Frequency modulation

For the application of frequency modulation to radio broadcasting, seeFM broadcasting.

"NFM" redirects here. For other uses, seeNFM (disambiguation).

A signal may be carried by anAMor FM radio wave.

FM has better noise (RFI) rejection than AM, as shown in this dramatic New York publicity demonstration byGeneral Electricin 1940. The radio has both AM and FM receivers. With a million-voltelectric arcas a source of interference behind it, the AM receiver produced only a roar ofstatic,while the FM receiver clearly reproduced a music program from Armstrong's experimental FM transmitterW2XMNin New Jersey.

Frequency modulation(FM) is the encoding ofinformationin acarrier waveby varying theinstantaneous frequencyof the wave. The technology is used intelecommunications,radio broadcasting,signal processing,andcomputing.

Inanalogfrequency modulation, such as radio broadcasting, of an audio signal representing voice or music, the instantaneousfrequency deviation,i.e. the difference between the frequency of the carrier and its center frequency, has a functional relation to the modulating signal amplitude.

Digital datacan be encoded and transmitted with a type of frequency modulation known asfrequency-shift keying(FSK), in which the instantaneous frequency of the carrier is shifted among a set of frequencies. The frequencies may represent digits, such as '0' and '1'. FSK is widely used in computermodems,such asfax modems,telephonecaller IDsystems, garage door openers, and other low-frequency transmissions.Radioteletypealso uses FSK.

Frequency modulation is widely used forFM radiobroadcasting.It is also used intelemetry,radar,seismic prospecting, and monitoringnewbornsfor seizures viaEEG,two-way radiosystems,sound synthesis,magnetic tape-recording systems and some video-transmission systems. In radio transmission, an advantage of frequency modulation is that it has a largersignal-to-noise ratioand therefore rejectsradio frequency interferencebetter than an equal poweramplitude modulation (AM)signal. For this reason, most music is broadcast overFM radio.

However, under severe enoughmultipathconditions it performs much more poorly than AM, with distinct high frequency noise artifacts that are audible with lower volumes and less complex tones.With high enough volume and carrier deviation audio distortion starts to occur that otherwise wouldn't be present without multipath or with an AM signal.

Frequency modulation andphase modulationare the two complementary principal methods ofangle modulation;phase modulation is often used as an intermediate step to achieve frequency modulation. These methods contrast withamplitude modulation,in which theamplitudeof the carrier wave varies, while the frequency and phase remain constant.

Theory

If the information to be transmitted (i.e., thebaseband signal) is$x_{m}(t)$and thesinusoidalcarrier is$x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,$,wheref_cis the carrier's base frequency, andA_cis the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:

${\begin{aligned}y(t)&=A_{c}\cos \left(2\pi \int _{0}^{t}f(\tau )d\tau \right)\\&=A_{c}\cos \left(2\pi \int _{0}^{t}\left[f_{c}+f_{\Delta }x_{m}(\tau )\right]d\tau \right)\\&=A_{c}\cos \left(2\pi f_{c}t+2\pi f_{\Delta }\int _{0}^{t}x_{m}(\tau )d\tau \right)\\\end{aligned}}$

where$f_{\Delta }=K_{f}A_{m}$,$K_{f}$being the sensitivity of the frequency modulator and$A_{m}$being the amplitude of the modulating signal or baseband signal.

In this equation,$f(\tau )\,$is theinstantaneous frequencyof the oscillator and$f_{\Delta }\,$is thefrequency deviation,which represents the maximum shift away fromf_cin one direction, assumingx_m(t) is limited to the range ±1.

It is important to realize that this process of integrating the instantaneous frequency to create an instantaneous phase is quite different from what the term "frequency modulation" naively implies, namely directly adding the modulating signal to the carrier frequency

${\begin{aligned}y(t)&=A_{c}\cos \left(2\pi \left[f_{c}+f_{\Delta }x_{m}(t)\right]t\right)\end{aligned}}$

which would result in a modulated signal that has spurious local minima and maxima that do not correspond to those of the carrier.

While most of the energy of the signal is contained withinf_c±f_Δ,it can be shown byFourier analysisthat a wider range of frequencies is required to precisely represent an FM signal. Thefrequency spectrumof an actual FM signal has components extending infinitely, although their amplitude decreases and higher-order components are often neglected in practical design problems.

