6.2 Frequency Modulation, Phase Modulation, and FM Detectors

6.2 Frequency Modulation, Phase Modulation, and FM Detectors

Unlike amplitude modulation—where the carrier envelope conveys intelligence while its frequency remains constant—Angle Modulation maintains a strictly constant carrier amplitude while varying the instantaneous phase or frequency. Angle modulation encompasses two closely interrelated processes: Frequency Modulation (FM) and Phase Modulation (PM). Celebrated for its remarkable immunity to amplitude-dependent atmospheric and man-made noise, FM serves as the premier analog modulation mode for amateur VHF and UHF repeaters, simplex mobile communications, and aerospace links.

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1. Angle Modulation Fundamentals: FM vs. PM

General Angle Modulation Representation

An unmodulated radio frequency carrier is defined by:

$$v_c(t) = V_c \cos[\theta(t)] = V_c \cos[\omega_c t + \phi(t)]$$

where $V_c$ is the constant carrier amplitude, $\theta(t)$ is the total instantaneous phase angle in radians, and $\phi(t)$ is the instantaneous phase departure from nominal.

The instantaneous angular frequency $\omega_i(t)$ is the time derivative of instantaneous phase:

$$\omega_i(t) = \frac{d\theta(t)}{dt} = \omega_c + \frac{d\phi(t)}{dt}$$ $$f_i(t) = \frac{1}{2\pi} \frac{d\theta(t)}{dt} = f_c + \frac{1}{2\pi}\frac{d\phi(t)}{dt}$$

Frequency Modulation (FM)

In true Frequency Modulation, the instantaneous frequency deviation of the carrier from its unmodulated center frequency is directly proportional to the instantaneous amplitude of the modulating signal $v_m(t)$, regardless of the modulating signal's frequency:

$$f_i(t) = f_c + k_f v_m(t)$$

where $k_f$ is the frequency sensitivity factor in $\text{Hz/Volt}$.

For a sinusoidal modulating signal $v_m(t) = V_m \cos(2\pi f_m t)$:

$$f_i(t) = f_c + k_f V_m \cos(2\pi f_m t) = f_c + \Delta f \cos(2\pi f_m t)$$

where $\Delta f = k_f V_m$ is the peak frequency deviation. Note that in FM, $\Delta f$ depends exclusively on the modulating audio amplitude $V_m$ and is completely independent of the audio frequency $f_m$.

Integrating $f_i(t)$ over time gives the instantaneous phase $\theta(t)$:

$$\theta(t) = 2\pi \int f_i(t) \, dt = 2\pi f_c t + \frac{\Delta f}{f_m} \sin(2\pi f_m t)$$

Defining the FM modulation index $\beta = \frac{\Delta f}{f_m}$:

$$v_{\text{FM}}(t) = V_c \cos[2\pi f_c t + \beta \sin(2\pi f_m t)]$$

Phase Modulation (PM)

In Phase Modulation, the instantaneous phase deviation $\phi(t)$ of the carrier from its unmodulated state is directly proportional to the instantaneous amplitude of the modulating signal:

$$\phi(t) = k_p v_m(t)$$

where $k_p$ is the phase sensitivity factor in $\text{radians/Volt}$.

For a sinusoidal modulating signal $v_m(t) = V_m \cos(2\pi f_m t)$:

$$\theta(t) = 2\pi f_c t + k_p V_m \cos(2\pi f_m t) = 2\pi f_c t + \Delta \theta \cos(2\pi f_m t)$$

where $\Delta \theta = k_p V_m$ is the peak phase deviation.

To find the equivalent instantaneous frequency in a PM wave, differentiate phase with respect to time:

$$f_i(t) = \frac{1}{2\pi} \frac{d\theta(t)}{dt} = f_c - k_p V_m f_m \sin(2\pi f_m t)$$

The equivalent peak frequency deviation in a PM system is:

$$\Delta f_{\text{PM}} = k_p V_m f_m$$

Critical Engineering Distinction: In Frequency Modulation (FM), frequency deviation $\Delta f$ depends only on audio amplitude ($V_m$). In Phase Modulation (PM), frequency deviation $\Delta f$ is proportional to both audio amplitude and audio frequency ($V_m \times f_m$). A higher-pitched audio tone of the same amplitude produces a proportionately greater frequency deviation in a PM transmitter.

