Electronics Guide

Mathematical Description of FM and PM

A general sinusoidal carrier can be expressed as v(t) = A cos(theta(t)), where theta(t) is the instantaneous phase angle. For an unmodulated carrier at frequency f_c, theta(t) = 2 pi f_c t + phi₀, where phi₀ is the initial phase. The instantaneous frequency is defined as the time derivative of phase divided by 2 pi:

f(t) = (1/2 pi) d theta(t)/dt

In phase modulation, the instantaneous phase varies directly with the modulating signal m(t):

theta(t) = 2 pi f_c t + k_p m(t)

where k_p is the phase sensitivity in radians per volt. The resulting PM signal is:

v_PM(t) = A cos(2 pi f_c t + k_p m(t))

In frequency modulation, the instantaneous frequency varies directly with the modulating signal:

f(t) = f_c + k_f m(t)

where k_f is the frequency sensitivity in Hz per volt. Integrating to find the phase and substituting gives the FM signal:

v_FM(t) = A cos(2 pi f_c t + 2 pi k_f integral of m(t) dt)

The maximum frequency deviation Delta f = k_f m_max represents the peak excursion of the instantaneous frequency from the carrier frequency.

Modulation Index and Bandwidth

The modulation index is a key parameter that characterizes the extent of angle modulation. For sinusoidal modulation m(t) = A_m cos(2 pi f_m t), the FM modulation index beta is defined as:

beta = Delta f / f_m

where Delta f is the peak frequency deviation and f_m is the modulating frequency. For PM with the same modulating signal, the modulation index equals the peak phase deviation in radians.

Unlike AM, where the bandwidth is simply twice the modulating frequency, FM and PM signals have theoretically infinite bandwidth due to the nonlinear nature of angle modulation. However, practical bandwidth can be estimated using Carson's rule:

B = 2(Delta f + f_m) = 2 f_m(beta + 1)

For narrowband FM (beta much less than 1), the bandwidth approaches 2 f_m, similar to AM. For wideband FM (beta much greater than 1), the bandwidth approaches 2 Delta f. Broadcast FM radio uses beta values around 5, with 75 kHz maximum deviation and 15 kHz maximum modulating frequency, yielding approximately 180 kHz bandwidth.

Spectrum of Angle-Modulated Signals

The spectrum of an FM or PM signal with sinusoidal modulation consists of the carrier and an infinite series of sidebands at frequencies f_c plus or minus n f_m, where n is a positive integer. The amplitude of each spectral component is proportional to Bessel functions of the first kind:

v(t) = A sum over n from minus infinity to infinity of J_n(beta) cos(2 pi (f_c + n f_m) t)

where J_n(beta) is the Bessel function of order n and argument beta. For small modulation indices, only J₀ and J₁ have significant values, resulting in a spectrum similar to AM. As beta increases, higher-order sidebands become significant, spreading the signal energy over a wider bandwidth.

An interesting property of Bessel functions is that J₀(beta) = 0 for certain values of beta (approximately 2.4, 5.5, 8.7, etc.), meaning the carrier component vanishes at these modulation indices. This behavior has practical implications for carrier recovery and can be used for modulation index measurement.

Relationship Between FM and PM

The mathematical relationship between FM and PM allows conversion between the two modulation types. Since the FM phase variation is the integral of the frequency variation, an FM signal can be generated by first integrating the modulating signal and then applying it to a phase modulator:

FM = PM with input integral of m(t)

Conversely, PM can be generated by differentiating the modulating signal and applying it to a frequency modulator:

PM = FM with input d m(t)/dt

This relationship is fundamental to the indirect FM generation method developed by Armstrong, where a crystal-controlled oscillator maintains frequency stability while phase modulation is applied and then converted to FM through integration of the modulating signal.

From a practical standpoint, FM and PM produce similar signals when the modulating signal contains a wide range of frequencies. The primary difference appears in the frequency response: FM maintains constant deviation regardless of modulating frequency, while PM deviation increases linearly with modulating frequency for constant amplitude input. This distinction affects pre-emphasis and de-emphasis circuit design in broadcast systems.

Voltage-Controlled Oscillators

The voltage-controlled oscillator (VCO) is the fundamental building block for direct FM generation. A VCO produces an output frequency that varies linearly with an applied control voltage. When the modulating signal is applied to the control input, the output frequency varies in proportion to the modulating signal, producing FM directly.

LC oscillators can be converted to VCOs by including a varactor diode in the frequency-determining network. The varactor's capacitance varies with reverse bias voltage, typically following a relationship C = C₀(1 + V/V₀)^-n, where n ranges from 0.3 to 2 depending on the diode junction profile. The modulating signal varies the varactor capacitance, which in turn varies the oscillation frequency.

For linear frequency-versus-voltage characteristics, the varactor should contribute only a fraction of the total tank capacitance. This linearization reduces the maximum deviation achievable but improves the distortion performance. Alternatively, specially designed hyperabrupt varactors with n approximately equal to 2 can provide nearly linear frequency variation when operated with appropriate bias levels.

Common VCO topologies for FM generation include Colpitts, Hartley, and Clapp oscillators modified with varactor tuning. The choice depends on the frequency range, required tuning linearity, and stability requirements. Clapp oscillators offer good frequency stability because the resonant frequency depends primarily on series inductance and capacitance rather than transistor parameters.

