Frequency Modulation (FM)

The Principle of Frequency Modulation

Frequency modulation (FM) carries information on a high-frequency carrier wave. AM varies the carrier amplitude, while FM keeps the carrier amplitude constant and varies its instantaneous frequency in proportion to the information signal.

A positive value of the modulating signal raises the instantaneous frequency above the carrier frequency, while a negative value lowers it. The signal amplitude remains constant, so changes in the information level appear as changes in the spacing between successive cycles.

FM Signal Components

The following description separates the modulating signal, the carrier, and the resulting FM signal.

• Modulating signal: The top waveform is the information to transmit, such as an audio sine wave. It has its own amplitude and frequency.
• Carrier wave: The middle waveform is the high-frequency carrier. Before modulation, its frequency and amplitude are constant.
• Modulated FM signal:The bottom waveform is the result. Its amplitude is constant, equal to the carrier's, while its frequency changes:
  • Where the modulating signal reaches its peak (maximum positive amplitude), the FM waves are closest together, indicating the highest frequency.
  • Where the modulating signal reaches its trough (maximum negative amplitude), the FM waves have the greatest spacing, indicating the lowest frequency.
  • When the modulating signal crosses the zero-axis, the FM wave's frequency is exactly equal to the original carrier frequency.

The Mathematics of FM

These relationships can be written using the following signals:

• Information (modulating) signal: $x_m(t)$
• Carrier wave signal: $x_c(t) = A \cdot \cos(2\pi f_c t)$, where $A$ is the constant amplitude and $f_c$ is the constant carrier frequency.

The resulting modulated FM signal, $y(t)$, is given by the formula:

Assume a normalized message with $|x_m(t)|\leq 1$. The parameter $f_{\Delta}$ is the peak frequency deviation in hertz, $t$ and $t_0$ are measured in seconds, and $\phi_0$ is the initial phase in radians. Every argument of the cosine is therefore dimensionless.

The derivative of the phase divided by $2\pi$ gives the instantaneous frequency $f_i(t)$ of the modulated signal at time $t$:

This equation shows a linear change in instantaneous frequency around the carrier frequency ($f_c$) according to the value of the modulating signal ($x_m(t)$). The parameter $f_{\Delta}$ is the frequency deviation, which sets the frequency shift for a unit value of $x_m(t)$.

The Modulation Index

In FM, the modulation index describes the relationship between frequency deviation and the bandwidth of the information signal. It is called the modulation index (denoted by $h$ or $\beta$). It is the dimensionless ratio of the maximum frequency deviation to the highest frequency of the modulating signal. For a single tone, this is $\beta=\Delta f/f_m$; for a band-limited signal, $f_m$ is replaced by its message bandwidth.

• ${\Delta f}_{\text{max}}$ is the maximum frequency deviation, occurring when the modulating signal reaches its peak amplitude.
• ${f_m}_{\text{max}}$ is the highest frequency present in the information signal spectrum (for example, approximately 3.4 kHz for speech and 15 to 20 kHz for Hi-Fi music).
• Both quantities in the ratio are frequencies measured in hertz, so the modulation index has no unit.

Bandwidth and Carson's Rule

The required FM bandwidth depends on the modulation index $h$. Two basic cases are distinguished:

• Narrowband FM (NBFM): Occurs when $h \ll 1$. The bandwidth of an NBFM signal is approximately twice the highest frequency of the modulating signal: $BW \approx 2 \cdot {f_m}_{max}$. This is similar to the bandwidth of an AM signal.
• Wideband FM (WBFM): Occurs when $h \gg 1$. WBFM requires more bandwidth but can provide a better signal-to-noise ratio with a suitable receiver. This variant is used, among other applications, for FM radio broadcasting.

Estimating FM Bandwidth: Carson's Rule

Carson's Rule is a practical approximation for a single-tone or band-limited FM signal. Under its usual 98% power criterion, about 98% of the signal power lies within the bandwidth estimated by Carson's Rule.

The actual occupied bandwidth depends on the modulation waveform and the selected power-containment criterion.

Example: FM Broadcast Parameters

For an example FM broadcast system in North America, use:

• Maximum frequency deviation: ${\Delta f}_{max} = 75 \text{ kHz}$
• Highest modulating frequency for Hi-Fi audio: ${f_m}_{max} = 15 \text{ kHz}$

The modulation index is:

$h = \frac{75 \text{ kHz}}{15 \text{ kHz}} = 5$

Since $h = 5$exceeds 1, this is a Wideband FM system. We then use Carson's Rule to calculate the required bandwidth:

$BW = 2 \cdot (75 \text{ kHz} + 15 \text{ kHz}) = 2 \cdot (90 \text{ kHz}) = 180 \text{ kHz}$

In this example, the radio channel is 200 kHz wide, for example from 98.1 MHz to 98.3 MHz. Compared with the approximate 180 kHz occupied bandwidth, 20 kHz remains in total, or 10 kHz on each side of the center frequency, reducing the risk of interference.

Advantages and Disadvantages of FM

This is an analog FM model, so bandwidth, signal-to-noise ratio, and distortion are the primary metrics. BER applies after an additional digital decision stage, such as FSK.