Interconversion of FM and PM

Because frequency is the derivative of phase, FM and PM transmitters can be interconverted using simple passive filtering networks:

• PM from an FM Modulator: Passing the audio modulating signal through a differentiator ($+6\text{ dB/octave}$ high-pass filter) prior to a frequency modulator produces Phase Modulation.
• FM from a Phase Modulator (Armstrong Indirect Method): Passing the audio modulating signal through an integrator ($-6\text{ dB/octave}$ low-pass filter) prior to a phase modulator produces true Frequency Modulation.

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2. Modulation Index, Deviation Ratio, and Bessel Sideband Spectra

The Modulation Index ($\beta$)

The modulation index in angle-modulated systems represents the maximum phase displacement of the carrier in radians:

$$\beta = \frac{\Delta f}{f_m}$$

where $\Delta f$ is the peak frequency deviation and $f_m$ is the modulating frequency.

• Deviation Ratio ($D$): A special case of the modulation index defined under worst-case maximum design parameters:

$$D = \frac{\Delta f_{\text{max}}}{f_{m,\text{max}}}$$

• Narrowband FM (NBFM): Defined when $\beta \le 0.5$ to $1.0$. The spectrum resembles AM, containing essentially only the carrier and a single pair of significant sidebands.
• Wideband FM (WBFM): Defined when $\beta \gg 1.0$ (such as commercial broadcast FM where $\Delta f = \pm 75\text{ kHz}, f_{m,\text{max}} = 15\text{ kHz} \implies D = 5.0$). A vast constellation of sidebands is produced.

Bessel Functions and Spectral Distribution

Expanding the equation $v_{\text{FM}}(t) = V_c \cos[\omega_c t + \beta \sin(\omega_m t)]$ into discrete sinusoidal frequencies requires the application of Bessel functions of the first kind of order $n$, denoted $J_n(\beta)$:

$$v_{\text{FM}}(t) = V_c \left\{ J_0(\beta) \cos(\omega_c t) + \sum_{n=1}^\infty J_n(\beta) [\cos((\omega_c + n \omega_m)t) + (-1)^n \cos((\omega_c - n \omega_m)t)] \right\}$$

This expansion reveals vital physical properties of FM signals:

1. Theoretical Infinite Bandwidth: The summation index $n$ extends to infinity. In theory, an FM signal generates an infinite number of sideband pairs spaced at integer multiples of the modulating frequency ($f_c \pm f_m, f_c \pm 2f_m, f_c \pm 3f_m, \dots$).
2. Carrier Amplitude Varies with Modulation: The amplitude of the carrier component at $f_c$ is multiplied by $J_0(\beta)$. As modulation index $\beta$ increases, carrier amplitude drops and can reach zero.
3. Constant Total Power: Unlike AM (where transmitted power increases with modulation depth), total power in an FM signal is strictly constant:

$$P_{\text{total}} = \frac{V_c^2}{2R} = \frac{V_c^2}{2R} \left[ J_0^2(\beta) + 2 \sum_{n=1}^\infty J_n^2(\beta) \right]$$

Energy is simply redistributed from the carrier into the sideband pairs as modulation increases.

Bessel Coefficients Jn(beta) for Various Modulation Indices:

beta      J0 (Carrier)   J1 (1st SB)   J2 (2nd SB)   J3 (3rd SB)   J4 (4th SB)
----------------------------------------------------------------------------
0.0           1.00          0.00          0.00          0.00          0.00
0.5           0.94          0.24          0.03          0.00          0.00
1.0           0.77          0.44          0.11          0.02          0.00
2.0           0.22          0.58          0.35          0.13          0.03
2.405         0.00 (Null!)  0.52          0.43          0.20          0.06
5.0          -0.18         -0.33          0.05          0.36          0.39
5.520         0.00 (Null!) -0.34         -0.12          0.25          0.40

The Carrier Null Phenomenon and Transmitter Calibration

At specific mathematical values of $\beta$, the zeroth-order Bessel function $J_0(\beta)$ passes through zero, causing the carrier at $f_c$ to disappear completely from the RF spectrum:

• First Carrier Null: $\beta \approx 2.4048$
• Second Carrier Null: $\beta \approx 5.5201$
• Third Carrier Null: $\beta \approx 8.6537$

Practical Measurement: With undistorted single-tone FM, observe a protected sample of the RF output and increase modulation from zero to the first carrier null. The first zero of $J_0$ occurs at $\beta\approx2.4048$, so deviation at that test condition is:

$$\Delta f = 2.4048 \times f_m$$

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3. Carson's Bandwidth Rule and Amateur Channel Spacing

Carson's Bandwidth Rule

An FM spectrum has infinitely many sidebands. Carson's rule estimates a practical bandwidth commonly described as containing about $98\%$ of the signal power:

$$B\approx2(\Delta f+f_{m,\text{max}})$$

Here $\Delta f$ is peak deviation and $f_{m,\text{max}}$ is the highest modulating frequency. At maximum design deviation, $D=\Delta f_{\text{max}}/f_{m,\text{max}}$, so $B\approx2f_{m,\text{max}}(D+1)$. For a single tone, use $\beta=\Delta f/f_m$ and $B\approx2f_m(\beta+1)$. The rule is an engineering estimate, not proof that an emission mask is met.