Reactance Modulator Circuits

The reactance modulator is a classic circuit for direct FM generation that simulates a variable reactance controlled by the modulating signal. A transistor or FET is configured so that its input impedance appears as either a variable capacitance or inductance when viewed from the oscillator tank circuit.

In a typical reactance modulator, a transistor is connected across the oscillator tank with a phase-shifting network that creates a 90-degree relationship between the voltage across and current through the device. This quadrature relationship makes the transistor appear reactive rather than resistive. The transconductance variation with bias modulates the effective reactance.

The modulating signal is applied to the transistor base or gate, varying its transconductance and hence the effective reactance presented to the tank. As this reactance changes, the oscillator frequency changes proportionally. The circuit can be designed to appear as either capacitive or inductive reactance, adding or subtracting from the tank's inherent components.

Reactance modulators can achieve moderate frequency deviations with reasonable linearity. However, they are sensitive to transistor parameter variations and require careful design to minimize distortion. The circuit's effectiveness depends on maintaining the proper phase relationship across the frequency range and modulation extremes.

Crystal Oscillator FM

Direct frequency modulation of crystal oscillators provides excellent frequency stability while allowing small frequency deviations. The crystal can be pulled slightly off its nominal frequency by adding variable reactance in series or parallel, with the pulling range limited by the crystal's high Q factor.

Typical crystal pulling ranges are on the order of 0.01% to 0.1% of the crystal frequency, depending on the crystal type and circuit configuration. This limited deviation suits applications requiring high stability with small modulation, such as narrowband FM radio systems and some telemetry applications.

Voltage-controlled crystal oscillators (VCXOs) incorporate varactor diodes to enable electronic frequency control. The varactor is typically placed in series with the crystal, where it contributes to the total series capacitance and hence the operating frequency. Temperature compensation can be included to maintain stability across environmental conditions.

For applications requiring both crystal stability and wide deviation, direct crystal FM may be followed by frequency multiplication stages that increase both the center frequency and the deviation proportionally. This approach combines the stability advantages of crystal control with the wide deviation needed for broadcast-quality FM.

Frequency Stability Considerations

The primary disadvantage of direct FM generation is frequency instability. Since the oscillator center frequency is not locked to a stable reference, it can drift with temperature, supply voltage variations, and component aging. This drift appears directly in the output signal, potentially causing reception problems if the drift exceeds the receiver's tuning range.

Several techniques address the stability problem in direct FM systems. Automatic frequency control (AFC) uses feedback from a frequency discriminator to correct slow drift. The AFC loop bandwidth must be much lower than the lowest modulating frequency to avoid affecting the intended modulation.

Temperature compensation reduces drift by using components with opposing temperature coefficients that cancel each other's effects. Negative-temperature-coefficient (NTC) capacitors or thermistors can compensate for positive drift in other components, maintaining the center frequency within acceptable limits.

Phase-locked loop (PLL) stabilization locks the VCO's average frequency to a crystal reference while allowing the FM modulation to pass through. The PLL bandwidth is set low enough that modulation frequencies are not attenuated, while frequencies below the loop bandwidth are corrected by the feedback action. This approach is widely used in modern FM transmitters.

Armstrong Indirect FM Transmitter

The Armstrong indirect FM transmitter begins with a crystal oscillator operating at a frequency much lower than the final output frequency. The crystal provides excellent frequency stability but operates at a low center frequency where the required phase deviation is manageable.

Phase modulation is applied to the crystal oscillator output using a balanced modulator and combining network. The modulating signal is first passed through an integrator (to convert the desired FM to equivalent PM), then fed to the phase modulator. The resulting narrowband FM signal has the correct modulation characteristic but at too low a frequency and with too little deviation.

Frequency multiplier stages increase both the center frequency and the frequency deviation by the multiplication factor. Multiple stages may be cascaded to achieve the required multiplication. A typical broadcast FM transmitter might use a crystal at around 200 kHz and multiply by 432 times (various combinations of doublers and triplers) to reach 88-108 MHz with 75 kHz deviation.

The multiplication process also multiplies any residual phase noise or spurious signals, so the initial modulator must be designed for low distortion and noise. Additionally, the multiplication chain requires careful filtering between stages to suppress harmonics and ensure that only the desired frequency reaches the final amplifier.

Phase Modulator Design

Phase modulators for indirect FM can be implemented using several techniques. The balanced modulator approach combines the carrier with a 90-degree phase-shifted component that has been amplitude-modulated by the (integrated) modulating signal. When the modulation index is small, the resulting vector sum approximates pure phase modulation.

In the balanced modulator phase modulator, the carrier passes through two paths. The direct path provides the reference carrier component. The quadrature path shifts the carrier by 90 degrees, applies amplitude modulation from the (integrated) modulating signal, and recombines with the direct path. The vector sum has constant amplitude (for small modulation) with phase varying according to the modulating signal.

The small-angle approximation limits the phase deviation to approximately plus or minus 0.5 radians (about 30 degrees) for acceptable distortion. Larger deviations cause the amplitude to vary noticeably, introducing AM distortion into what should be a pure PM signal. Higher deviations require more sophisticated modulator designs or multiple stages of small-deviation modulation.

