Frequency modulation

Passband modulation
Analog modulation
AM FM PM QAM SM SSB
Digital modulation
ASK APSK CPM FSK MFSK MSK OOK PPM PSK QAM SC-FDE TCM WDM
Hierarchical modulation
QAM WDM
Spread spectrum
CSS DSSS FHSS THSS
See also
Capacity-approaching codes Demodulation Line coding Modem AnM PoM PAM PCM PDM PWM ΔΣM OFDM FDM Multiplexing
v t e

RFI) rejection than AM, as shown in this dramatic New York publicity demonstration by General Electric in 1940. The radio has both AM and FM receivers. With a million-volt electric arc as a source of interference behind it, the AM receiver produced only a roar of static, while the FM receiver clearly reproduced a music program from Armstrong's experimental FM transmitter W2XMN

in New Jersey.

Frequency modulation (FM) is the encoding of

.

In analog frequency modulation, such as radio broadcasting, of an audio signal representing voice or music, the instantaneous frequency 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 data can be encoded and transmitted with a type of frequency modulation known as frequency-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 computer modems, such as fax modems, telephone caller ID systems, garage door openers, and other low-frequency transmissions.^[1] Radioteletype also uses FSK.^[2]

Frequency modulation is widely used for

.

However, under severe enough multipath conditions it performs much more poorly than AM, with distinct high frequency noise artifacts that are audible with lower volumes and less complex tones.^{[citation needed]} 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.^{[citation needed]}

Frequency modulation and phase modulation are the two complementary principal methods of angle modulation; phase modulation is often used as an intermediate step to achieve frequency modulation. These methods contrast with amplitude modulation, in which the amplitude of the carrier wave varies, while the frequency and phase remain constant.

Theory

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If the information to be transmitted (i.e., the

) is ${\displaystyle x_{m}(t)}$ and the carrier is ${\displaystyle x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,}$, where f_c is the carrier's base frequency, and A_c is the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:^{[4] [citation needed]}

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

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

In this equation, ${\displaystyle f(\tau )\,}$ is the

of the oscillator and ${\displaystyle f_{\Delta }\,}$ is the frequency deviation, which represents the maximum shift away from f_c in one direction, assuming x_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

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

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 within f_c ± f_Δ, it can be shown by

Sinusoidal baseband signal

Mathematically, a baseband modulating signal may be approximated by a sinusoidal continuous wave signal with a frequency f_m. This method is also named as single-tone modulation. The integral of such a signal is:

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

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

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

where the amplitude ${\displaystyle A_{m}\,}$ of the modulating sinusoid is represented in the peak deviation ${\displaystyle f_{\Delta }=K_{f}A_{m}}$ (see frequency deviation).

The

; 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 the

carrier frequency

:

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

where ${\displaystyle f_{m}\,}$ is the highest frequency component present in the modulating signal x_m(t), and ${\displaystyle \Delta {}f\,}$ is the peak frequency-deviation – i.e. the maximum deviation of the

instantaneous frequency

from 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 ${\displaystyle h\ll 1}$, the modulation is called narrowband FM (NFM), and its bandwidth is approximately ${\displaystyle 2f_{m}\,}$. Sometimes modulation index ${\displaystyle h<0.3}$ 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:

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

where ${\displaystyle T_{s}\,}$ is the symbol period, and ${\displaystyle f_{m}={\frac {1}{2T_{s}}}\,}$ 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 highest fundamental of the modulating binary waveform. In the case of digital modulation, the carrier ${\displaystyle f_{c}\,}$ is never transmitted. Rather, one of two frequencies is transmitted, either ${\displaystyle f_{c}+\Delta f}$ or ${\displaystyle f_{c}-\Delta f}$, depending on the binary state 0 or 1 of the modulation signal.

If ${\displaystyle h\gg 1}$, the modulation is called wideband FM and its bandwidth is approximately ${\displaystyle 2f_{\Delta }\,}$. While wideband FM uses more bandwidth, it can improve the signal-to-noise ratio significantly; for example, doubling the value of ${\displaystyle \Delta {}f\,}$, while keeping ${\displaystyle f_{m}}$ constant, results in an eight-fold improvement in the signal-to-noise ratio.^[6] (Compare this with chirp 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.^[7] For example, narrowband FM (NFM) is used for two-way radio systems such as Family 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 for FM 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

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind, as a function of the sideband number 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.^[5]

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

A rule of thumb, Carson's rule states that nearly all (≈98 percent) of the power of a frequency-modulated signal lies within a bandwidth ${\displaystyle B_{T}\,}$ of:

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

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

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

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 improved signal-to-noise ratio (SNR), as compared for example with AM. 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.^[8][9]

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 a voltage-controlled oscillator.
• For indirect FM modulation, the message signal is integrated to generate a phase-modulated signal. This is used to modulate a crystal-controlled oscillator, and the result is passed through a frequency multiplier to 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.^[10]

Demodulation

Many FM detector circuits exist. A common method for recovering the information signal is through a

(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 echolocating

in 1968.

Magnetic tape storage

FM is also used at

timebase 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; by Bessel analysis, 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.^[11]

Sound

FM is also used at audio frequencies to synthesize sound. This technique, known as FM synthesis, was popularized by early digital synthesizers and became a standard feature in several generations of personal computer sound cards.

Radio

Edwin Howard Armstrong (1890–1954) was an American electrical engineer who invented wideband frequency modulation (FM) radio.^[12] He patented the regenerative circuit in 1914, the superheterodyne receiver in 1918 and the super-regenerative circuit in 1922.^[13] 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 the Institute of Radio Engineers on November 6, 1935. The paper was published in 1936.^[14]

As the name implies, wideband FM (WFM) requires a wider

was a problem in early (or inexpensive) receivers; inadequate selectivity may affect any tuner.

A wideband FM signal can also be used to carry a stereo signal; this is done with multiplexing and demultiplexing before and after the FM process. The FM modulation and demodulation process is identical in stereo and monaural processes.

FM is commonly used at

, narrowband FM (NBFM) is used to conserve bandwidth for land mobile, marine mobile and other radio services.

A high-efficiency radio-frequency

.

There are reports that on October 5, 1924, Professor

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

References

Further reading

• Carlson, A. Bruce (2001). Communication Systems. Science/Engineering/Math (4th ed.). McGraw-Hill.
  ISBN 978-0-07-011127-1
  .
• Frost, Gary L. (2010). Early FM Radio: Incremental technology in twentieth-century America. Baltimore, MD: Johns Hopkins University Press.
  ISBN 978-0-8018-9440-4
  .
• Seymour, Ken (2005) [1996]. "Frequency Modulation". The Electronics Handbook (2nd ed.). CRC Press. pp. 1188–1200.
  ISBN 0-8493-8345-5
  .

External links

• Analog Modulation online interactive demonstration using Python in Google Colab Platform, by C Foh.