Frequency modulation
Frequency modulation 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.
In analog frequency modulation, such as radio broadcasting of voice and 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 using a form of frequency modulation known as frequency-shift keying, in which the frequency of a carrier is switched among a discrete set of values. In its simplest form, binary FSK, two frequencies represent binary symbols 0 and 1. FSK is widely used in low to moderate data-rate applications because of its simplicity and robustness. Common uses include early computer modems, telephone caller-ID systems, garage-door openers, remote keyless entry systems, and radioteletype.
Frequency modulation is widely used for FM broadcasting. It is also used in telemetry, radar, seismic prospecting, and monitoring newborns for seizures via EEG, two-way radio systems, sound synthesis, magnetic tape-recording systems and some video-transmission systems. In radio transmission, an advantage of frequency modulation is that it has a larger signal-to-noise ratio and therefore rejects radio frequency interference better than an equal power amplitude modulation signal. For this reason, most music is broadcast over FM radio.
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.
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:
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:
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
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 is:In this case, the expression for y above simplifies to:
where the amplitude of the modulating sinusoid is represented in the peak 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
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:where is the highest frequency component present in the modulating signal xm, and 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, the modulation is called narrowband FM, and its bandwidth is approximately. Sometimes modulation index is considered NFM and other modulation indices are considered wideband FM.
For digital modulation systems, for example, binary frequency shift keying, where a binary signal modulates the carrier, the modulation index is given by:
where is the symbol period, and 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 is never transmitted. Rather, one of two frequencies is transmitted, either or, depending on the binary state 0 or 1 of the modulation signal.
If, the modulation is called wideband FM and its bandwidth is approximately. While wideband FM uses more bandwidth, it can improve the signal-to-noise ratio significantly; for example, doubling the value of, while keeping constant, results in an eight-fold improvement in the signal-to-noise ratio..
With a tone-modulated FM wave, if the modulation frequency is held constant and the modulation index is increased, the 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 than the signal frequency. For example, narrowband FM 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
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.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.
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 or a 13.2 kHz required bandwidth.
Carson's rule
A rule of thumb, Carson's rule states that the frequency-modulated signal lies within a bandwidth of:where, as defined above, is the peak deviation of the instantaneous frequency from the center carrier frequency, is the modulation index which is the ratio of frequency deviation to highest frequency in the modulating signal, and is the highest frequency in the modulating signal.
Carson's rule can only be applied to sinusoidal signals. For non-sinusoidal signals:
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.