# Frequency modulation: Map

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In telecommunications, frequency modulation (FM) conveys information over a carrier wave by varying its frequency (contrast this with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant). In analog applications, the instantaneous frequency of the carrier is directly proportional to the instantaneous value of the input signal. Digital data can be sent by shifting the carrier's frequency among a set of discrete values, a technique known as frequency-shift keying.

## Theory

Suppose the baseband data signal (the message) to be transmitted is

x_m(t)\,

and is restricted in amplitude to be

\left| x_m(t) \right| \le 1, \,

and the sinusoidal carrier is

x_c(t) = A_c \cos (2 \pi f_c t)\,

where fc is the carrier's base frequency and Ac is the carrier's amplitude.The modulator combines the carrier with the baseband data signal to get the transmitted signal,

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).

In this equation, f(\tau)\, is the instantaneous frequency of the oscillator and f_{\Delta}\, is the frequency deviation, which represents the maximum shift away from fc in one direction, assuming xm(t) is limited to the range ±1.

Although it may seem that this limits the frequencies in use to fc ± fΔ, this neglects the distinction between instantaneous frequency and spectral frequency. The frequency spectrum of an actual FM signal has components extending out to infinite frequency, although they become negligibly small beyond a point.

The harmonic distribution of a sine wave carrier modulated by a sine wave signal can be represented with Bessel functions - this provides a basis for a mathematical understanding of frequency modulation in the frequency domain.

### Modulation index

As with other modulation indices, this quantity indicates by how much the modulated variable varies around its unmodulated level. It relates to the variations in the frequency of the carrier signal:

h = \frac{\Delta{}f}{f_m} = \frac{f_\Delta |x_m(t)|}{f_m} \

where f_m\, is the highest frequency component present in the modulating signal xm(t), and \Delta{}f\, is the Peak frequency-deviation, i.e the maximum deviation of the instantaneous frequency from the carrier frequency. If h \ll 1, the modulation is called narrowband FM, and its bandwidth is approximately 2 f_m\,. If h \gg 1, the modulation is called wideband FM and its bandwidth is approximately 2 f_\Delta\,. While wideband FM uses more bandwidth, it can improve signal-to-noise ratio significantly.

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 stays 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.

### Carson's rule

A rule of thumb, Carson's rule states that nearly all (~98%) of the power of a frequency-modulated signal lies within a bandwidth B_T\, of

\ B_T = 2(\Delta f +f_m)\,

where \Delta f\,, as defined above, is the peak deviation of the instantaneous frequency f(t)\, from the center carrier frequency f_c\,.

## Noise quieting

The noise power decreases as the signal power increases, therefore the SNR goes up significantly.

## Bessel functions

The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals. Based on the Bessel functions.
Modulation index 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
2.41 0 0.52 0.43 0.20 0.06 0.02
2.5 −.05 0.50 0.45 0.22 0.07 0.02 0.01
3.0 −.26 0.34 0.49 0.31 0.13 0.04 0.01
4.0 −.40 −.07 0.36 0.43 0.28 0.13 0.05 0.02
5.0 −.18 −.33 0.05 0.36 0.39 0.26 0.13 0.05 0.02
5.53 0 −.34 −.13 0.25 0.40 0.32 0.19 0.09 0.03 0.01
6.0 0.15 −.28 −.24 0.11 0.36 0.36 0.25 0.13 0.06 0.02
7.0 0.30 0.00 −.30 −.17 0.16 0.35 0.34 0.23 0.13 0.06 0.02
8.0 0.17 0.23 −.11 −.29 −.10 0.19 0.34 0.32 0.22 0.13 0.06 0.03
8.65 0 0.27 0.06 −.24 −.23 0.03 0.26 0.34 0.28 0.18 0.10 0.05 0.02
9.0 −.09 0.25 0.14 −.18 −.27 −.06 0.20 0.33 0.31 0.21 0.12 0.06 0.03 0.01
10.0 −.25 0.04 0.25 0.06 −.22 −.23 −.01 0.22 0.32 0.29 0.21 0.12 0.06 0.03 0.01
12.0 0.05 −.22 −.08 0.20 0.18 −.07 −.24 −.17 0.05 0.23 0.30 0.27 0.20 0.12 0.07 0.03 0.01

## Implementation

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

A common method for recovering the information signal is through a Foster-Seeley discriminator.

