The Principle of Frequency Modulation

Frequency Modulation (FM) is a method of impressing information onto a high-frequency carrier wave. Unlike AM where the carrier's strength (amplitude) changes, in FM, the amplitude of the carrier wave remains constant. Instead, its frequency is varied in direct proportion to the instantaneous amplitude of the information signal (the modulating signal).

Imagine the carrier wave as a steady musical note. To encode an information signal (like your voice), FM changes the pitch of this note. When your voice is loud (high amplitude), the pitch goes up (higher frequency). When your voice is quiet (low amplitude), the pitch goes down (lower frequency). The loudness of the note itself never changes, only its pitch.

Interactive FM – Time Domain and Instantaneous Frequency

Waveforms

Instantaneous frequency fi(t) = fc + fΔ·xm(t)

Visualizing FM

The relationship between the modulating signal, the carrier, and the resulting FM signal is best understood visually.

• Modulating Signal: The top waveform is the information we want to send (e.g., 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. Notice that its amplitude is constant, identical to the carrier's. However, its frequency changes:
  • Where the modulating signal reaches its peak (maximum positive amplitude), the FM waves are packed most tightly together, indicating the highest frequency.
  • Where the modulating signal reaches its trough (maximum negative amplitude), the FM waves are most spread out, 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

We can describe these relationships mathematically. Let's define the signals:

• Information (modulating) signal: $x_m(t)$
• Carrier wave signal: $x_c(t) = A \cdot \sin(2\pi f_c t)$, 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:

While this formula looks complex, its derivative reveals the core principle. The instantaneous frequency, $f_i(t)$, of the modulated signal at any moment in time $t$ is much simpler:

This key equation tells us that the instantaneous frequency varies linearly around the central carrier frequency ($f_c$) in sync with the modulating signal's amplitude ($x_m(t)$). The parameter $f_{\Delta}$ is the Frequency Deviation, which acts as a sensitivity factor, determining how much the frequency shifts for a given amplitude of $x_m(t)$.

Measuring Modulation Depth: The Modulation Index

Similar to AM, we use a factor to describe how strongly the carrier wave is affected by the modulating signal. In FM, this is called the Modulation Index (denoted by $h$ or $\beta$). It is defined as the ratio of the maximum frequency deviation to the highest frequency component of the modulating signal.

• ${\Delta f}_{\text{max}}$ 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 spectrum of the information signal (e.g., approx. 3.4 kHz for human speech, 15-20 kHz for Hi-Fi music).

Bandwidth: The Price of Quality

One of the most significant characteristics of FM is its bandwidth requirement. Depending on the modulation index $h$, we distinguish between two main types:

• 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 a significantly wider bandwidth but in return offers a substantial improvement in signal-to-noise ratio. This is the technique used for high-quality FM radio broadcasting.

Estimating FM Bandwidth: Carson's Rule

A widely used and practical approximation for the bandwidth of an FM signal is known as Carson's Rule.

Real-World Example: FM Radio Broadcasting

Let's apply these concepts to standard FM radio broadcasting (e.g., in North America or Europe). The parameters set by regulatory bodies like the FCC are:

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

First, we calculate the modulation index:

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

Since $h = 5$ is much greater than 1, this is clearly a Wideband FM system. Now, we use Carson's Rule to calculate the required bandwidth:

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

This is why FM radio channels are typically allocated 200 kHz of spectrum space (e.g., from 98.1 MHz to 98.3 MHz), providing a 20 kHz guard band around the 180 kHz signal to prevent interference.

Advantages and Disadvantages of FM