxford University Press.

Recall that in amplitude-modulated signals, the information content of the baseband (message) signal $m(t)$ appears in the amplitude variations of the carrier. In angle modulation discussed in this article, the information content of $m(t)$ is carried by the angle (which includes its frequency and phase) variations of the carrier.

Angle-modulated signals

Angle modulation also goes by the name exponential modulation. The generalized angle-modulated (or exponentially modulated) carrier can be described as $${\phi }_{\mathrm{EM}}(t)=A\cos [{\omega }_{c}t+k\psi (t)]$$ where $k$ is an arbitrary constant and $\psi (t)$ is some measure of $m(t)$.

By selecting different form of $\psi (t)$, we get various angle modulation methods. Among them the two most famous one are phase modulation (PM) and frequency modulation (FM)¹.

Notice that ${\phi }_{\mathrm{EM}}(t)$ does not have a constant frequency. Thus, we call ${\phi }_{\mathrm{EM}}(t)$ a generalized sinusoidal signal, instead of a traditional sinusoidal signal, whose frequency must be constant.

Generalized sinusoidal signals

Let us consider a generalized sinusoidal signal $\phi (t)$ given by $$\phi (t)=A\cos \theta (t)$$ where $\theta (t)$, the generalized angle, is a function of $t$. We then define instantaneous frequency of $\theta (t)$ as its first-order derivative with respect to time $t$: $${\omega }_{\mathrm{i}}(t)=\frac{d\theta (t)}{dt}.$$ Also, we have $$\begin{array}{}\text{(1)}& \theta (t)={\int }_{-\mathrm{\infty }}^t{\omega }_{\mathrm{i}}(\alpha )d\alpha .\end{array}$$

By the definition of generalized sinusoidal signals, we obtain $${\phi }_{\mathrm{EM}}(t)=A\cos [{\omega }_{c}t+k\psi (t)]=A\cos \theta (t)$$ where $\theta (t)={\omega }_{c}t+k\psi (t)$.

The instantaneous frequency is $$\begin{array}{}\text{(2)}& {\omega }_{\mathrm{i}}(t)=\frac{d\theta (t)}{dt}={\omega }_c+k\dot{\psi }(t).\end{array}$$

Phase modulation (PM)

For PM, the instantaneous frequency ${\omega }_{\mathrm{i}}(t)$ is given by $${\omega }_{\mathrm{i}}(t)={\omega }_c+k\dot{m}(t)$$

which varies linearly with $\dot{m}(t)$. Compare it with $\text{(2)}$, we get $\dot{\psi }(t)=\dot{m}(t)$. From $\text{(1)}$, $$\psi (t)={\int }_{-\mathrm{\infty }}^t\dot{m}(\alpha )d\alpha .$$ Thus, a phase-modulated (PM) signal is given by $${\phi }_{\mathrm{PM}}(t)=A\cos [{\omega }_{c}t+k_{p}m(t)]$$ where $k_p$ is just the constant $k$, the subscript $p$ is for "phase".

Frequency modulation (FM)

For FM, ${\omega }_{\mathrm{i}}(t)$ is given by $${\omega }_{\mathrm{i}}(t)={\omega }_c+km(t)$$

which varies linearly with $m(t)$. Compare it with $\text{(2)}$, we get $\dot{\psi }(t)=m(t)$. From $\text{(1)}$, $$\psi (t)={\int }_{-\mathrm{\infty }}^{t}m(\alpha )d\alpha .$$ Thus, a frequency-modulated (FM) signal is given by $${\phi }_{\mathrm{FM}}(t)=A\cos [{\omega }_{c}t+k_f{\int }_{-\mathrm{\infty }}^{t}m(\alpha )d\alpha ]$$ where $k_f$ is just the constant $k$, the subscript $f$ is for "frequency".

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1. Although in digital communication the use of phase and frequency modulation is common, so-called broadcast FM is not FM in the classical sense, but is a generalized angle modulation because of the inclusion of a preemphasis filter used to improve its noise-suppressing abilities.↩︎