FM Synthesis Craig Stuart Sapp 23 March 1995

Transcription

1 FM Synthesis Craig Stuart Sapp 23 March 995 Background of FM Synthesis Frequency Modulation for Radios In frequency odulation (FM), unlike AM, the aplitude of the carrier is kept constant, but its frequency is altered in accordance with variations in the audio signal being sent. This for of odulation was developed by the Aerican electrical engineer Edwin H. Arstrong during the early 93s in an effort to overcoe interference and noise that affect AM radio reception. FM is less susceptible than is AM to certain kinds of interference, such as that caused by thunderstors as well as rando electrical currents fro achinery and other related sources. These noise-producing signals affect the aplitude of a radio wave but not its frequency, and so an FM signal reains virtually unchanged. FM is better adapted than is AM to the transission of stereophonic sound, audio signals for television progras, and long-distance telephone calls by icrowave radio relay. Coercial FM broadcasting stations are assigned higher frequencies than are AM stations. The assigned frequencies, spaced 2 khz apart, range fro 88 to 8 MHz. In 933 Arstrong secured four patents on advanced circuits that were to solve this last basic proble. They revealed an entirely new radio syste, fro transitter to receiver. Instead of varying the aplitude, or power, of radio waves to carry voice or usic, as in all radio before then, the new syste varied, or odulated, the waves frequency (nuber of waves per second) over a wide band of frequencies. This created a carrier wave that natural static--an aplitude phenoenon created by electrical stors--could not break into. As a result, FM s wide frequency range ade possible the first clear, practical ethod of high-fidelity broadcasting. Because the new syste required a basic change in transitters and receivers, it was not ebraced with any alacrity by the established radio industry. Arstrong had to build the first full-scale FM station hiself in 939 at a cost of ore than $3, to prove its worth. He then had to develop and proote the syste, sustain it through World War II (while he again turned to ilitary research), and fight off postwar regulatory attepts to hobble FM s growth. When FM slowly established itself, Arstrong again found hiself entrapped in another interinable patent suit to retain his invention. Ill and aging in 954, with ost of his wealth gone in the battle

2 2 aboutfm.nb for FM, he took his own life. The years have brought increasing recognition of Arstrong s place in science and invention. FM is now the preferred syste in radio, the required sound channel in all television, and the doinant ediu in obile radio, icrowave relay, and space-satellite counications. Posthuously, Arstrong was elected to the pantheon of electrical greats by the International Telecounications Union, to join such figures as André- Marie Apère, Alexander Graha Bell, Michael Faraday, and Guglielo Marconi. (L.P.L.) Fro Encyclopaedia Brittanica Online ( FM synthesis Frequency Modulation Synthesis was first described by John Chowning in this paper: Chowning, John M. "The Synthesis of Coplex Audio Spectra by Means of Frequency Modulation," Journal of the Audio Engineering Society, volue 2, nuber 7 (Septeber 973). pp FM produces a wide variety of coplex tibres by rapidly varying the frequency of one wavefor in proportion to the aplitude of another wavefor. Coputationally, FM synthesis is very cheap, so it was good for aking sound synthesizers One of the earliest and best-known digital synthesizers was the Yaaha DX-7, which was based on the results of Chowning s research in FM Synthesis. Introduced in 983, the DX-7 was polyphonic, had a five-octave touch-sensitive keyboard, and offered a wide choice of tibres, which the player could adjust. Siple FM Synthesis c::"usage" = "The carrier frequency or average frequency."; ::"usage" = "The odulating frequency."; d::"usage" = "The peak deviation fro the carrier frequency."; A::"usage" = "The aplitude of the sound in any units."; Exaple values of the variables (Sounds siilar to a Clarinet):

3 aboutfm.nb 3 c = 9.; = 6.; d = 6.; A =.; fwave@c_, _, d_, t_d := NAA SinA2 π ct+ d Sin@2 π td EE Clear@tD flist = Table@fwave@c,, k, td, 8k,, 6 6, 2<D; Here is a graph of tie versus aplitude of an FM function that starts out with no inner sine ter, then gradually the odulation increases, until the odulation index equals 6: tieplot = JPlotAEvaluate@#D, 8t,,.4<, PlotPoints 35, AspectRatio, PlotRange All, Frae TrueE&Nê@ 5 flist; << "FMPlot." fclarinetspectra = Table@Join@88,.<<, 88, <<, AbsPositiveFMSpectru@9, 6, id, 888, <<, 888,.<<D, 8i,, 6,.2<D; fclarinetspectrapt = 88,.<, 8, <, 89,.<, 88, <, 88,.<<; HSpectralPlot@#D&Lê@fClarinetSpectra;