Sinusoidal baseband signal

Mathematically, a baseband modulating signal may be approximated by asinusoidalcontinuous wavesignal with a frequencyf_m.This method is also named as single-tone modulation. The integral of such a signal is:

$\int _{0}^{t}x_{m}(\tau )d\tau ={\frac {\sin \left(2\pi f_{m}t\right)}{2\pi f_{m}}}\,$

In this case, the expression for y(t) above simplifies to:

$y(t)=A_{c}\cos \left(2\pi f_{c}t+{\frac {f_{\Delta }}{f_{m}}}\sin \left(2\pi f_{m}t\right)\right)\,$

where the amplitude$A_{m}\,$of the modulatingsinusoidis represented in the peak deviation$f_{\Delta }=K_{f}A_{m}$(seefrequency deviation).

Theharmonicdistribution of asine wavecarrier modulated by such asinusoidalsignal can be represented withBessel functions;this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.

Modulation index

As in other modulation systems, the modulation index indicates by how much the modulated variable varies around its unmodulated level. It relates to variations in thecarrier frequency:

$h={\frac {\Delta {}f}{f_{m}}}={\frac {f_{\Delta }\left|x_{m}(t)\right|}{f_{m}}}$

where$f_{m}\,$is the highest frequency component present in the modulating signalx_m(t), and$\Delta {}f\,$is the peak frequency-deviation – i.e. the maximum deviation of theinstantaneous frequencyfrom the carrier frequency. For a sine wave modulation, the modulation index is seen to be the ratio of the peak frequency deviation of the carrier wave to the frequency of the modulating sine wave.

If$h\ll 1$,the modulation is callednarrowband FM(NFM), and its bandwidth is approximately$2f_{m}\,$.Sometimes modulation index$h<0.3$is considered NFM and other modulation indices are considered wideband FM (WFM or FM).

For digital modulation systems, for example, binary frequency shift keying (BFSK), where a binary signal modulates the carrier, the modulation index is given by:

$h={\frac {\Delta {}f}{f_{m}}}={\frac {\Delta {}f}{\frac {1}{2T_{s}}}}=2\Delta {}fT_{s}\$

where$T_{s}\,$is the symbol period, and$f_{m}={\frac {1}{2T_{s}}}\,$is used as the highest frequency of the modulating binary waveform by convention, even though it would be more accurate to say it is the highestfundamentalof the modulating binary waveform. In the case of digital modulation, the carrier$f_{c}\,$is never transmitted. Rather, one of two frequencies is transmitted, either$f_{c}+\Delta f$or$f_{c}-\Delta f$,depending on the binary state 0 or 1 of the modulation signal.

If$h\gg 1$,the modulation is calledwideband FMand its bandwidth is approximately$2f_{\Delta }\,$.While wideband FM uses more bandwidth, it can improve thesignal-to-noise ratiosignificantly; for example, doubling the value of$\Delta {}f\,$,while keeping$f_{m}$constant, results in an eight-fold improvement in the signal-to-noise ratio.(Compare this withchirp spread spectrum,which uses extremely wide frequency deviations to achieve processing gains comparable to traditional, better-known spread-spectrum modes).

With a tone-modulated FM wave, if the modulation frequency is held constant and the modulation index is increased, the (non-negligible) bandwidth of the FM signal increases but the spacing between spectra remains the same; some spectral components decrease in strength as others increase. If the frequency deviation is held constant and the modulation frequency increased, the spacing between spectra increases.

Frequency modulation can be classified as narrowband if the change in the carrier frequency is about the same as the signal frequency, or as wideband if the change in the carrier frequency is much higher (modulation index > 1) than the signal frequency.For example, narrowband FM (NFM) is used fortwo-way radiosystems such asFamily Radio Service,in which the carrier is allowed to deviate only 2.5 kHz above and below the center frequency with speech signals of no more than 3.5 kHz bandwidth. Wideband FM is used forFM broadcasting,in which music and speech are transmitted with up to 75 kHz deviation from the center frequency and carry audio with up to a 20 kHz bandwidth and subcarriers up to 92 kHz.