Varactor phase modulators offer another implementation approach. A varactor diode in a transmission line or filter structure shifts the phase of the passing signal according to the control voltage. By applying the modulating signal to the varactor bias, phase modulation results directly. This approach can achieve wider deviation with lower distortion than balanced modulator designs.

Frequency Multiplication

Frequency multipliers increase the output frequency as an integer multiple of the input frequency while also multiplying any frequency deviation. If a narrowband FM signal with deviation Delta f passes through an n-times multiplier, the output has deviation n Delta f. This property is essential for indirect FM generation, where small initial deviations must be increased to broadcast levels.

Frequency doublers and triplers use the nonlinearity of diodes or transistors to generate harmonics of the input signal. A tuned circuit at the output selects the desired harmonic while rejecting the fundamental and other harmonics. Class C amplifiers driven into saturation and cutoff naturally produce the required harmonics.

Varactor frequency multipliers use the voltage-dependent capacitance of varactor diodes to generate harmonics. The input signal modulates the varactor capacitance, producing mixing products at harmonic frequencies. These multipliers can be very efficient, with conversion efficiencies exceeding 70% for doubling at microwave frequencies.

Step recovery diodes (SRDs) offer very efficient high-order multiplication. When a SRD switches from forward to reverse conduction, it generates a fast current pulse rich in harmonics. Proper circuit design can efficiently extract a specific high-order harmonic, enabling single-stage multiplication factors of 10 or more. SRD multipliers are common in microwave frequency synthesizers.

Heterodyne Methods

When simple multiplication cannot achieve the desired output frequency efficiently, heterodyne (mixing) methods combine multiplication with frequency translation. The modulated signal is multiplied to an intermediate frequency, then mixed with another oscillator to translate it to the final output frequency.

Consider a target output of 100 MHz with 75 kHz deviation. Direct multiplication from a 100 kHz crystal with 17.4 Hz deviation would require multiplication by 1000 times, yielding 100 MHz carrier and 17.4 kHz times 1000 = 17.4 kHz deviation, far short of the 75 kHz goal. Using heterodyne techniques, the signal can be multiplied to a convenient intermediate frequency, then mixed to the final frequency while maintaining the deviation achieved at the intermediate stage.

The mixing process adds the phases of the two input signals. If one input is unmodulated (a crystal oscillator), the output retains the modulation of the other input but at a new carrier frequency. This allows flexibility in achieving the desired final frequency without being constrained to simple multiplication ratios.

Care must be taken to filter the mixer output properly, as mixing produces both sum and difference frequencies along with various spurious products. Image rejection and spurious suppression are important considerations in heterodyne FM transmitter design.

Direct Phase Modulation

Direct phase modulation shifts the phase of a carrier signal according to the modulating signal. Unlike FM, which inherently integrates the modulating signal, PM preserves the modulating waveform directly in the phase variations. For analog modulation, this means PM has increasing deviation with increasing modulating frequency, the opposite of FM's behavior.

Vector modulator circuits can produce PM by controlling both in-phase (I) and quadrature (Q) components of the carrier. The desired phase angle theta produces I = cos(theta) and Q = sin(theta). By generating these components with balanced modulators and combining them, any phase angle can be achieved. This I/Q modulation approach is fundamental to modern digital communication systems.

Transmission line phase shifters provide another direct PM approach. A signal passing through a transmission line experiences phase shift proportional to the line's electrical length. By varying the effective length through switched sections or variable reactances, the phase can be modulated. PIN diode switches and varactor tuning enable electronic phase control at RF and microwave frequencies.

Reflection-type phase modulators use varactors in a reflective termination to vary the phase of a reflected signal. The varactor's impedance variation changes the reflection coefficient phase while maintaining nearly constant magnitude. These modulators can achieve 180 degrees or more of phase shift with good linearity and are widely used in phased array systems.

Digital Phase Modulation

Digital phase modulation encodes digital data by switching the carrier phase among discrete states. Binary phase shift keying (BPSK) uses two phase states separated by 180 degrees to represent binary 0 and 1. Quadrature phase shift keying (QPSK) uses four phase states at 90-degree intervals, encoding two bits per symbol.

Higher-order PSK schemes such as 8-PSK and 16-PSK encode more bits per symbol by using more closely spaced phase states. However, the reduced spacing between adjacent states increases susceptibility to noise and phase errors. The choice of modulation order balances spectral efficiency against error rate performance.

Differential phase modulation encodes data in the phase change between successive symbols rather than the absolute phase. Differential BPSK (DBPSK) and differential QPSK (DQPSK) simplify demodulation by eliminating the need for carrier phase recovery, as only phase changes matter. The trade-off is slightly reduced noise performance compared to coherent detection.

Implementation of digital phase modulators typically uses I/Q baseband processing. The data bits select the appropriate I and Q values from a lookup table, which are then applied to a quadrature upconverter to produce the RF output. Digital signal processing enables precise control of the constellation points and allows implementation of pulse shaping for spectrum control.

Quadrature Amplitude Modulation

While strictly a combination of amplitude and phase modulation, quadrature amplitude modulation (QAM) evolved from phase modulation concepts and shares many implementation techniques. QAM varies both the amplitude and phase of the carrier, placing symbol points anywhere in the I-Q plane rather than only on a constant-amplitude circle.