## Applications

FM is commonly used at VHF radio frequencies for high-fidelity broadcasts of music and speech (see FM broadcasting). Normal (analog) TV sound is also broadcast using FM. A narrow band form is used for voice communications in commercial and amateur radio settings. The type of FM used in broadcast is generally called wide-FM, or W-FM. In two-way radio, narrowband narrow-fm (N-FM) is used to conserve bandwidth. In addition, it is used to send signals into space.

### Hardware

FM is also used at intermediate frequencies by all analog VCR systems, including VHS, to record both the luminance (black and white) and the chrominance portions of the video signal. FM is the only feasible method of recording video to and retrieving video from magnetic tape without extreme distortion, as video signals have a very large range of frequency components — from a few hertz to several megahertz, too wide for equalizers to work with due to electronic noise below -60 dB. FM also keeps the tape at saturation level, and therefore acts as a form of noise reduction, and a simple limiter can mask variations in the playback output, and the FM capture effect removes print-through and pre-echo. A continuous pilot-tone, if added to the signal — as was done on V2000 and many Hi-band formats — can keep mechanical jitter under control and assist 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, while the second sidebands are on 13 MHz and (-1) MHz. The result is a sideband of reversed phase on +1 MHz; on demodulation, this results in an unwanted output at (6 - 1) = 5 Mhz. The system must be designed so that this is at an acceptable level.

### 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 for several generations of personal computer sound cards.   Edwin 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.

As the name implies, wideband FM (W-FM) requires a wider signal bandwidth than amplitude modulation by an equivalent modulating signal, but this also makes the signal more robust against noise and interference. Frequency modulation is also more robust against simple signal amplitude fading phenomena. As a result, FM was chosen as the modulation standard for high frequency, high fidelity radio transmission: hence the term "FM radio" (although for many years the BBC called it "VHF radio", because commercial FM broadcasting uses a well-known part of the VHF band; in certain countries, expressions referencing the more familiar wavelength notion are still used in place of the more abstract modulation technique name).

FM receivers employ a special detector for FM signals and exhibit a phenomenon called capture effect, where the tuner is able to clearly receive the stronger of two stations being broadcast on the same frequency. Problematically however, frequency drift or lack of selectivity may cause one station or signal to be suddenly overtaken by another on an adjacent channel. Frequency drift typically constituted a problem on very old or inexpensive receivers, while inadequate selectivity may plague any tuner.

An FM signal can also be used to carry a stereo signal: see FM stereo. However, this is done by using multiplexing and demultiplexing before and after the FM process. The rest of this article ignores the stereo multiplexing and demultiplexing process used in "stereo FM", and concentrates on the FM modulation and demodulation process, which is identical in stereo and mono processes.

A high-efficiency radio-frequency switching amplifier can be used to transmit FM signals (and other constant-amplitude signals).For a given signal strength (measured at the receiver antenna), switching amplifiers use less battery power and typically cost less than a linear amplifier.This gives FM another advantage over other modulation schemes that require linear amplifiers, such as AM and QAM.

## Miscellaneous

Frequency modulation can be regarded as phase modulation where the carrier phase modulation is the time integral of the FM modulating signal.

Frequency-shift keying is the frequency modulation using only a discrete number of frequencies. Morse code transmission has been implemented this way, as were most early telephone-line modems. Radio teletype also use FSK.

By the phenomenon of slope detection whereby FM is converted to AM in a frequency-selective circuit tuned slightly away from the nominal signal frequency, AM receivers may detect some FM transmissions, though this does not provide an efficient method of detection for FM broadcasts.

FM modulation is also used in telemetry applications.

## References

• A. Bruce Carlson: "Communication systems, 2nd edition", McGraw-Hill, Inc, 1981, ISBN 0-07-085082-2