2.16: Frequency Modulation

Learning Objectives: Frequency Modulation

1. Understand how instantaneous phase and instantaneous frequency are related for FM signals.
2. Recognize the differences between narrowband and broadband FM spectra.

YouTube Videos

MIT Courseware Lecture 24: Modulation, Part 2 (Credit to Prof. Alan V. Oppenheim)

Summary

Frequency Modulation (FM)

Definition: Frequency Modulation (FM) → The modulating signal is used to control the frequency of a sinusoidal carrier

• With FM, the envelope of the carrier is constant
• FM systems are highly non-linear
• FM reception is typically better than AM reception

Consider a sinusoidal carrier: \[ c(t)=A\cos(\omega_c t+\theta_c)=\cos(\theta(t)) \]

\(\omega_c\): carrier's frequency
\(\theta_c\): carrier's phase

Angle Modulation: \(\theta(t)=\omega_c t+\theta_c\) corresponds to using the modulating signal to change/vary the phase \(\theta(t)\)

\[ y(t)=A\cos\!\big[\omega_c t+\theta_c(t)\big] \quad \text{where} \quad \theta_c(t)=\theta_0+k_p x(t) \]

If \(x(t)\) is constant, the phase of \(y(t)\), i.e., \(\theta_c(t)\), is constant and proportional to the amplitude of \(x(t)\)

\[ y(t)=A\cos[\theta(t)] \]

\[ \frac{d\theta(t)}{dt}=\omega_c+k_f x(t) \qquad \text{if } x(t)=\text{constant} \]

Frequency Modulation

\[ y(t)=A\cos\!\Big[(\omega_c+k_f x)t\Big] \]

\(y(t)\) follows a sinusoid with a frequency that is offset from the carrier frequency \(\omega_c\) by an amount proportional to the amplitude of \(x(t)\).

Note that phase modulation and frequency modulation are different forms of angle modulation

For phase modulation: \[ \frac{d\theta(t)}{dt}=\omega_c+k_p\frac{dx(t)}{dt} \]

Frequency modulation with a step corresponds to the frequency of the sinusoidal carrier changing instantaneously from one value to another when \(x(t)\) changes value at \(t=0\).

Figure 2.16.1: (a) Phase modulation if x(t) is a ramp signal; (b) Frequency modulation if x(t) is a ramp signal; and (c) Frequency modulation if x(t) is a unit step.

Instantaneous Frequency

When the frequency modulation is a ramp, the frequency changes linearly

\[ y(t)=A\cos\theta(t) \qquad \omega_i=\frac{d\theta(t)}{dt} \]

If \[ y(t)=A\cos[\omega_c t+\theta_0] \;\Rightarrow\; \omega_i=\frac{d}{dt}[\omega_c t+\theta_0]=\omega_c \]

For FM, \[ \theta(t)=(\omega_c+k_f x(t))t \;\Rightarrow\; \omega_i=\omega_c+k_f x(t) \]

For phase modulation \[ \omega_i=\omega_c+k_p\frac{dx(t)}{dt} \]

Narrowband Frequency Modulation

\[ x(t)=A\cos(\omega_m t) \]

\[ \omega_i=\omega_c+k_f A\cos(\omega_m t) \]

The instantaneous frequency \(\omega_i\) ranges from \(\omega_c-k_fA\) to \(\omega_c+k_fA\)

\[ \Delta\omega=\frac{\max-\min}{2}=k_fA \qquad \]

Thus \[ \omega_i=\omega_c+\Delta\omega\cos(\omega_m t) \]

\[ y(t)=\cos\!\left[\omega_c t+\int x(t)\,dt\right] =\cos\!\left[\omega_c t+\frac{\Delta\omega}{\omega_m}\sin(\omega_m t)+\theta_0\right] \]

m: modulating index

\[ y(t)=\cos\!\big[\omega_c t+m\sin\omega_m t+\theta_0\big] \] if \(m\) is small → narrowband FM

if \(\theta_0=0\)

\[ y(t)=\cos(\omega_c t)\cos(m\sin\omega_m t)-\sin(\omega_c t)\sin(m\sin\omega_m t) \]

If \(m \ll\), then \(\cos(m\sin\omega_m t)\approx 1\) and \(\sin(m\sin\omega_m t)\approx m\sin\omega_m t\) since\(\sin\theta\approx \theta\). Thus \[ y(t)\approx \cos(\omega_c t)-m\sin(\omega_c t)\sin(\omega_m t) \]

Figure 2.16.2: Narrowband FM spectrum approximation.

Wideband Frequency Modulation

\[ y(t)=\cos\!\big[\omega_c t+m\sin\omega_m t+\theta_0\big] \] if \(m\) is large → wideband FM

\[ y(t)=\cos(\omega_c t)\cos(m\sin\omega_m t)-\sin(\omega_c t)\sin(m\sin\omega_m t) \]

(\(\cos(\omega_c t)\cos(m\sin\omega_m t)\)): \(\cos(\omega_c t)\) carrier modulating by \(\cos(m\sin\omega_m t)\)
(\(\cos(m\sin\omega_m t)\)) and (\(\sin(m\sin\omega_m t)\)): periodic signals with frequency \(\omega_m\)
(\(\sin(\omega_c t)\sin(m\sin\omega_m t)\)): \(\sin(\omega_c t)\) carrier modulated by \(\sin(m\sin\omega_m t)\)

Figure 2.16.3: Wideband frequency modulation spectrum magnitude with \(m=11\): \(\cos(\omega_c t)\cos(m\sin\omega_m t)\) (top); \(\sin(\omega_c t)\sin(m\sin\omega_m t)\) (middle); \(\cos(\omega_c t+m\sin\omega_m t)\) (bottom).