In telecommunications and signal processing, frequency modulation (FM)
is the encoding of information in a carrier wave by varying the
instantaneous frequency of the wave.
In analog frequency modulation, such as
Contents 1 Theory 1.1
2 Noise reduction 3 Implementation 3.1 Modulation 3.2 Demodulation 4 Applications 4.1
5 See also 6 References 7 Further reading Theory[edit] This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2017) (Learn how and when to remove this template message) If the information to be transmitted (i.e., the baseband signal) is x m ( t ) displaystyle x_ m (t) and the sinusoidal carrier is x c ( t ) = A c cos ( 2 π f c t ) displaystyle x_ c (t)=A_ c cos(2pi 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:[citation needed] y ( t ) = A c cos ( 2 π ∫ 0 t f ( τ ) d τ ) = A c cos ( 2 π ∫ 0 t [ f c + f Δ x m ( τ ) ] d τ ) = A c cos ( 2 π f c t + 2 π f Δ ∫ 0 t x m ( τ ) d τ ) displaystyle begin aligned y(t)&=A_ c cos left(2pi int _ 0 ^ t f(tau )dtau right)\&=A_ c cos left(2pi int _ 0 ^ t left[f_ c +f_ Delta x_ m (tau )right]dtau right)\&=A_ c cos left(2pi f_ c t+2pi f_ Delta int _ 0 ^ t x_ m (tau )dtau right)\end aligned where f Δ displaystyle f_ Delta , = K f displaystyle K_ f A m displaystyle A_ m , K f displaystyle K_ f being the sensitivity of the frequency modulator and A m displaystyle A_ m being the amplitude of the modulating signal or baseband signal. In this equation, f ( τ ) displaystyle f(tau ), is the instantaneous frequency of the oscillator and f Δ displaystyle 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.
While most of the energy of the signal is contained within fc ± fΔ,
it can be shown by
∫ 0 t x m ( τ ) d τ = A m sin ( 2 π f m t ) 2 π f m displaystyle int _ 0 ^ t x_ m (tau )dtau = frac A_ m sin(2pi f_ m t) 2pi f_ m , In this case, the expression for y(t) above simplifies to: y ( t ) = A c cos ( 2 π f c t + A m f Δ f m sin ( 2 π f m t ) ) displaystyle y(t)=A_ c cos left(2pi f_ c t+ frac A_ m f_ Delta f_ m sin left(2pi f_ m tright)right), where the amplitude A m displaystyle A_ m , of the modulating sinusoid is represented by the peak deviation f Δ displaystyle f_ Delta , (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.
h = Δ f f m = f Δ
x m ( t )
f m
displaystyle h= frac Delta f f_ m = frac f_ Delta x_ m (t) f_ m where f m displaystyle f_ m , is the highest frequency component present in the modulating signal xm(t), and Δ f 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 h ≪ 1 displaystyle hll 1 , the modulation is called narrowband FM (NFM), and its bandwidth is approximately 2 f m displaystyle 2f_ m , . Sometimes modulation index h < 0.3 displaystyle h<0.3 radians is considered as NFM, otherwise 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 = Δ f f m = Δ f 1 2 T s = 2 Δ f T s
displaystyle h= frac Delta f f_ m = frac Delta f frac 1 2T_ s =2Delta fT_ s where T s displaystyle T_ s , is the symbol period, and f m = 1 2 T s 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 f c displaystyle f_ c , is never transmitted. Rather, one of two frequencies is transmitted, either f c + Δ f displaystyle f_ c +Delta f or f c − Δ f displaystyle f_ c -Delta f , depending on the binary state 0 or 1 of the modulation signal. If h ≫ 1 displaystyle hgg 1 , the modulation is called wideband FM and its bandwidth is approximately 2 f Δ displaystyle 2f_ Delta , . While wideband FM uses more bandwidth, it can improve the signal-to-noise ratio significantly; for example, doubling the value of Δ f displaystyle Delta f, , while keeping f m displaystyle f_ m constant, results in an eight-fold improvement in the signal-to-noise
ratio.[5] (Compare this with
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 −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 5.53 0 −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 8.65 0 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[edit] Main article: Carson bandwidth 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 B T displaystyle B_ T , of:
B T = 2 ( Δ f + f m ) displaystyle B_ T =2(Delta f+f_ m ), = 2 f m ( β + 1 ) displaystyle =2f_ m (beta +1) where Δ f displaystyle Delta f, , as defined above, is the peak deviation of the instantaneous frequency f ( t ) displaystyle f(t), from the center carrier frequency f c displaystyle f_ c , β displaystyle beta is the
f m displaystyle f_ m , is the highest frequency in the modulating signal. Condition for application of Carson's rule is only sinusoidal signals.
B T = 2 ( Δ f + W ) displaystyle B_ T =2(Delta f+W), = 2 W ( D + 1 ) displaystyle =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[edit]
FM provides improved
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.[9] Demodulation[edit]
See also: Detectors
Many FM detector circuits exist. A common method for recovering the
information signal is through a Foster-Seeley discriminator. A
phase-locked loop can be used as an FM demodulator. Slope detection
demodulates 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 of detection for FM broadcasts.
Applications[edit]
An American
Main article: FM broadcasting
References[edit] ^ Stan Gibilisco (2002). Teach yourself electricity and electronics.
McGraw-Hill Professional. p. 477.
ISBN 978-0-07-137730-0.
^ David B. Rutledge (1999). The Electronics of Radio. Cambridge
University Press. p. 310. ISBN 978-0-521-64645-1.
^ B. Boashash, editor, "Time-Frequency Signal Analysis and Processing
– A Comprehensive Reference", Elsevier Science, Oxford, 2003;
ISBN 0-08-044335-4
^ a b T.G. Thomas, S. C. Sekhar Communication Theory, Tata-McGraw Hill
2005, ISBN 0-07-059091-5 page 136
^ Der, Lawrence, Ph.D., Frequency
Further reading[edit] A. Bruce Carlson. Communication Systems, 4th edition. McGraw-Hill Science/Engineering/Math. 2001. ISBN 0-07-011127-8, ISBN 978-0-07-011127-1. Gary L. Frost. Early FM Radio: Incremental Technology in Twentieth-Century America. Baltimore: Johns Hopkins University Press, 2010. ISBN 0-8018-9440-9, ISBN 978-0-8018-9440-4. Ken Seymour, AT&T Wireless (Mobility). Frequency Modulation, The Electronics Handbook, pp 1188-1200, 1st Edition, 1996. 2nd Edition, 2005 CRC Press, Inc., ISBN 0-8493-8345-5 (1st Edition). v t e
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