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Spectra of FM signals

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Frequency modulation varies the frequency of a sine wave carrier depending on the source signal.

Frequency modulation - transmit signal
Transmit signal.

The difference between the instantaneous and center frequency of the carrier is proportional to the modulating signal.

Δf(t)= k M m(t) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaamOzaiaacIcacaWG0bGaaiykaiabg2da9iaadUgadaWgaaWcbaGaamytaaqabaGccaWGTbGaaiikaiaadshacaGGPaaaaa@38FF@

Where

Δf(t) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaamOzaiaacIcacaWG0bGaaiykaaaa@32BD@ frequency deviation

k M MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBaaaleaacaWGnbaabeaaaaa@3007@ sensitivity of the frequency modulator [ V Hz ] MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadaWcaaqaaiaadAfaaeaacaWGibGaamOEaaaaaiaawUfacaGLDbaaaaa@32C2@

m(t) MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacIcacaWG0bGaaiykaaaa@315D@ modulating signal


Modulating signalSpectrum
DC 0V Modulating Signal
FM Carrier
DC 1V Modulating Signal
FM Frequenz Deviation
Frequency modulation example - frequency deviation is 1 kHz for a 1V-DC modulating signal

In this experiment a sinewave signal is frequency modulated. Modulating signal and modulator parameters determine the spectrum of the resulting FM transmission signal.

FM

Start

Modulating signalSpectrum
Modulation index ß=2 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaaikdaaaa@313F@

ß= Δ f max f m = k M m ^ f m MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9maalaaabaGaeyiLdqKaamOzamaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaaGcbaGaamOzamaaBaaaleaacaWGTbaabeaaaaGccqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGaamytaaqabaGcceWGTbGbaKaaaeaacaWGMbWaaSbaaSqaaiaad2gaaeqaaaaaaaa@3E1B@

Where

ß MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aaaa@2F7D@ modulation index

k M MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBaaaleaacaWGnbaabeaaaaa@3007@ modulation constant

m ^ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaajaaaaa@2F1B@ modulating signal amplitude

f m MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGTbaabeaaaaa@3022@ modulating sinewave signal frequency

The modulation index for the initial setting is:

ß= k M m ^ f m = 1kHz/V2V 1kHz =2 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9maalaaabaGaam4AamaaBaaaleaacaWGnbaabeaakiqad2gagaqcaaqaaiaadAgadaWgaaWcbaGaamyBaaqabaaaaOGaeyypa0ZaaSaaaeaacaaIXaGaam4AaiaadIeacaWG6bGaai4laiaadAfacqGHflY1caaIYaGaamOvaaqaaiaaigdacaWGRbGaamisaiaadQhaaaGaeyypa0JaaGOmaaaa@44D5@

FM Modulator
The modulation index ß MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaac+daaaa@3146@ is the ratio of the maximum frequency deviation of the carrier to the frequency of the sinewave modulating signal.

The Bessel function values at the resulting modulation index determine the spectrum of the FM signal.

Experiment

Vary the modulating signal amplitude m ^ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaajaaaaa@2F1B@ .

Signal Generator

The modulation index is proportional to the modulating signal amplitude. In this setting the amplitude in Volts is the modulation index:

ß= k M m ^ f m = 1kHz/V m ^ 1kHz = m ^ /V MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9maalaaabaGaam4AamaaBaaaleaacaWGnbaabeaakiqad2gagaqcaaqaaiaadAgadaWgaaWcbaGaamyBaaqabaaaaOGaeyypa0ZaaSaaaeaacaaIXaGaam4AaiaadIeacaWG6bGaai4laiaadAfacqGHflY1ceWGTbGbaKaaaeaacaaIXaGaam4AaiaadIeacaWG6baaaiabg2da9iqad2gagaqcaiaac+cacaWGwbaaaa@4614@

The adjusted modulating signal amplitude determines the spectral amplitudes of the carrier and sideband frequencies. For some values the carrier or specific sideband frequencies disappear. This relates to zero crossings of the respective Bessel function at the corresponding modulation index.

Modulating signalSpectrum
Modulation index ß=1 MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaaigdaaaa@313E@
When does the carrier frequency disappear?

Modulation index ß=? MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaac+daaaa@3146@

When does the first sideband frequency disappear?

Modulation index ß=? MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam43aiabg2da9iaac+daaaa@3146@

FM signal spectra for sinewave modulation with different modulation indices.

Note

The carrier frequency is 0 Hz in this setting. It might be changed via the modulator properties.

Next steps

This simulation implements frequency modulation. The FM signal is generated for the chosen modulating signal. Its spectrum is shown in a spectrum analyzer. All parameters of the modulating signal and modulator can be adjusted.

To change the different settings click on the corresponding wiring:

Adjust parameters of input signal
Signal Generator
Adjust parameters of FM modulator

Left click on FM modulator:

FM Modulator
Open measure for transmission signal

Right click on s(t):

Measure Selector labAlive
Adjust parameters of FM


1. Order the spectra by modulation index! A cosine source signal is used.

    Modulation index

  • 0.88
  • 1.435
  • 1.84
  • 2.4

    Transmit spectrum