QAM constellations such as 16-QAM and 64-QAM achieve higher spectral efficiency than PSK by using amplitude variation to distinguish symbols that would otherwise require very closely spaced phase states. The rectangular QAM constellations (4x4 for 16-QAM, 8x8 for 64-QAM) are popular because they simplify implementation and allow independent I and Q data streams.

QAM modulators use the same I/Q architecture as PSK modulators, with independent control of both I and Q amplitudes. The baseband I and Q signals are generated digitally according to the desired constellation mapping, converted to analog, and applied to the quadrature upconverter. Linearity requirements are more stringent than for constant-envelope modulations because amplitude information must be preserved.

The trade-off for QAM's spectral efficiency is increased sensitivity to nonlinear distortion and amplitude errors. Power amplifiers must operate with sufficient back-off to maintain linearity, reducing efficiency. Adaptive predistortion and linearization techniques can improve performance, but the complexity of high-order QAM systems is substantially greater than constant-envelope alternatives.

Continuous Phase Modulation

Continuous phase modulation (CPM) maintains phase continuity at symbol transitions, avoiding the discontinuities present in abrupt phase-shift keying. This continuity constrains the modulated signal to a smoother trajectory through the phase plane, reducing spectral spreading and enabling constant-envelope transmission.

Minimum shift keying (MSK) is a special case of CPM that is also a form of continuous-phase FSK with modulation index 0.5. The phase changes by plus or minus 90 degrees over each symbol period, following a linear trajectory. MSK achieves the minimum bandwidth possible for binary orthogonal signaling while maintaining constant envelope.

Gaussian minimum shift keying (GMSK) passes the data signal through a Gaussian filter before the MSK modulator, further smoothing the phase trajectory and reducing bandwidth. GMSK is used in GSM cellular systems and achieves excellent spectral efficiency while allowing efficient nonlinear amplification.

CPM modulator implementation typically uses a frequency modulator driven by filtered and shaped data pulses. The instantaneous frequency is the derivative of the desired phase trajectory, which can be computed based on the data sequence and the chosen pulse shape. Digital implementations directly compute the phase trajectory and use it to address a lookup table for the I and Q samples.

Slope Detection

Slope detection, the simplest FM demodulation method, uses a tuned circuit offset from the carrier frequency so that the carrier operates on the slope of the circuit's frequency response. As the FM signal varies in frequency, it experiences varying attenuation from the tuned circuit, converting FM to AM. An envelope detector then recovers the modulating signal.

The tuned circuit is adjusted so the carrier frequency falls at a point where the amplitude response changes linearly with frequency. The FM signal, swinging above and below this center point, creates amplitude variations proportional to the frequency variations. After envelope detection, the output is a replica of the original modulating signal.

Slope detection suffers from poor linearity because the tuned circuit response is not perfectly linear over the required frequency range. The technique is also highly sensitive to AM on the input signal, as any amplitude variations pass directly to the output. These limitations make slope detection unsuitable for high-quality audio applications but acceptable for non-critical uses.

A balanced slope detector improves linearity by using two tuned circuits, one above and one below the carrier frequency. The difference between their detected outputs cancels even-order distortion and provides some AM rejection. This configuration, while more complex, achieves better performance than single-ended slope detection.

Foster-Seeley Discriminator

The Foster-Seeley discriminator is a classic FM demodulator circuit that converts frequency variations to voltage variations through the phase relationship between primary and secondary voltages in a transformer. At the center frequency, these voltages are in quadrature, and deviations from the center frequency create phase shifts that appear as amplitude differences after rectification.

In the Foster-Seeley circuit, the secondary winding is center-tapped and connected to two diode detectors. The primary voltage is applied to the center tap through a capacitor. At each diode, the detected voltage depends on the vector sum of the secondary voltage and the coupled primary voltage. When these are in quadrature (at the center frequency), both diodes detect equal voltages.

As the input frequency deviates above or below center, the phase relationship shifts, increasing the vector sum at one diode while decreasing it at the other. The difference between the two detector outputs is proportional to the frequency deviation, providing the demodulated output.

The Foster-Seeley discriminator has good linearity over a moderate frequency range and forms the basis for many FM receiver designs. Its primary limitation is sensitivity to amplitude variations, requiring a preceding limiter stage to remove any AM from the signal before demodulation. The limiter clips the signal to constant amplitude, eliminating AM while preserving FM.

Ratio Detector

The ratio detector, developed for consumer FM receivers, provides inherent AM rejection without requiring a separate limiter stage. The circuit is similar to the Foster-Seeley discriminator but with different diode polarities and the addition of a large capacitor that stabilizes the sum of the detected voltages.

In the ratio detector, both diodes are oriented in the same direction, and a large electrolytic capacitor connects across the entire detector output. This capacitor charges to the average value of the detected signals and cannot follow rapid amplitude variations. The output is taken as the difference between the two detector sections, but referenced to the stable voltage on the capacitor.

The ratio detector's AM rejection arises because amplitude variations equally affect both diodes, changing only the voltage across the stabilizing capacitor (which cannot follow rapid changes due to the long time constant). The ratio of the two detector outputs, and hence the demodulated signal, remains relatively constant despite AM on the input.