Bessel functions

Frequency spectrum andwaterfall plotof a 146.52MHz carrier, frequency modulated by a 1,000Hz sinusoid. The modulation index has been adjusted to around 2.4, so the carrier frequency has small amplitude. Several strong sidebands are apparent; in principle an infinite number are produced in FM but the higher-order sidebands are of negligible magnitude.

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated usingBessel functionsof the first kind, as a function of thesidebandnumber and the modulation index. The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals. For particular values of the modulation index, the carrier amplitude becomes zero and all the signal power is in the sidebands.

Since the sidebands are on both sides of the carrier, their count is doubled, and then multiplied by the modulating frequency to find the bandwidth. For example, 3 kHz deviation modulated by a 2.2 kHz audio tone produces a modulation index of 1.36. Suppose that we limit ourselves to only those sidebands that have a relative amplitude of at least 0.01. Then, examining the chart shows this modulation index will produce three sidebands. These three sidebands, when doubled, gives us (6 × 2.2 kHz) or a 13.2 kHz required bandwidth.

Modulation index	Sideband amplitude
	Carrier	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
0.00	1.00
0.25	0.98	0.12
0.5	0.94	0.24	0.03
1.0	0.77	0.44	0.11	0.02
1.5	0.51	0.56	0.23	0.06	0.01
2.0	0.22	0.58	0.35	0.13	0.03
2.40483	0.00	0.52	0.43	0.20	0.06	0.02
2.5	−0.05	0.50	0.45	0.22	0.07	0.02	0.01
3.0	−0.26	0.34	0.49	0.31	0.13	0.04	0.01
4.0	−0.40	−0.07	0.36	0.43	0.28	0.13	0.05	0.02
5.0	−0.18	−0.33	0.05	0.36	0.39	0.26	0.13	0.05	0.02
5.52008	0.00	−0.34	−0.13	0.25	0.40	0.32	0.19	0.09	0.03	0.01
6.0	0.15	−0.28	−0.24	0.11	0.36	0.36	0.25	0.13	0.06	0.02
7.0	0.30	0.00	−0.30	−0.17	0.16	0.35	0.34	0.23	0.13	0.06	0.02
8.0	0.17	0.23	−0.11	−0.29	−0.10	0.19	0.34	0.32	0.22	0.13	0.06	0.03
8.65373	0.00	0.27	0.06	−0.24	−0.23	0.03	0.26	0.34	0.28	0.18	0.10	0.05	0.02
9.0	−0.09	0.25	0.14	−0.18	−0.27	−0.06	0.20	0.33	0.31	0.21	0.12	0.06	0.03	0.01
10.0	−0.25	0.04	0.25	0.06	−0.22	−0.23	−0.01	0.22	0.32	0.29	0.21	0.12	0.06	0.03	0.01
12.0	0.05	−0.22	−0.08	0.20	0.18	−0.07	−0.24	−0.17	0.05	0.23	0.30	0.27	0.20	0.12	0.07	0.03	0.01

Carson's rule

Main article:Carson bandwidth rule

Arule of thumb,Carson's rulestates that nearly all (≈98 percent) of the power of a frequency-modulated signal lies within abandwidth$B_{T}\,$of:

$B_{T}=2\left(\Delta f+f_{m}\right)=2f_{m}(\beta +1)$

where$\Delta f\,$,as defined above, is the peak deviation of the instantaneous frequency$f(t)\,$from the center carrier frequency$f_{c}$,$\beta$is the Modulation index which is the ratio of frequency deviation to highest frequency in the modulating signal and$f_{m}\,$is the highest frequency in the modulating signal. Condition for application of Carson's rule is only sinusoidal signals. For non-sinusoidal signals:

$B_{T}=2(\Delta f+W)=2W(D+1)$

where W is the highest frequency in the modulating signal but non-sinusoidal in nature and D is the Deviation ratio which the ratio of frequency deviation to highest frequency of modulating non-sinusoidal signal.