Amateur VHF/UHF Repeaters: $25\text{ kHz}$ vs. $12.5\text{ kHz}$ Systems

The following illustrate common wide- and narrow-deviation settings. Use the applicable band plan and repeater specification to select channel spacing and deviation:

1. Standard / Wide FM ($25\text{ kHz}$ Channel Spacing):
   • Peak deviation: $\Delta f = \pm 5.0\text{ kHz}$
   • Audio bandwidth: $f_{m,\text{max}} = 3.0\text{ kHz}$
   • Deviation ratio: $D=5.0/3.0\approx1.67$
   • Carson's bandwidth:

$$B = 2 (5.0\text{ kHz} + 3.0\text{ kHz}) = 2 \times 8.0\text{ kHz} = 16.0\text{ kHz}$$

The $16.0\text{ kHz}$ estimate is narrower than $25\text{ kHz}$ channel spacing. The difference is not an empty guard band: smaller sidebands remain outside the Carson estimate. Adjacent-channel performance also depends on audio shaping, deviation and transmitter/receiver filtering.

1. Narrowband FM (NBFM, $12.5\text{ kHz}$ Channel Spacing):
   • Peak deviation: $\Delta f = \pm 2.5\text{ kHz}$
   • Audio bandwidth: $f_{m,\text{max}} = 3.0\text{ kHz}$
   • Deviation ratio: $D=2.5/3.0\approx0.833$
   • Carson's bandwidth:

$$B = 2 (2.5\text{ kHz} + 3.0\text{ kHz}) = 2 \times 5.5\text{ kHz} = 11.0\text{ kHz}$$

The estimate is $11.0\text{ kHz}$ for this setting. A $12.5\text{ kHz}$ raster provides twice as many nominal channel centres as a $25\text{ kHz}$ raster over the same span, but filtering and spectral compliance must be checked separately.

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4. Worked Calculation: Carson's Bandwidth and Deviation Calibration

Worked Example 6.2: Deviation and Carrier Null Calibration

An amateur radio club operates a $146.700\text{ MHz}$ 2-metre repeater. The transmitter audio deviation limiter must be calibrated to ensure standard wide-deviation limits ($\,\Delta f = \pm 5.0\text{ kHz}\,)$ are not exceeded.

1. Calculate the occupied bandwidth under voice modulation ($f_{m,\text{max}} = 3.0\text{ kHz}$):

$$B = 2 (\Delta f + f_{m,\text{max}}) = 2 (5.0\text{ kHz} + 3.0\text{ kHz}) = 16.0\text{ kHz}$$

1. Determine the exact audio frequency ($f_m$) needed to produce the first carrier null ($\,\beta = 2.4048\,$) at precisely $\pm 5.0\text{ kHz}$ deviation:

$$\beta = \frac{\Delta f}{f_m} \implies f_m = \frac{\Delta f}{\beta} = \frac{5000\text{ Hz}}{2.4048} \approx 2079.2\text{ Hz}$$

1. Measurement and limiter checks:
   • Feed the transmitter into a rated dummy load and couple a suitably attenuated RF sample to the analyser. Use the manufacturer's test input and settings.
   • Apply a low-distortion $2079.2\text{ Hz}$ tone and increase modulation from zero while avoiding audio clipping or limiting. Resolve the carrier and identify its first null. For sinusoidal FM, that null corresponds to approximately $5.0\text{ kHz}$ peak deviation.
   • This checks one modulation condition. Set and verify the deviation limiter separately using the manufacturer's procedure, measuring positive and negative peak deviation over the specified audio-frequency and input-level range. A distorted tone or a single null cannot certify maximum voice deviation.

