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PDFFrequency modulation.
Major Edwin Howard Armstrong’s 1933 invention of wideband frequency modulation gave the world, for the first time, static-free radio. The price to pay for static-free radio is most often bandwidth. Whereas commercial AM radio has a bandwidth of about 10 kHz, FM radio has a bandwidth of 100 kHz, which includes a guard band. Likewise, there is also another toll: complexity. Both the FM transmitter and the FM receiver require more components, thus, they are more costly than their amplitude modulation (AM) counterparts. Aside from the technical considerations, there is a dominant theoretical difference. In principle, AM is a linear modulation process, while FM is very nonlinear. This complicates the analysis, the design, and the implementation of FM devices.
Because the fundamental equation for AM shows the amplitude of the carrier signal and the two sideband signals as a function of the modulation index, m, how can we determine similar carrier and sideband amplitudes for FM? The following equation (1.1) is the fundamental—and much more complicated than AM—equation for FM.
or
or, in its simple, fundamental form,
(1-1)where the modulation index mf is somewhat analogous to m in AM, and the amplitude factor A is normalized to unity. (Different references use either a sine or a cosine function for the signal term.)
The FM modulation index can be defined as:
Equation (1-1) shows the first (carrier) signal sine term is a function of a second, and more complicated, t + ksin (or kcos) term, where k is a constant. To solve (1-1) you will need some higher math called Bessel functions. Most texts on communications show a table listing the FM sideband-frequency values as a function of their Bessel functions, Jn, and mf. Likewise, the values in these tables are usually also shown in graphical form.
Table 1 shows the FM sideband functions that can be produced using mf and the Bessel factors Jn. The factors in this table are the tabulated form of m-file scripts created using The MathWorks, Inc. MATLAB® software. They are used to produce the sideband curves in Figure 1, which are shown below in Table 1.
|
Modulation
mf |
Carrier
J0 |
Side b and 1
J1 |
Side b and 2
J2 |
|
0.25
|
0.98
|
0.12
|
0.01
|
|
0.50
|
0.94
|
0.24
|
0.03
|
|
1.00
|
0.77
|
0.44
|
0.11
|
|
1.50
|
0.51
|
0.56
|
0.23
|
|
2.00
|
0.22
|
0.56
|
0.35
|
|
2.40
|
0.00
|
0.52
|
0.43
|
|
3.00
|
-0.25
|
0.34
|
0.49
|
|
4.00
|
-0.40
|
-0.07
|
0.36
|
|
5.00
|
-0.18
|
-0.33
|
0.05
|
|
5.50
|
0.00
|
-0.34
|
-0.12
|
|
6.00
|
0.15
|
-0.28
|
-0.24
|
|
7.00
|
0.30
|
0.00
|
-0.30
|
|
8.00
|
0.17
|
0.23
|
-0.11
|
|
8.65
|
0.00
|
0.27
|
0.06
|
Figure 1. Bessel Functions of Each Waveshape
Figure 1 shows the magnitude (on the y-axis) of the FM carrier in addition to the first two sidebands. The x-axis of Figure 1 is the frequency modulation index. This modulation index, mf, is roughly analogous to the AM modulation index, m.
MATLAB® is a registered trademark of The MathWorks, Inc.
Refer to the RF & Communications Resources page for additional information about RF terminology, fundamentals, and National Instruments RF products.
Additional References
Helpful Web Sites:
Information Contributed By: Bob Libbey, Retired RCA Engineer and Adjunct Professor, New Jersey Institute of Technology.
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