0.3 Frequency modulation (fm) mathematics  (Page 2/2)

Download the LabVIEW VI demonstrated in the video: fm_demo1.vi . Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

Fm spectrum

The following trigonometric identity facilitates quantitative understanding of the spectrum produced by the basic FM equation of :

The term $J_k(a)$ defines a Bessel function of the first kind of order $k$ evaluated at the value $a$ .

Note how the left-hand side of the identity possesses exactly the same form as the basic FM equation of . Therefore, the right-hand side of the identity explains where the spectral components called sidebands are located, and indicates the amplitude of each spectral component. The screencast video continues the discussion by explaining the significance of each part of , especially the location of the sideband spectral components.

As discussed video, the basic FM equation produces an infinite number of sideband components; this is also evident by noting that the summation of runs from k=1 to infinity. However, the amplitude of each sideband is controlled by the Bessel function, and non-zero amplitudes tend to cluster around the central carrier frequency.The screencast video continues the discussion by examining the behavior of the Bessel function $J_k(a)$ when its two parameters are varied, and shows how these parameters link to the modulation index and sideband number.

Now that you have developed a better quantitative understanding of the spectrum produced by the basic FM equation, the screencast video revisits the earlier audio demonstration of the FM equation to relate the spectrum to its quantitative explanation.

Harmonicity ratio

The basic FM equation generates a cluster of spectral components centered about the carrier frequency

$f_c$ with cluster density controlled by the modulation frequency

$f_m$ . Recall that we perceive multiple spectral components to be a single tone when the components are located at integer multiples of a fundamental frequency, otherwise we perceive multiple tones with different pitches. The harmonicity ratio $H$ provides a convenient way to choose the modulation frequency to produce either harmonic or inharmonic tones. Harmonicity ratio is defined as:

Harmonicity ratios that involve an integer, i.e., $H=N$ or $H=1/N$ for $N\ge 1$ , result in sideband spacing that follows a harmonic relationship. On the other hand, non-integer-based harmonicity ratios, especially using irrational numbers such as $\pi$ and $\sqrt{2}$ , produce interesting inharmonic sounds.

Try experimenting with the basic FM equation yourself. The LabVIEW VI fm_demo2.vi provides front-panel controls for carrier frequency, modulation index, and harmonicity ratio. You can create an amazingly wide variety of sound effects by strategically choosing specificvalues for these three parameters. The screencast video illustrates how to use the VI and provides some ideas about how to choose the parameters.Refer to TripleDisplay to install the front-panel indicator used to view the signal spectrum.

References

• Moore, F.R., "Elements of Computer Music," Prentice-Hall, 1990, ISBN 0-13-252552-6.
• Dodge, C., and T.A. Jerse, "Computer Music: Synthesis, Composition, and Performance," 2nd ed., Schirmer Books, 1997, ISBN 0-02-864682-7.