4 4 aboutfm.nb Aplitude Frequency Note that these spectra are created with only two sinewaves. If additive synthesis were used, ore sinewave generators would need to be used. In the case of Index = 2, 6 sinwaves would be needed: Clear@tD index2 = Plus@@ H#P2T Sin@2 π #PT td&lê@ AbsPositiveFMSpectru@9, 6, 2D Sin@6 p td Sin@8 p td Sin@3 p td Sin@42 p td Sin@54 p td Sin@66 p td Plot@index2, 8t,,.4<D; Sin[x + Sin[x]] can be transfored into a series of Bessel functions, which akes it easier to see the nuber of individual haronics in the resulting wavefor: d = 2; Clear@tD

5 aboutfm.nb 5 besselindex2 = BesselJA, d E Sin@2 π ctd+ BesselJA, d EHSin@2 π Hc + L td Sin@2 π Hc LDL + BesselJA2, d EHSin@2 π Hc + 2LtD Sin@2 π Hc 2LDL + BesselJA3, d EHSin@2 π Hc + 3LtD Sin@2 π Hc 3LDL + BesselJA4, d EHSin@2 π Hc + 4LtD Sin@2 π Hc 4LDL; Plot@besselIndex2, 8t,,.4<D; How FM Spectra are Calculated

6 6 aboutfm.nb yd yd, 8x,, 2<, 8y,, 2<, PlotPoints 2, AxesLabel 8"Order", "Index", "Ap"<, FaceGrids None, BoxRatios 8,,.5<, Mesh True, FaceGrids All, Shading False, ViewPoint ,.93,.294<, PlotRange All, DefaultFont 8"Helvetica Bold", 8<D; 2 5 Order 5.5 Ap 5 Index 5 2

7 aboutfm.nb 7 Cross sections of the 3D plot above: Slicing at the far back (Order Bessel J): Clear@xD Plot@BesselJ@, xd, 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks 8Range@2D, Autoatic<D; Slicing the plot perpendicular to the last plot give the aplitudes of the individual haronics in the sound that has an index of. In the plot there is only one haronic present at :

8 8 aboutfm.nb 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks Autoatic<D; Order Bessel J: Plot@BesselJ@, xd, 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks 8Range@2D, Autoatic<D; Slicing the plot perpendicular to the last plot give the aplitudes of the individual haronics in the sound that has an index of. Only the values at the integers exist, So there is lots of the first haronic () and the second haronic (), plus a little bit at the third (2) and fourth (3) haronics:

9 aboutfm.nb 9 Plot@BesselJ@x, D, 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks 8Range@2D, Autoatic<D; Order 2 Bessel J: Plot@BesselJ@2, xd, 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks 8Range@2D, Autoatic<D; Slicing the plot perpendicular to the last plot give the aplitudes of the individual haronics in the sound that has an index of. Only the values at the integers exist. The plot below this one shows soe of the haronics created.

10 aboutfm.nb 2D, 8x,, 2<, PlotRange 88, 2<, 8, <<, Ticks Autoatic<D; Note the siilarity in shape to the previous plot: data = Drop@FMSpectru@, 5, 2D, 22D; SpectralPlot@ Join@88, <<, 88, <<, data, 88, <<, 88, <<D,.7D;.5 Aplitude Frequency

11 aboutfm.nb FM synthesis sounds in Matheatica The Matheatica Technical Report Guide to Standard Matheatica Packages version 2.2 describes how to do FM synthesis in the section on Miscellaneous Audio ( page 23). Sound generation in Matheatica isn t too efficient (so it takes a while to generate a sound). Needs[Miscellaneous Audio ]? Miscellaneous Audio * AplitudeModulation PrintHeader Square Cascade ReadSoundfile Standard FrequencyModulation Ring Triangle ListWavefor Sawtooth Type Overtones Sinusoid Wavefor Parallel SetOptions@FrequencyModulation, DisplayFunction $SoundDisplayFunction, SapleRate 825, SapleDepth 8D 8SapleRate Æ 825, SapleDepth Æ 8, PlayRange Æ All, DisplayFunction Æ HDisplay@$SoundDisplay, #D&L, Type Æ Standard<? FrequencyModulation FrequencyModulation[fc, {f, pd}, d, opts] returns a Sound object that is a frequency odulated sinusoid, where fc, f, and pd are the carrier, odulator, and peak deviation frequencies in Hertz, and d is the duration in seconds. The option Type can be set to Standard (default), Cascade, or Parallel. For Cascade and Parallel, the second arguent ust be a list of pairs of the for {{f, pd}, {f2, pd2},...}, where pdi is the peak deviation associated with odulating frequency fi, and both values are easured in Hertz. FrequencyModulation@99, 866, 66 <,.D; FrequencyModulation@99, 866, 66 3<,.D; FrequencyModulation@99, 866, 66 2<,.D;