While the ratio detector offers simpler receiver design by eliminating the limiter, its AM rejection is not perfect, and its audio quality is somewhat inferior to a properly designed limiter-discriminator combination. Nevertheless, the ratio detector was widely used in broadcast FM receivers where cost was a primary consideration.

Quadrature Detector

The quadrature detector, widely used in integrated circuit FM demodulators, creates a 90-degree phase-shifted version of the FM signal and multiplies it with the original. At the center frequency, the multiplication of two signals in quadrature produces zero average output. Frequency deviations shift the phase relationship, producing a non-zero output proportional to the deviation.

The quadrature phase shift is typically produced by a tuned circuit, either external or integrated on-chip. At the circuit's resonant frequency, the voltage across it is in quadrature with the current through it. By deriving one multiplier input from the voltage and the other from the current (or a signal in phase with the current), the required quadrature relationship is established at the center frequency.

The multiplication produces output components at DC (proportional to the phase difference, hence frequency deviation) and at twice the carrier frequency. A low-pass filter removes the double-frequency component, leaving the demodulated baseband signal. The inherent limiting action of most multiplier implementations provides excellent AM rejection.

IC quadrature detectors such as the MC3359 and similar devices combine the limiter, quadrature multiplier, and low-pass filter in a single package. These integrated solutions provide good performance with minimal external components, typically requiring only an external quadrature coil and a few capacitors. The quadrature coil can often be a simple LC tank or even a ceramic resonator.

Phase-Locked Loop FM Demodulation

Phase-locked loops provide excellent FM demodulation performance, particularly for high-quality audio applications. When a PLL locks to an FM signal, its control voltage tracks the input frequency variations, directly providing the demodulated output. The feedback action continuously adjusts the VCO to match the instantaneous input frequency.

The PLL's control voltage represents the demodulated signal because this voltage must vary to make the VCO frequency equal to the instantaneous input frequency. If the input frequency increases, the control voltage must increase (for positive VCO sensitivity) to pull the VCO up to match. This varying control voltage is the recovered baseband signal.

PLL FM demodulators offer several advantages: excellent linearity (limited mainly by VCO linearity), inherent AM rejection through the limiting action of the phase detector, and the ability to track signals through brief noise bursts or signal fades. The loop bandwidth can be optimized for the expected modulation bandwidth, providing additional noise filtering beyond the limiter.

Integrated circuit PLL FM demodulators such as the NE565 and LM565 provide complete demodulation with few external components. More modern devices integrate the VCO and loop filter components, further reducing the external part count. The demodulated output linearity depends on the VCO linearity, which in IC implementations can be quite good.

Coherent Phase Detection

Coherent phase detection multiplies the received signal by a locally generated carrier at the same frequency and phase. When the phases are aligned, the detector output reflects the modulation. Any phase offset between the received carrier and local reference appears as a DC offset or rotation in the detected signal.

For BPSK demodulation, the multiplier output is positive when the received and reference phases are aligned, negative when they are 180 degrees apart. After low-pass filtering to remove the double-frequency component, a decision circuit determines whether the bit is 0 or 1 based on the sign of the output.

QPSK demodulation requires two parallel coherent detectors, one in phase (I) and one in quadrature (Q) with the carrier reference. Each detector output represents one of the two bits encoded in each QPSK symbol. The I and Q outputs are separately decided to recover the transmitted bit pairs.

The critical challenge in coherent detection is generating the correct local carrier reference. Carrier recovery circuits must extract the carrier phase from the received signal, which may not contain an explicit carrier component (depending on the modulation format). Squaring loops, Costas loops, and decision-directed loops are commonly used carrier recovery techniques.

Carrier Recovery Techniques

The squaring loop recovers the carrier from a suppressed-carrier signal by squaring the received signal. Squaring a BPSK signal produces a component at twice the carrier frequency, regardless of the data modulation. This doubled-frequency component is filtered, divided by two, and used as the coherent reference.

The Costas loop uses two parallel demodulation paths with quadrature local oscillators. The I and Q outputs are multiplied together to produce an error signal that is independent of the data but proportional to the phase error of the local oscillator. This error signal drives the oscillator phase toward the correct value.

Decision-directed carrier recovery uses the detected data to remove the modulation effect from the error computation. After making tentative decisions on the received symbols, the expected symbol phase is subtracted from the received phase, leaving a phase error estimate. This error is filtered and used to correct the oscillator phase.

All carrier recovery loops exhibit a phase ambiguity: the BPSK squaring loop can lock at either 0 or 180 degrees phase, and QPSK loops can lock at any multiple of 90 degrees. Differential encoding of the data resolves this ambiguity, ensuring that the demodulated data is correct regardless of which phase state the carrier recovery loop has chosen.

Differential Phase Detection

Differential phase detection compares the phase of each received symbol to the phase of the previous symbol, recovering information encoded in the phase change rather than the absolute phase. This approach eliminates the need for carrier recovery, simplifying the receiver at the cost of some performance degradation.

In differential detection, the current received symbol is multiplied by the complex conjugate of the previous symbol (equivalently, by the previous symbol delayed and phase-reversed). The product's phase equals the phase difference between successive symbols, which directly represents the transmitted data in differential encoding.

For DBPSK, the phase difference is either 0 degrees (no change, representing one bit value) or 180 degrees (inversion, representing the other bit value). The sign of the multiplier output indicates the transmitted bit. No explicit carrier reference is required because only relative phase matters.

