FM Modulation Notes

Modulation Index and Deviation Ratio of FM

• Modulation Index (β):

  • Definition: The ratio of the frequency deviation to the maximum modulating signal frequency.

  • Formula: $β = \frac{Frequency\ deviation}{Maximum\ modulating\ frequency}$ or $β = \frac{δ}{f_m(max)}$

• Deviation Ratio (D):

  • Definition: The ratio of the maximum frequency deviation to the maximum modulating frequency.

  • Formula: $D = \frac{δ<em>{max}}{f</em>m(max)}$

• Time Domain Representation of FM Waveform:

  • A sketch illustrating the relationship between the carrier signal, modulating signal, and the resulting FM signal over time.

Comparison of Frequency Modulation (FM) and Phase Modulation (PM)

FM Wave Calculations

• Given Equation: $V = 20 sin[10^8 t + 4 sin(1200π t)]$

  • General form: $E<em>c sin[ω</em>c t + β sin(ω_m t)]$

  • $E_c = 20 V$

  • Carrier Frequency (f_c):

    • $ω_c = 10^8$

    • $2π f_c = 10^8$

    • $f_c = \frac{10^8}{2π} ≈ 15.9 MHz$

  • Modulating Frequency (f_m):

    • $ω_m = 1200π$

    • $2π f_m = 1200π$

    • $f_m = \frac{1200π}{2π} ≈ 600 Hz$

  • Modulation Index (β):

    • $β = 4$

  • Maximum Deviation (δ):

    • $δ = β × f_m$

    • $δ = 4 × 600 = 2400 Hz$

  • Power Dissipated (P) in an 8-Ω Resistor:

    • $P = \frac{E_c^2}{2R}$

    • $P = \frac{(20)^2}{2 × 8} = \frac{400}{16} = 25 Watts$

FM Generation by Armstrong Method

• Block Diagram:

  • Crystal Oscillator: Generates a stable carrier frequency.

  • Buffer: Provides isolation between stages.

  • 90° Phase Shifter: Shifts the carrier signal by 90 degrees.

  • Balanced Modulator: Suppresses the carrier and produces sidebands (DSB-SC).

  • Audio Equalizer: Processes the modulating signal (applied before balanced modulator).

  • Combining Network: Adds the carrier and sidebands to achieve phase modulation.

  • Frequency Multipliers & Mixers: Boost the operating signal.

  • Power Amplifier: Amplifies the FM signal for transmission.

  • Antenna: Radiates the FM signal.

• Phaser Diagram:

  • Illustrates the phase relationships between the carrier and modulating signals.

  • $V_m$ refers presumably to the voltage of the modulating signal.

  • $V_c$ may represent the voltage of the carrier signal.

• Construction:

  • Crystal Oscillator generates a carrier signal (typically around 1 MHz).

  • The carrier is fed to a 90° phase shifter and a combining network via a buffer circuit.

  • The audio frequency modulating signal is passed through an audio equalizer.

  • A balanced modulator receives the phase-shifted carrier and equalized audio signal.

  • Multipliers and mixers boost the operating signal.

• Working:

  • Crystal Oscillator: Generates a stable carrier frequency (e.g., 1 MHz).

    • This frequency is lower than the transmission frequency and is boosted using frequency multipliers.

  • Buffer: Provides isolation between stages.

  • 90° Phase Shifter:

    • Shifts the carrier signal by 90°.

    • The phase-shifted carrier is fed to one input of the balanced modulator.

  • Balanced Modulator:

    • Generates a Double Sideband Suppressed Carrier (DSB-SC) signal.

    • Receives the 90° phase-shifted carrier and the equalized audio modulating signal.

    • Output consists of two sidebands.

    • The resultant of the sideband voltages is out of phase with the carrier vector.

  • Combining Network:

    • An adder circuit that combines the zero-degree phased unmodulated carrier vector and the output of the balanced modulator.

    • Achieves phase modulation at the output.

FM with Sinusoidal Signal

• A carrier is frequency modulated with a 2 kHz sinusoidal signal, resulting in a 6 kHz frequency deviation.

  • Original modulating signal:

    • Frequency: 2 kHz

    • Deviation: 6 kHz

  • Modified modulating signal:

    • Amplitude: Increased by a factor of 2.

    • Frequency: Halved (2 kHz / 2 = 1 kHz).

  • Maximum deviation (δ):

    • $δ$ is proportional to the amplitude of the modulating signal.

    • Since amplitude is doubled, new deviation = 6 kHz * 2 = 12 kHz

  • Modulation index (β):

    • $β = \frac{δ}{f_m}$

    • New $β = \frac{12 kHz}{1 kHz} = 12$

  • Bandwidth (BW):

    • Using Carson's rule, $BW = 2(δ + f_m)$

    • New $BW = 2(12 kHz + 1 kHz) = 26 kHz$

Pre-emphasis in FM

• Definition:

  • The process of boosting high-frequency signals before modulation at the transmitter in an FM system.

