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4. Analog frequency modulation and demodulation

## Introduction

This Java applet allows you to experience the basics of frequency modulation and demodulation visually. It’s interesting to look at the spectrogram and try changing the various parameters. In this way you can understand the background to the characteristics of the signal.

## An outline of frequency modulation and demodulation

With analog frequency modulation, the modulating signal is an analog signal, and it’s given the name FM (frequency modulation). Frequency modulation is used in many fields including transceivers and FM radio broadcasting. There are various reasons why FM is used, but the main reasons are as follows.

1, It has good tone quality
2, It’s resistant to noise
3, The hardware is cheap
4, It has a good track record with reliable communication
5, You can set an appropriate modulation factor to improve signal to noise
7, The technology is well established
8, Component parts are readily available
9, Other

As far as tone quality is concerned, the same level can be achieved with other modulation methods too, and it isn’t a quality inherent to FM. It’s simply determined by the bandwidth assigned by the Radio Act. Whether analog modulation or digital modulation is used, the main reason why frequency modulation radios are common is that they offer good performance for the price.

## The theory of frequency modulation and demodulation

◆Frequency modulation shown as a formula
With frequency modulation, the frequency of the carrier wave is shifted proportionally to the amplitude of the modulating signal. Therefore the frequency of the modulated wave is shifted continuously, and since a different spectrum from the modulating signal occurs centered on the carrier frequency, modulation is nonlinear.

In frequency modulation, the modulated wave Sfm can be expressed with the following equation.

Kfm: Constant

With frequency modulation, the carrier wave phase Φc is shifted proportionally to the integral of the modulating signal m(t). The formula shows a form similar to phase modulation, while frequency modulation integrates the modulating signal m(t) before phase modulation.

The maximum frequency shift Δf of the instantaneous frequency is as follows. Δf is also called deviation.

When the modulating signal m(t) is a single sine wave

Therefore

Accordingly

When the initial phase Φc of the carrier wave is Φc = 0, the frequency modulated wave is

Specifying the frequency modulation index m gives the following.

When the frequency modulation index m is less than 1, it’s known as narrowband FM, and if it’s greater than 1, it’s known as wideband FM. Wideband FM has good signal to noise ratio when it’s demodulated at the receiver, so it’s used widely in FM radios and so on. The higher the modulation index, the wider the bandwidth must be for transmission. Higher deviation during demodulation improves signal to noise, but this increases the bandwidth required for transmission. The priority given here depends on the system requirements. Improving the signal to noise by increasing the deviation is called FM gain.

◆FM modulated wave spectrum
The spectrum of the frequency modulation modulated wave occurs above and below the carrier wave Fc, and the frequency is the integeral multiple of the modulating signal Fm. In the spectrum, the modulation signal frequency Fm, deviation Δf, and frequency modulation index m are related as follows. (When the modulating signal is a single sine wave)
The spectrum interval is the modulation signal frequency Fm, and it spreads in an infinite frequency band. Deviation Δf is the difference in the center frequency Fc of the carrier wave and the frequency of the modulation index number from Fc.

◆The required bandwidth for frequency modulated waves
The envelope amplitude of frequency modulated waves is said to be constant, but this is only when all spectrum components are collected, and actually, amplitude fluctuation appears due to frequency band restrictions.

The bandwidth B required so that amplitude fluctuation doesn’t become a problem for demodulation is as follows. Δf is the maximum frequency shift, m is the modulation index, and Fm is the modulation signal frequency.

When the maximum frequency of the modulating signal is fixed, a wide transmission band is required if the modulation index m is high. When the maximum frequency shift Δf is fixed, the spectral interval becomes narrow if the modulation index m is high.

For example, the occupied band width of FM radio is 200 kHz and the maximum frequency shift Δf is 75 kHz, so the maximum frequency Fm and modulation index m of the modulated wave are as follows.
Fm = B/2-Δf = 200/2-75 = 25 kHz
m = Δf/Fm = 75/25 = 3

The modulation index is greater than 1, so it is wideband FM.

◆The frequency modulated wave spectrum is a Bessel function of the first kind
The frequency modulated wave spectra appear chaotic and difficult to analyze theoretically, but the modulation index m of the spectrum is expressed as a Bessel function of the first kind Jn(m)

The figure below is an applet for calculating the frequency modulated wave spectrum You can see that when the frequency modulation index changes, the shape of the spectrum also changes. The frequency modulation and demodulation applet finds and displays the modulated wave spectrum modulated by the carrier wave and modulating signal using fast fourier transform, but if you compare this spectrum with a spectrum obtained with a Bessel function of the first kind, you can see that they are the same shape. This applet sets the degree at 10, but the spectrum actually expands indefinitely. For actual radio communication, you must limit the frequency band to the bandwidth set by the Radio Act.

## Actual frequency modulation and demodulation

◆Modulation
There are two ways to generate a frequency modulation signal, direct FM and indirect FM. With frequency modulation, the information is incorporated in the frequency, so a nonlinear power amplifier can be used which ensures good electrical efficiency.

・Direct FM
With direct FM, the modulating signal is input in a VCO (voltage control oscillator) which generates a frequency shift proportionally to the added voltage. The signal input of the VCO has a variable capacitance diode, and when a modulating signal voltage is applied here, the capacitance of the variable capacitance diode changes and the oscillation frequency (carrier frequency) changes.

・ Indirect FM
We won’t go into detail here. Please refer to a good book on the subject.

◆Demodulation
The frequency discrimination method or PLL detection method can be used to demodulate the frequency modulation signal. Demodulation in this applet uses envelope detection after differentiation of the modulated wave.

With frequency modulation, information is carried by the frequency of the carrier wave, so it’s said to be resistant to noise in the amplitude direction. FM detectors detect frequencies and convert them into voltages, and since they respond to amplitude fluctuations, these cause errors. The received signal is affected by fading and outside noise which affects the amplitude direction and appears as distortion or noise in the detector output signal. Therefore a limiter circuit is built in before the detector in order to eliminate amplitude fluctuations.

## Frequency modulation and demodulation applet

A frequency modulation and demodulation applet Click the “Explain” button at the bottom right of the applet for how to use it.