Noise reduction

FM provides improvedsignal-to-noise ratio(SNR), as compared for example withAM.Compared with an optimum AM scheme, FM typically has poorer SNR below a certain signal level called the noise threshold, but above a higher level – the full improvement or full quieting threshold – the SNR is much improved over AM. The improvement depends on modulation level and deviation. For typical voice communications channels, improvements are typically 5–15 dB. FM broadcasting using wider deviation can achieve even greater improvements. Additional techniques, such as pre-emphasis of higher audio frequencies with corresponding de-emphasis in the receiver, are generally used to improve overall SNR in FM circuits. Since FM signals have constant amplitude, FM receivers normally have limiters that remove AM noise, further improving SNR.

Implementation

Modulation

FM signals can be generated using either direct or indirect frequency modulation:

• Direct FM modulation can be achieved by directly feeding the message into the input of avoltage-controlled oscillator.
• For indirect FM modulation, the message signal is integrated to generate aphase-modulated signal.This is used to modulate acrystal-controlled oscillator,and the result is passed through afrequency multiplierto produce an FM signal. In this modulation, narrowband FM is generated leading to wideband FM later and hence the modulation is known as indirect FM modulation.

Demodulation

See also:Detector (radio) § Frequency and phase modulation detectors

FM modulation

Many FM detector circuits exist. A common method for recovering the information signal is through aFoster–Seeley discriminatororratio detector.Aphase-locked loopcan be used as an FM demodulator.Slope detectiondemodulates an FM signal by using a tuned circuit which has its resonant frequency slightly offset from the carrier. As the frequency rises and falls the tuned circuit provides a changing amplitude of response, converting FM to AM. AM receivers may detect some FM transmissions by this means, although it does not provide an efficient means ofdetectionfor FM broadcasts. InSoftware-Defined Radioimplementations the demodulation may be carried out by using theHilbert transform(implemented as a filter) to recover the instantaneous phase, and thereafter differentiating this phase (using another filter) to recover the instantaneous frequency. Alternatively, a complex mixer followed by a bandpass filter may be used to translate the signal to baseband, and then proceeding as before.

Applications

Doppler effect

When an echolocatingbatapproaches a target, its outgoing sounds return as echoes, which are Doppler-shifted upward in frequency. In certain species of bats, which produce constant frequency (CF)echolocationcalls, the bats compensate for theDoppler shiftby lowering their call frequency as they approach a target. This keeps the returning echo in the same frequency range of the normal echolocation call. This dynamic frequency modulation is called theDoppler Shift Compensation(DSC), and was discovered byHans Schnitzlerin 1968.

Magnetic tape storage

FM is also used atintermediate frequenciesby analogVCRsystems (includingVHS) to record theluminance(black and white) portions of the video signal. Commonly, thechrominancecomponent is recorded as a conventional AM signal, using the higher-frequency FM signal asbias.FM is the only feasible method of recording the luminance ( "black-and-white" ) component of video to (and retrieving video from)magnetic tapewithout distortion; video signals have a large range of frequency components – from a fewhertzto severalmegahertz,too wide forequalizersto work with due to electronic noise below −60dB.FM also keeps the tape at saturation level, acting as a form ofnoise reduction;alimitercan mask variations in playback output, and theFM captureeffect removesprint-throughandpre-echo.A continuous pilot-tone, if added to the signal – as was done onV2000and many Hi-band formats – can keep mechanical jitter under control and assisttimebase correction.

These FM systems are unusual, in that they have a ratio of carrier to maximum modulation frequency of less than two; contrast this with FM audio broadcasting, where the ratio is around 10,000. Consider, for example, a 6-MHz carrier modulated at a 3.5-MHz rate; byBesselanalysis, the first sidebands are on 9.5 and 2.5 MHz and the second sidebands are on 13 MHz and −1 MHz. The result is a reversed-phase sideband on +1 MHz; on demodulation, this results in unwanted output at 6 – 1 = 5 MHz. The system must be designed so that this unwanted output is reduced to an acceptable level.

Sound

FM is also used ataudio frequenciesto synthesize sound. This technique, known asFM synthesis,was popularized by early digitalsynthesizersand became a standard feature in several generations ofpersonal computersound cards.