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5. Pre-emphasis and De-emphasis Networks

The Triangular Noise Characteristic

FM demodulators (discriminators and ratio detectors) operate by detecting frequency changes over time. When white Gaussian thermal noise (which has a flat power spectral density $S_n(f) = N_0 / 2$) enters an FM detector, the demodulated output noise voltage ($e_n$) increases linearly with audio frequency:

$$e_n(f) \propto f$$

Because noise power is proportional to voltage squared ($P_n \propto e_n^2$), the demodulated noise power spectral density increases quadratically:

$$S_{\text{noise}}(f) \propto f^2$$

This phenomenon is known as the triangular noise spectrum:

Demodulated Noise Power
      ^
      |                                * (High noise at 3 kHz)
      |                             *
      |                          *
      |                       *
      |                    *
      |               *  (Triangular Noise: Pn proportional to f^2)
      |          *
      +----+----+----+----+----+----+----> Audio Frequency (f)
          500  1000 1500 2000 2500 3000 Hz

Consequently, the high-frequency components of human speech ($2000\text{ Hz}$ to $3000\text{ Hz}$, which contain low acoustic energy but are critical for consonant intelligibility) suffer a catastrophic drop in signal-to-noise ratio compared to low-frequency voice fundamentals.

Pre-emphasis (Transmitter)

Pre-emphasis raises high-frequency speech components before modulation so that matching receiver de-emphasis can reduce high-frequency detector noise. Narrowband two-way voice systems commonly use approximately $+6\text{ dB/octave}$ ($+20\text{ dB/decade}$) over the $300$–$3000\text{ Hz}$ voice band, with band limiting and deviation control.

For example, doubling audio frequency from $1$ to $2\text{ kHz}$ gives about twice the voltage gain in the emphasis slope:

$$20\log_{10}(2)\approx6.02\text{ dB}$$

The extra boost uses deviation headroom, so the audio limiter and following low-pass filter must prevent excessive deviation and unwanted spectrum. The $50$/$75\,\mu\text{s}$ regional constants associated with FM broadcasting are not a universal amateur voice setting.

De-emphasis (Receiver)

Matching de-emphasis applies approximately $-6\text{ dB/octave}$ over the intended voice band. Together the two responses restore the intended speech response while the receiver roll-off attenuates high-frequency detector noise.

The noise improvement depends on the audio/noise spectra, bandwidth and permitted deviation. It is not a fixed $10$–$13\text{ dB}$ gain for every voice link. Use the radio's specified audio path; a flat data port may intentionally bypass voice emphasis.

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6. FM Generation Circuits: Direct vs. Indirect

Direct: Audio ---> Varactor-controlled oscillator ---> FM

Indirect (Armstrong):
Audio ---> Integrator ---> phi(t) -----------+
                                             v
Crystal ---> Quadrature RF sin(wc*t) ---> [Multiplier] ---> -phi*sin -+
    |                                                                 |
    +------> Carrier cos(wc*t) --------------------------------------> [Sum]
                                                                      |
                                                                      v
                                                   Limiter ---> Multipliers ---> FM

The indirect diagram uses a small phase excursion and normalized amplitudes. The sum is approximately $\cos(\omega_ct)-\phi(t)\sin(\omega_ct)$, the small-angle form of $\cos[\omega_ct+\phi(t)]$.

1. Direct FM Generation (Varactor Diode Tuning)

In direct FM, the modulating audio voltage directly modulates the reactance of an oscillator's frequency-determining resonant tank:

• A varactor (varicap) diode is connected across an LC tank circuit or quartz crystal.
• The audio signal reverse-biases the varactor diode; because junction depletion capacitance varies inversely with reverse voltage ($C_j \propto V_R^{-\gamma}$), the total tank capacitance shifts with the audio signal.
• Advantage: Highly sensitive; easily produces large frequency deviations.
• Challenge: Difficult to maintain carrier frequency stability. Standard LC oscillators drift with temperature. Direct FM designs can use an Automatic Frequency Control (AFC) loop or a Phase-Locked Loop (PLL) synthesizer for stability; mixing a low-frequency modulated oscillator with a stable crystal oscillator is another approach.

2. Indirect FM Generation (Armstrong Method)

Invented by Edwin Howard Armstrong, indirect FM achieves crystal-controlled center frequency stability by phase-modulating a fixed crystal oscillator:

1. A stable quartz crystal oscillator generates the carrier.
2. Modulating audio is passed through an integrator ($-6\text{ dB/octave}$ network) to convert phase modulation into frequency modulation.
3. A balanced modulator forms a product of the integrated audio and a quadrature carrier. Combine these small sidebands with the unshifted carrier to approximate narrowband angle modulation; a limiter removes the residual amplitude variation. Keep phase excursion small. With integrated audio, the result approximates NBFM (for example, $50\text{ Hz}$ peak deviation).
4. A cascade of Class C frequency multiplier stages (e.g. triplers and doublers totalling $\times 48$ or $\times 96$) multiplies both the carrier frequency and the frequency deviation up to the final operating band:

$$f_{\text{final}} = N \cdot f_{\text{osc}}, \quad \Delta f_{\text{final}} = N \cdot \Delta f_{\text{osc}}$$