DQPSK detection similarly compares successive symbol phases, with four possible phase changes (typically plus or minus 45 degrees and plus or minus 135 degrees) representing the four possible two-bit combinations. The I and Q components of the differential detector output are separately decided to recover both bits.

The performance penalty for differential detection compared to coherent detection is typically 1-3 dB, depending on the modulation format and noise conditions. This penalty arises because noise affects both the current symbol and the reference symbol (the previous symbol), effectively doubling the noise power in the detection process.

I/Q Demodulation

I/Q demodulation separately recovers the in-phase and quadrature components of the received signal, providing a complete representation of the signal's amplitude and phase in Cartesian coordinates. This approach is fundamental to modern digital receivers and enables a wide variety of signal processing techniques.

The I/Q demodulator uses two mixers driven by local oscillator signals in quadrature (90 degrees apart). The I mixer multiplies the received signal by cos(2 pi f_c t), while the Q mixer multiplies by -sin(2 pi f_c t). After low-pass filtering, the I and Q outputs represent the baseband in-phase and quadrature components.

The magnitude sqrt(I² + Q²) represents the signal amplitude, while the phase arctan(Q/I) represents the instantaneous phase. For FM demodulation, the phase can be differentiated to obtain the instantaneous frequency. For PM demodulation, the phase directly represents the modulating signal (after unwrapping any 2 pi discontinuities).

Digital I/Q demodulation samples the I and Q outputs with analog-to-digital converters and performs subsequent processing digitally. This approach offers flexibility to implement various demodulation algorithms in software or firmware, adapting to different modulation formats without hardware changes. Modern software-defined radios are built around this I/Q digitization concept.

Varactor Diode Selection and Application

Varactor diodes are the primary tuning element in most modern FM modulators and frequency synthesizers. Key selection parameters include capacitance range, tuning ratio (maximum to minimum capacitance), Q factor, and temperature stability.

Abrupt junction varactors provide moderate tuning ratio (typically 3:1 to 5:1) with relatively linear capacitance-voltage characteristics. They are suitable for general-purpose VCO tuning where moderate deviation is required. The capacitance varies approximately as (V_R)^-0.5 where V_R is the reverse voltage.

Hyperabrupt junction varactors provide higher tuning ratio (up to 20:1) and more linear frequency-versus-voltage characteristics. Their capacitance varies approximately as (V_R)^-2, which compensates for the square-root relationship between capacitance and frequency in an LC resonator, yielding nearly linear frequency tuning. These are preferred for wide-deviation FM modulators.

For best linearity and lowest distortion in FM modulators, the varactor should be operated in the most linear region of its characteristic, typically avoiding both very low bias (where nonlinearity is highest) and very high bias (where capacitance change per volt is minimal). Careful selection of bias point and varactor type can minimize harmonic distortion in the modulated output.

Limiter Circuit Design

Limiters remove amplitude variations from FM signals before demodulation, improving noise performance and eliminating AM interference. The limiter should provide sufficient gain to drive into saturation over the expected range of input signal levels while maintaining wide bandwidth to pass the FM sidebands undistorted.

Differential amplifier limiters use a cascade of differential gain stages that progressively limit the signal. Each stage provides moderate gain and limiting action; the cascade achieves the required total limiting. This approach maintains good symmetry and minimizes AM-to-PM conversion that could distort the FM signal.

The limiting threshold should be set to accommodate the expected range of input signal levels with adequate margin. Too high a threshold allows weak signals to pass without limiting, degrading noise performance. Too low a threshold may cause strong signals to overdrive the limiter, potentially creating spurious responses.

Bandwidth considerations are critical in limiter design. The limiter must pass all significant FM sidebands without attenuation or phase distortion. For broadcast FM with approximately 200 kHz occupied bandwidth, the limiter bandwidth should exceed 1 MHz to provide adequate margin. Insufficient bandwidth causes amplitude distortion of the outer sidebands, which converts to phase distortion after mixing in the demodulator.

Demodulator Alignment and Calibration

Foster-Seeley and ratio detector circuits require careful alignment of the discriminator transformer for optimal performance. The primary and secondary tuning must be adjusted for proper center frequency alignment and symmetrical response on either side of center.

Alignment typically uses an FM signal generator set to produce a sweep through the discriminator passband, with the demodulated output displayed on an oscilloscope. The familiar S-curve response should be centered at the carrier frequency with equal amplitude excursions above and below. Adjusting the tuning controls achieves symmetry and proper center frequency.

Quadrature detector alignment involves tuning the quadrature coil for zero output with an unmodulated carrier at the center frequency. This adjustment is typically less critical than discriminator alignment because the integrated circuit multiplier provides inherent centering as long as the quadrature phase shift is correct at the center frequency.

PLL demodulator adjustment involves setting the VCO free-running frequency close to the expected carrier frequency and selecting loop filter components for the desired bandwidth. The free-running frequency should be within the lock range for reliable acquisition. Loop bandwidth should be wide enough to track the modulation without distortion but narrow enough to provide good noise filtering.