• Circuit Diagram:

  • A high-pass filter circuit consisting of a resistor and capacitor.

• Frequency Response:

  • Signals with higher modulation frequencies have lower SNR.

  • Pre-emphasis enhances high-frequency signals to compensate, thus allowing higher frequency signals to modulate the carrier at a higher level, causing more frequency deviation.

  • The pre-emphasis network acts as a differentiator or high-pass filter.

  • A pre-emphasis circuit produces an increase in the amplitude of the modulating signal with an increase in frequency.

NBFM and WBFM

Pre-emphasis and De-emphasis in Detail

• Pre-emphasis:

  • The process of boosting high frequencies before modulation at the transmitter in an FM system.

• Circuit Diagram:

  • Similar to a high-pass filter circuit.

  • Signals with higher modulation frequencies exhibit lower SNR.

  • The pre-emphasis network allows higher frequency signals to modulate the carrier at higher levels, causing more frequency deviation.

  • Pre-emphasis circuit increases the amplitude of the modulating signal with an increase in frequency.

• De-emphasis:

  • The inverse process of pre-emphasis, used to compensate for the boosted frequencies at the receiver after demodulation.

  • Pre-emphasis and de-emphasis together help produce a more uniform SNR throughout the modulating signal frequency spectrum.

  • De-emphasis circuit consists of a low-pass filter.

• By the use of an active pre-emphasis network, we can reduce the signal loss & distortion with the increase of SNR.

Superheterodyne FM Receiver

• Block Diagram:

  • RF Amplifier: Amplifies the received RF signal.

  • Mixer: Converts the RF signal to an intermediate frequency (IF).

  • Local Oscillator: Generates a signal to mix with the RF signal.

  • IF Amplifier: Amplifies the IF signal.

  • FM Demodulator: Extracts the audio signal from the FM signal.

  • Audio Amplifier: Amplifies the audio signal.

  • Speaker: Reproduces the audio.

Balanced Slope Detector

• Phasor Diagram:

  • $Vc$: the voltage of the carrier signal

  • $V_{out}$: the voltage of the output signal

  • $f_c$: the frequency of the carrier signal

  • $f$: the frequency of the signal at a specific time

• Description:

  • Also known as Travis Detector, triple-tuned discriminator, or amplitude discriminator.

  • Consists of two slope detectors connected back-to-back to opposite ends of a center-tapped transformer.

  • The input to these two slope detectors is fed 180° out of phase.

  • When the input frequency is between the extremes, the overall output has some intermediate value.

  • The polarity of the output depends on which side of fc the input frequency lies.

• Limitations of balanced slope detector:

• Alignment is difficult.

• Amplitude limiting is not provided.

• Linearity is insufficient.

FM Signal Parameters

• Given FM signal: $ac(t) = 10 cos [2π × 10^8 t] + 5 sin [2π × 200 t]$

• Standard FM equation: $Ec cos [ω<em>c t + mf sin(ω</em>m t)]$

• Parameters:

  • $Ec = 10$

  • $mf = 5$

  • Carrier Frequency (fc):

    • $ω_c = 2π × 10^8$

    • $fc = 10^8 Hz$

  • Modulating Frequency (fm):

    • $ω_m = 2π × 200$

    • $fm = 200 Hz$

  • Peak Frequency Deviation (δ):

    • $δ = mf × fm$

    • $δ = 5 × 200 = 1000 Hz$

  • Modulation Index (mf):

    • $mf = 5$

Angle Modulation

• Definition:

  • Modulation in which the phase or frequency of the carrier is varied in accordance with the amplitude of the modulating signal.

Relationship between FM & PM

• Frequency is the rate of change of phase (derivative of phase).

• Modulation index is proportional to phase deviation and inversely proportional to modulating frequency.

• As signal frequency increases, phase changes more rapidly.

FM Generation - Direct and Indirect Methods

• Direct Method:

  • Generates Wideband FM (WBFM) directly.

  • Uses a Voltage Controlled Oscillator (VCO).

  • VCO produces an output frequency proportional to the input signal voltage.

• Indirect Method (Armstrong Method):

  • (Refer to previous notes for details of Armstrong method)

Block Diagram of Direct Generation of FM Wave using VCO
Modulating Signal -> Voltage Controlled Oscillator -> FM Wave

VCO can be designed using

*Reactance Modulator method
*Varactor Diode Method

Balanced Slope Detector

• Circuit Diagram:

  • (Refer to diagrams, showing a center-tapped transformer and two diodes)

• Construction:

  • Two slope detectors connected back-to-back to the opposite ends of the center-tapped transformer

  • Input to the two slope detectors is fed 180° out of phase.

  • One slope detector is tuned above the intermediate frequency (IF).

  • The other slope detector is tuned below the IF (fc - $δ$f).

Working

A frequency modulated signal will have frequency variations. The balanced slope detector converts these to amplitude variations, and then rectifies and filters them to recover the original audio.