Angle Modulation

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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 \[ \color{orange}{\varphi_{\mathrm{EM}}(t)=A \cos \left[\omega_c t+k \psi(t)\right]} \] 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 \(\varphi_{\mathrm{EM}}(t)\) does not have a constant frequency. Thus, we call \(\varphi_{\mathrm{EM}}(t)\) 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 \(\varphi(t)\) given by \[ \varphi(t)=A \cos \theta(t) \] 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 \(t\): \[ \omega_{\mathrm{i}}(t)=\frac{d \theta (t)}{d t} . \] Also, we have \[ \begin{equation} \label{eq1} \theta(t)=\int_{-\infty}^t \omega_{\mathrm{i}}(\alpha) d \alpha . \end{equation} \]

By the definition of generalized sinusoidal signals, we obtain \[ \varphi_{\mathrm{EM}}(t)=A \cos \left[\omega_c t+k \psi(t)\right] = A \cos \theta(t) \] where \(\theta(t) = \omega_c t+k \psi(t)\).

The instantaneous frequency is \[ \begin{equation} \label{eq2} \omega_{\mathrm{i}}(t)=\frac{d \theta (t)}{d t} = \omega_c +k \dot \psi(t) . \end{equation} \]

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 \(\eqref{eq2}\), we get \(\dot \psi(t)= \dot m(t)\). From \(\eqref{eq1}\), \[ \psi(t) = \int_{-\infty}^{t} \dot m(\alpha) d \alpha . \] Thus, a phase-modulated (PM) signal is given by \[ \color{teal} {\varphi_{\mathrm{PM}}(t)=A \cos \left[\omega_c t+k_p m(t)\right]} \] 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 \[ \omega_{\mathrm{i}}(t)=\omega_c+k {m}(t) \]

which varies linearly with \({m}(t)\). Compare it with \(\eqref{eq2}\), we get \(\dot \psi(t)= m(t)\). From \(\eqref{eq1}\), \[ \psi(t) = \int_{-\infty}^{t} m(\alpha) d \alpha . \] Thus, a frequency-modulated (FM) signal is given by \[ \color{salmon} {\varphi_{\mathrm{FM}}(t)=A \cos \left[\omega_c t+k_f \int_{-\infty}^t m(\alpha) d \alpha\right]} \] where \(k_f\) is just the constant \(k\), the subscript \(f\) is for "frequency".

  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.↩︎