Pre-emphasis and De-emphasis

Broadcast FM systems use pre-emphasis at the transmitter and de-emphasis at the receiver to improve the signal-to-noise ratio for high-frequency audio components. Pre-emphasis boosts high frequencies before transmission, and de-emphasis restores flat response at the receiver while reducing high-frequency noise that would otherwise degrade audio quality.

The FM noise spectrum has triangular shape, with noise power increasing proportionally with offset frequency from the carrier. High-frequency audio components, which naturally have lower amplitude than low frequencies, suffer worse signal-to-noise ratio without compensation. Pre-emphasis boosts these high frequencies so they better compete with the triangular noise spectrum.

Standard pre-emphasis curves are defined by time constants: 75 microseconds in North America and 50 microseconds in Europe and much of Asia. The pre-emphasis filter provides approximately 6 dB per octave boost above the corner frequency (2.1 kHz for 75 microseconds, 3.2 kHz for 50 microseconds). De-emphasis provides the complementary 6 dB per octave roll-off.

Pre-emphasis circuit implementation uses a simple RC high-pass filter section, while de-emphasis uses the corresponding RC low-pass filter. The time constant equals R times C for both circuits. Broadcast-quality implementations pay careful attention to component tolerances and temperature stability to maintain accurate response matching.

FM Broadcasting

FM broadcasting uses wideband frequency modulation in the VHF band (88-108 MHz in most regions) to deliver high-fidelity audio programming. The 75 kHz maximum deviation and 15 kHz audio bandwidth provide excellent audio quality while maintaining reasonable channel spacing. The 200 kHz channel allocation (250 kHz in some regions) allows for guard bands between stations.

FM stereo broadcasting uses a composite baseband signal that maintains compatibility with monaural receivers while carrying the stereo difference information. The left-plus-right (L+R) sum signal occupies 0-15 kHz, compatible with mono receivers. A 19 kHz pilot tone and the left-minus-right (L-R) difference signal on a suppressed carrier at 38 kHz enable stereo recovery in equipped receivers.

Additional subcarriers can be added to the composite signal for subsidiary communications. The Radio Data System (RDS) uses a 57 kHz subcarrier (the third harmonic of the stereo pilot) to transmit station identification, program type, traffic information, and other data. Some stations also carry additional audio services on separate subcarriers.

FM broadcast transmitters typically use indirect FM generation from crystal oscillators for frequency stability, with high-power amplifier chains to achieve the required effective radiated power. Modern transmitters use digital signal processing to generate the composite baseband signal and may use digital modulation techniques for improved performance and flexibility.

Two-Way Radio Communications

Narrowband FM serves land mobile, marine, and aviation two-way radio applications, where spectral efficiency requirements demand smaller channel spacing than broadcast FM. Typical narrowband FM systems use plus or minus 5 kHz deviation with 25 kHz channel spacing, or plus or minus 2.5 kHz deviation with 12.5 kHz channel spacing for higher-density applications.

Narrowband FM sacrifices some audio quality for spectral efficiency, with audio bandwidth limited to approximately 3 kHz. This bandwidth suffices for voice communications, where intelligibility is more important than fidelity. Pre-emphasis and de-emphasis improve the noise performance of the restricted bandwidth.

Continuous Tone-Coded Squelch System (CTCSS) uses sub-audible tones (typically 67-250 Hz) to provide selective calling capability. The receiver squelch opens only when both the carrier and the correct CTCSS tone are present, allowing multiple user groups to share a channel without hearing each other's communications. Digital Coded Squelch (DCS) provides similar functionality using digital codes.

Modern narrowband FM systems increasingly incorporate digital voice and data capabilities. Standards such as P25 (public safety) and DMR (Digital Mobile Radio) provide improved audio quality, better noise performance, and integrated data services while maintaining compatibility with existing channel plans and infrastructure.

Satellite Communications

Satellite communication systems use various forms of phase and frequency modulation depending on the application requirements. Analog video transmission traditionally used wideband FM, while digital services increasingly use phase shift keying modulation formats for better spectral efficiency.

QPSK and 8PSK are widely used for satellite digital television transmission. The constant envelope property allows efficient Class C or saturated amplifier operation in the satellite transponder, maximizing power efficiency where power is at a premium. Turbo codes and LDPC codes provide error correction that enables operation at signal levels only a few dB above theoretical limits.

Higher-order modulation schemes including 16APSK and 32APSK combine amplitude and phase modulation to achieve higher spectral efficiency when link conditions permit. These formats require more linear amplification and have lower noise immunity, so they are used selectively when signal strength is adequate.

Satellite navigation systems including GPS use binary phase modulation to spread the navigation signals across wide bandwidths. The spreading codes enable multiple satellites to share the same frequencies while providing ranging accuracy. The receivers use correlation processing to extract the navigation data and pseudo-range measurements from the weak, spread-spectrum signals.

Digital Wireless Systems

Modern digital wireless systems use sophisticated modulation formats that build on FM and PM fundamentals while adding techniques for spectral efficiency and robustness. Understanding the evolution from analog FM to digital modulation illustrates how fundamental concepts scale to meet increasing capacity demands.

GSM cellular systems use GMSK (Gaussian Minimum Shift Keying), a form of continuous-phase frequency modulation that maintains constant envelope while achieving good spectral efficiency. The Gaussian filtering of the data pulses before modulation reduces the FM bandwidth while the minimum shift keying (modulation index 0.5) provides orthogonal signaling.

