326 Practical MATLAB® Applications for Engineers
Then its RMS value can be evaluated by
F
ac c cn
RMS 02
12
2222 22 2
Note that the RMS value can be easily evaluated using MATLAB as indicated in
the following:
let
Facc c (^22) n
1
(^0122)
...
*
Then the MATLAB command, Frms = norm(F), returns the RMS value of f(t).
R.4.18 Harmonic distortion is an index that indicates the discrepancy between the series
approximation of f(t), and the actual waveform of f(t). The percentage distortion due
to a particular Fourier component (harmonic) is given by
Percentage distortion for the n-harmonic component (PDn)
cc
cn
1(^) * 100 %
R.4.19 The percentage of total harmonic distortion (PTHD) is given by
PTHD
c
cc
cc
c2 n
12
3
1212
100
* %
R.4.20 Gibb’s* phenomena states that if f(t) presents some discontinuities, then the FS
approximation at the discontinuities would show a signifi cant amount of ripple,
and the synthesized function converges to the average value at the point of the
discontinuity, if a suffi ciently large number of terms are employed in its series
approximation. Recall that a discontinuity refers to any point of f(t), whose ampli-
tude abruptly changes from one value to another (step).
Josiah Willard Gibbs fi rst published the preceding observation in 1899.
R.4.21 Recall that the word synthesis means that the sum of the parts constitutes the
whole.
For the case of the Fourier analysis, the term synthesis means that the recombi-
nation of the terms of the FS, usually the fi rst fi ve or six terms, represent a good
approximation of the original wave.
R.4.22 The coeffi cients F 0 , F 1 , F− 1 , F 2 , F− 2 , ..., Fn of the exponential FS represent the
magnitude of the fundamental and all the harmonic frequencies at 0, w 0 , −w 0 , 2 w 0 ,
− 2 w 0 , ..., ±nw 0 , respectively.
*^ Josiah Willard Gibbs (1839–1903), a physicist–chemist at Yale University, where he served honorarily for
10 years; he also served with a reduced salary at John Hopkins University.