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7. FM Demodulation Circuit Architectures

Demodulating an FM signal requires converting instantaneous frequency deviations into proportional voltage fluctuations. Several distinct circuit topologies have evolved:

1. The Slope Detector

The simplest and crude FM detector consists of an ordinary LC tank tuned slightly off resonance (on the skirt or "slope" of the selectivity curve) followed by a diode envelope detector. As incoming frequency swings, the detuned tank's impedance changes, translating frequency shifts into amplitude variations. It is non-linear, has poor audio fidelity, and cannot reject amplitude noise; it is never used in high-performance equipment.

2. The Foster-Seeley Discriminator

A classic double-tuned transformer circuit where primary and secondary are coupled both inductively and through a center-tap capacitor. At center frequency, secondary voltage is $90^\circ$ out of phase with primary voltage. As frequency deviates, this phase angle swings above and below $90^\circ$. Two diodes rectify the vector sum, producing an output voltage proportional to frequency departure. It provides excellent linearity but responds directly to amplitude fluctuations; it must always be preceded by a hard limiter stage to strip AM noise.

3. The Ratio Detector

A brilliant refinement of the Foster-Seeley discriminator. One of the diodes is reversed, and a large electrolytic reservoir capacitor ($C \approx 2 - 10\ \mu\text{F}$) is shunted across the diode load. The capacitor charges to the total average carrier voltage, clamping the sum voltage constant. Output audio is taken from the center point as a ratio of the two diode voltages.

• Key Advantage: The stabilizing capacitor absorbs rapid amplitude transients, providing inherent amplitude limiting and AM rejection. A ratio detector operates cleanly without a preceding limiter stage.

4. The Quadrature Detector

The standard analog IC detector (e.g. MC3361, MC3362). The amplified IF signal splits into two paths: one feeds an internal Gilbert-cell phase comparator; the other passes through an external high-Q phase-shift tank (or ceramic discriminator) tuned to produce an exact $90^\circ$ phase shift at center frequency. Frequency departures shift this phase angle, and the phase detector outputs a DC error voltage directly proportional to instantaneous frequency deviation. It requires only a single external tuned coil or ceramic resonator, making it the dominant circuit in compact FM receivers.

5. Phase-Locked Loop (PLL) Demodulator

A Voltage-Controlled Oscillator (VCO) inside a PLL is locked to the incoming FM intermediate frequency. As the carrier frequency deviates with modulation, the phase detector outputs an error voltage that forces the VCO to track the incoming carrier frequency. This internal error control voltage is a direct, highly linear reproduction of the modulating audio signal.

• Advantages: Superior linearity, widest dynamic range, and threshold extension capabilities (operating down to low carrier-to-noise ratios).

FM In ---->| Phase     |----[ Loop Filter ]----+----> Demodulated Audio Out
           | Detector  |                       |
           +-----------+                       |
                 ^                             |
                 |       +-------------+       |
                 +-------| Voltage-    |<------+
                         | Controlled  |
                         | Osc (VCO)   |
                         +-------------+

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8. Comparison of FM Demodulator Architectures

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9. Receiver Limiting, Quieting, and the Capture Effect

IF Limiting and Receiver Quieting

A high-gain IF limiter clips the positive and negative peaks, giving a nearly constant-amplitude waveform while preserving its zero-crossing timing. It rejects unwanted amplitude variation, but phase/frequency fluctuations remain and can produce detector noise. Deep fading and multipath distortion are not automatically removed.

With no wanted signal, an unsquelched FM receiver produces hiss. As carrier-to-noise ratio increases, noise causes smaller phase excursions relative to the wanted carrier and the recovered audio becomes quieter. This is receiver quieting. A sensitivity specification such as 20 dB quieting states the RF input needed for a defined reduction in output noise; it is not evidence that a limiter erases phase noise.

The Capture Effect

With two co-channel signals, an FM receiver generally favours the stronger and suppresses the weaker. The capture ratio is the required input-level difference for a specified suppression under the test conditions:

$$\text{Capture ratio}=20\log_{10}\left(\frac{V_{\text{strong}}}{V_{\text{weak}}}\right)$$

A ratio of a few decibels is typical of good FM reception, but the stated ratio belongs to a specified suppression test, not a guarantee of perfectly undisturbed audio. Near-equal signals can interfere or switch dominance as their levels change. On a repeater, simultaneous co-channel speech is commonly called doubling; mobile multipath fading can cause rapid flutter.