CDMA systems use QPSK or similar phase modulation combined with direct sequence spreading. The spreading codes distinguish different users sharing the same frequency band while providing processing gain against interference and multipath. The underlying QPSK modulation transmits two bits per symbol period.

LTE and 5G systems use OFDM (Orthogonal Frequency Division Multiplexing), which divides the available bandwidth into many narrow subcarriers, each carrying QAM-modulated data. The many parallel narrowband transmissions can adapt individually to frequency-selective fading, and the orthogonality enables efficient spectrum utilization. Higher-order QAM constellations (up to 256-QAM or beyond) achieve high data rates when conditions permit.

FM Noise Performance

FM exhibits a characteristic noise behavior where the demodulated signal-to-noise ratio depends strongly on the modulation index and on whether the input signal-to-noise ratio is above or below a threshold. Above threshold, FM provides significant noise improvement over AM; below threshold, performance degrades rapidly.

The demodulated signal-to-noise ratio for wideband FM above threshold is:

SNR_out = 3 beta² (B/f_m) SNR_in

where beta is the modulation index, B is the IF bandwidth, and f_m is the maximum modulating frequency. This relationship shows that increasing the modulation index (and hence the bandwidth) improves the output signal-to-noise ratio at the expense of using more spectrum. This is the essence of the FM bandwidth-versus-noise trade-off.

The FM threshold occurs at input signal-to-noise ratios of approximately 10 dB for typical broadcast parameters. Below this level, noise impulses cause the PLL or discriminator to experience cycle slips or clicks, producing impulse noise in the demodulated audio that is far more objectionable than the steady noise above threshold. Threshold extension techniques can lower the threshold by a few dB.

Pre-emphasis and de-emphasis improve the effective output signal-to-noise ratio by approximately 13 dB for 75-microsecond pre-emphasis, taking advantage of the triangular noise spectrum to improve high-frequency audio SNR. This improvement is in addition to the wideband FM improvement and contributes significantly to FM broadcast audio quality.

Phase Noise and Jitter

Phase noise is the random variation in the phase of an oscillator or modulated signal, typically characterized by its spectral density at various offset frequencies from the carrier. Phase noise degrades the performance of phase-modulated systems by introducing errors in phase detection.

In a PSK system, phase noise causes the received constellation points to spread into clouds rather than precise points. When the phase noise spreads constellation points beyond the decision boundaries, errors result. Higher-order PSK and QAM systems with closer decision boundaries are more susceptible to phase noise degradation.

Phase noise from local oscillators in the transmitter and receiver both contribute to the total phase noise affecting the communication link. The dominant contribution typically comes from whichever oscillator has worse phase noise performance. Specifications for phase noise are usually given as dBc/Hz at specified offset frequencies.

Jitter, the time-domain manifestation of phase noise, is particularly important in digital transmission systems. Timing jitter in the recovered clock causes sampling point variations that can produce bit errors. Clock recovery circuits must filter jitter while tracking legitimate frequency variations, requiring careful loop bandwidth selection.

Distortion in FM Systems

FM systems can exhibit distortion from several sources including modulator nonlinearity, bandwidth limiting, and multipath propagation. Understanding these distortion mechanisms enables design optimization for minimum distortion.

Modulator nonlinearity occurs when the VCO frequency-versus-voltage characteristic deviates from linearity. The resulting harmonic distortion of the modulating signal appears in the demodulated output. Varactor diodes are a common source of modulator nonlinearity; hyperabrupt varactors or linearization techniques can reduce this distortion.

Bandwidth limiting distorts FM signals by attenuating or phase-shifting the outer sidebands. Because the FM spectrum has significant energy in these sidebands (especially for high modulation indices), their distortion appears as distortion in the demodulated signal. IF filters must provide adequate bandwidth with minimal in-band phase variation.

Multipath propagation causes frequency-selective fading that attenuates different spectral components of the FM signal by varying amounts. This effectively bandwidth-limits the signal in a time-varying manner, producing distortion that varies with the multipath environment. Receivers in mobile environments are particularly affected.

Spectral Efficiency Comparison

Different modulation schemes achieve different trade-offs between spectral efficiency (bits per second per hertz of bandwidth) and power efficiency (energy required per bit for a given error rate). Understanding these trade-offs guides modulation selection for specific applications.

Analog FM bandwidth depends on the modulation index, with wideband FM requiring many times the bandwidth of the original signal. This spectral inefficiency is accepted in exchange for improved noise performance, making FM suitable for broadcast applications where excellent audio quality justifies the bandwidth usage.

BPSK achieves 1 bit per second per hertz of bandwidth with moderate power efficiency. QPSK achieves 2 bits per second per hertz with approximately the same power efficiency as BPSK, effectively doubling spectral efficiency without power penalty. Higher-order PSK and QAM achieve progressively higher spectral efficiency but require increasingly higher signal-to-noise ratio.

Constant-envelope modulations such as MSK and GMSK achieve spectral efficiency comparable to QPSK while allowing efficient nonlinear amplification. This combination makes them attractive for power-limited applications such as portable and satellite equipment. The spectral efficiency of GMSK is approximately 1.35 bits/s/Hz, somewhat less than QPSK but achieved with simpler, more efficient transmitter hardware.