Radio

Main article:FM broadcasting

An American FM radio transmitter in Buffalo, NY atWEDG

Edwin Howard Armstrong(1890–1954) was an American electrical engineer who invented wideband frequency modulation (FM) radio. He patented the regenerative circuit in 1914, the superheterodyne receiver in 1918 and the super-regenerative circuit in 1922.Armstrong presented his paper, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation", (which first described FM radio) before the New York section of theInstitute of Radio Engineerson November 6, 1935. The paper was published in 1936.

As the name implies, wideband FM (WFM) requires a widersignal bandwidththanamplitude modulationby an equivalent modulating signal; this also makes the signal more robust againstnoiseandinterference.Frequency modulation is also more robust against signal-amplitude-fading phenomena. As a result, FM was chosen as the modulation standard for high frequency,high fidelityradiotransmission, hence the term "FM radio"(although for many years theBBCcalled it "VHF radio" because commercial FM broadcasting uses part of theVHFband – theFM broadcast band). FMreceiversemploy a specialdetectorfor FM signals and exhibit a phenomenon known as thecapture effect,in which thetuner"captures" the stronger of two stations on the same frequency while rejecting the other (compare this with a similar situation on an AM receiver, where both stations can be heard simultaneously).Frequency driftor a lack ofselectivitymay cause one station to be overtaken by another on anadjacent channel.Frequencydriftwas a problem in early (or inexpensive) receivers; inadequate selectivity may affect any tuner.

A wideband FM signal can also be used to carry astereosignal; this is done withmultiple xingand demultiple xing before and after the FM process. The FM modulation and demodulation process is identical in stereo and monaural processes.

FM is commonly used atVHFradio frequenciesforhigh-fidelitybroadcastsof music andspeech.In broadcast services, where audio fidelity is important, wideband FM is generally used. Analog TV sound is also broadcast using FM. Narrowband FM is used for voice communications in commercial andamateur radiosettings. Intwo-way radio,narrowband FM (NBFM) is used to conserve bandwidth for land mobile, marine mobile and other radio services.

A high-efficiency radio-frequencyswitching amplifiercan be used to transmit FM signals (and otherconstant-amplitude signals). For a given signal strength (measured at the receiver antenna), switching amplifiers useless battery powerand typically cost less than alinear amplifier.This gives FM another advantage over other modulation methods requiring linear amplifiers, such as AM andQAM.

There are reports that on October 5, 1924, ProfessorMikhail A. Bonch-Bruevich,during a scientific and technical conversation in theNizhny Novgorod Radio Laboratory,reported about his new method of telephony, based on a change in the period of oscillations. Demonstration of frequency modulation was carried out on the laboratory model.

Hearing assistive technology

Frequency modulated systems are a widespread and commercially availableassistive technologythat make speech more understandable by improving the signal-to-noise ratio in the user's ear. They are also calledauditory trainers,a term which refers to any sound amplification system not classified as ahearing aid.They intensify signal levels from the source by 15 to 20 decibels.FM systems are used by hearing-impaired people as well as children whose listening is affected by disorders such asauditory processing disorderorADHD.For people withsensorineural hearing loss,FM systems result in better speech perception than hearing aids. They can be coupled with behind-the-ear hearing aids to allow the user to alternate the setting.FM systems are more convenient and cost-effective than alternatives such ascochlear implants,but many users use FM systems infrequently due to their conspicuousness and need for recharging.

See also

• Amplitude modulation
• Continuous-wave frequency-modulated radar
• Chirp
• FM broadcasting
• FM stereo
• FM-UWB(FM and Ultra Wideband)
• History of radio
• Modulation,for a list of other modulation techniques
• Phase modulation

Further reading

• Carlson, A. Bruce (2001).Communication Systems.Science/Engineering/Math (4th ed.). McGraw-Hill.ISBN978-0-07-011127-1.
• Frost, Gary L. (2010).Early FM Radio: Incremental technology in twentieth-century America.Baltimore, MD: Johns Hopkins University Press.ISBN978-0-8018-9440-4.
• Seymour, Ken (2005) [1996]. "Frequency Modulation".The Electronics Handbook(2nd ed.). CRC Press. pp. 1188–1200.ISBN0-8493-8345-5.

External links

• Analog Modulation online interactive demonstrationusing Python inGoogle Colab Platform,by C Foh.