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Frequency modulation

Frequency modulation (FM) is a signal modulation technique used in electronic communication, originally for transmitting messages with a radio wave. In frequency modulation a carrier wave is varied in its instantaneous frequency in proportion to a property, primarily the instantaneous amplitude, of a message signal, such as an audio signal. The technology is used in telecommunications, radio broadcasting, signal processing, and computing.

FM Signal
According to Paul Nahin, "To apply the baseband signal of a microphone output directly to the transmitter antenna won't work, because ...a quarter-wavelength antenna at audio frequencies is physically enormous. To have a reasonably sized antenna requires a transmitter signal at frequencies considerably higher than those of the bandwidth spectrum; that is, the baseband spectrum must be upshifted to the radio frequencies." This is called signal modulation. According to Ron Bertrand, "Frequency modulation is a method of modulating a carrier wave whereby the modulating audio causes the instantaneous frequency of the carrier to change. Without modulation, an FM transmitter produces a single carrier frequency." The FM signal produced by a sinusoidal carrier of frequency ωc, modulated by an audio tone of frequency ωa with amplitude A, can be written as: :\begin{align} e(t) &= \sin \left(\omega_c t + kA \sin\left(\omega_at\right)\right) \\ \end{align} We need the instantaneous frequency, which describes a frequency varying above and below the carrier frequency at the audio tone frequency, which we derive by using Carson's time derivative method: :\begin{align} \frac{d}{dt}\left(\omega_c t + kA \sin\left(\omega_at\right)\right) &= \omega_c + kA \omega_a\cos\left(\omega_at\right) \\ \end{align} The amplitude factor kAωa defines the maximum Frequency deviation around ωc. Dividing by ωa, gives us the modulation index kA, which "is the ratio of the amount of frequency deviation to the audio modulating frequency." While most of the energy of the signal is contained within fc ± fΔ, it can be shown by Fourier analysis that a wider range of frequencies is required to precisely represent an FM signal. The frequency spectrum of 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==
Sinusoidal baseband signal
Mathematically, a baseband modulating signal may be approximated by a sinusoidal continuous wave signal with a frequency fm. This method is also named as single-tone modulation. The integral of such a signal x_m(t) = \cos(2\pi f_m t) 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 modulating sinusoid is represented in the peak deviation f_\Delta = K_f A_m (see frequency deviation). The harmonic distribution of a sine wave carrier modulated by such a sinusoidal signal can be represented with Bessel functions; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain. ==Modulation index==
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: :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 signal xm(t), and \Delta{}f\, is the peak frequency-deviationi.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 h \ll 1, the modulation is called narrowband FM (NFM), and its bandwidth is approximately 2f_m\,. Sometimes modulation index h  is considered NFM and other modulation indices are considered wideband FM (WFM or FM). (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. 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==
Bessel functions
of 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. In his 1922 FM paper, Carson pointed out an infinite number of side frequencies are generated when a carrier frequency is modulated by a signal frequency, the amplitudes expressed as Bessel functions. The separation is determined by the frequency of the modulating signal, and the amplitude dependent upon the modulation index. A table of Bessel functions of the first kind is used to determine the side frequency amplitudes. 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. ==Carson's rule==
Carson's rule
A rule of thumb, ''Carson's rule'' states that the frequency-modulated signal lies within a bandwidth B_T\, of: :B_T = 2\left(\Delta f + f_m\right) = 2f_m(h + 1) where \Delta f\,, as defined above, is the peak deviation of the instantaneous frequency f(t)\, from the center carrier frequency f_c, h 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. Carson's rule can only be applied to 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 is the ratio of frequency deviation to highest frequency of modulating non-sinusoidal signal. ==Noise reduction==
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. ==Implementation==
{{anchor|Practical Implementation}}Implementation
Modulation FM signals can be generated using either direct or indirect frequency modulation: • Direct FM modulation can be achieved by directly feeding the modulating audio voltage into a voltage-controlled oscillator. Demodulation Many FM detector circuits exist. A common method for recovering the information signal is through a Foster–Seeley discriminator or ratio detector. A phase-locked loop can be used as an FM demodulator. If the demodulated signal is sampled at or above Nyquist, this allows for recovery of near-instantaneous phase changes. ==Applications==
Applications
Doppler effect In 1968, Schnitzler noted certain bats lower the animal echolocation emission frequency by 13 to 16 kHz, compensating for Doppler shifts caused by the bat’s own movement. Doppler shift compensation, dynamic frequency modulation, ensures that the returning echo frequency is optimally adjusted for the bat's auditory fovea. Magnetic tape storage FM is also used at intermediate frequencies by analog VCR systems (including VHS) to record the luminance (black and white) portions of the video signal. Commonly, the chrominance component is recorded as a conventional AM signal, using the higher-frequency FM signal as bias. FM is the only feasible method of recording the luminance ("black-and-white") component of video to (and retrieving video from) magnetic tape without distortion; video signals have a 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, acting as a form of noise reduction; a limiter can mask variations in 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 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 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 in Buffalo, New York 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 the Institute of Radio Engineers on November 6, 1935. The paper was published in 1936. The first experimental station, W2XMN, went on the air in 1937. As the name implies, wideband FM (WFM) requires a wider signal bandwidth than amplitude modulation by an equivalent modulating signal; this also makes the signal more robust against noise and interference. 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 fidelity radio transmission, hence the term "FM radio" (although for many years the BBC called it "VHF radio" because commercial FM broadcasting uses part of the VHF bandthe FM broadcast band). FM receivers employ a special detector for FM signals and exhibit a phenomenon known as the capture effect, in which the tuner "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 drift or a lack of selectivity may cause one station to be overtaken by another on an adjacent channel. Frequency drift 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 VHF radio frequencies for high-fidelity broadcasts of music and speech. 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 and amateur radio settings. In two-way radio, narrowband FM (NBFM) is used to conserve bandwidth for land mobile, marine mobile and other radio services. 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 methods requiring linear amplifiers, such as AM and QAM. There are reports that on October 5, 1924, Professor Mikhail A. Bonch-Bruevich, during a scientific and technical conversation in the Nizhny 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 available assistive technology that make speech more understandable by improving the signal-to-noise ratio in the user's ear. They are also called auditory trainers, a term which refers to any sound amplification system not classified as a hearing 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 as auditory processing disorder or ADHD. For people with sensorineural 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 as cochlear implants, but many users use FM systems infrequently due to their conspicuousness and need for recharging. ==See also==
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