Re: Are harmonics real?

Tim Wescott <tim@xxxxxxxxxxxxxxxx> wrote:
(snip)

< Do you understand the mathematics behind the Fourier transform and the
< FFT? Harmonics in a signal are as real as those mathematics, and in the
< case of the Fourier transform that's pretty darn real -- you can sum up
< all the energy in a signal in your choice of real time or the Fourier
< frequency domain, and you get exactly the same number (see Parseval's
< Theorem). If the energy balance works, then its hard to argue with the
< physics.

This is true, but it doesn't explain why harmonics, that is,
sines and cosines as basis functions, are so useful.

One reason is that many physical systems generate harmonically
related signals, though not all. The normal modes of most
bells are not harmonically related, though they are still sinusoids.
The modes of a drum head with uniform tension are bessel functions.

Another reason is that analog filters (RLC, or mass and spring)
can be described in terms of their response to sinusoidal inputs.

< The FFT case is a bit problematical, because the FFT is only exact if you
< happen to be dealing with a periodic and sampled signal. What makes the
< FFT fail to be exact isn't because the transform itself isn't exact, it's
< because you're "telling" the math that you're giving it one cycle of a
< sampled periodic wave, and chances are that's not really what you have.

Sometimes you can reduce the effect by extending the time, also
known as zero padding. If the signal decays smoothly to zero,
then the amplitude of the harmonics usually decreases with
frequency and the result isn't so bad. If the system is continually
driven that likely won't work. If it is driven periodically, then
do the transform on that period.

-- glen
.

Relevant Pages

  • Re: Are harmonics real?
    ... Harmonics in a signal are as real as those mathematics, and in the < case of the Fourier transform that's pretty darn real -- you can sum up < all the energy in a signal in your choice of real time or the Fourier < frequency domain, and you get exactly the same number. ... related signals, though not all. ... < The FFT case is a bit problematical, because the FFT is only exact if you < happen to be dealing with a periodic and sampled signal. ...
    (comp.dsp)
  • Re: Discrete Deconvolution
    ... I'm analyzing some time signals for periodicities, ... a window function, and s is the actual source signal. ... I know the FFT ... The FFT can be used as an approximation of the Fourier transform of a signal in sampled time with an infinite number of points, which is in turn an approximation of a signal in continuous time that extends into infinity. ...
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  • Re: FFT frequency shift
    ... The FFT may ... Are you trying to show the average power around each of the harmonics? ... We are acquiring signals from ... > entire signal consists in multiple pulses in multiple frequencies ...
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  • Re: Large FFT vs Many FFTs
    ... I've been asked to research a DSP system to take an FFT to detect harmonic ... Because of the very large number of samples to be averaged, to find the average signal levels you could use a fairly crude approach of averaging the absolute values of the filtered signal samples, for each of the five filters. ... Also, as the fundamental will have the same scaling error as the harmonics, you could probably safely leave out applying a scaling factor to convert 'average value' to RMS. ... If you really wanted the maximum possible precision you would convert the signal to analytic, use a bank of complex filters, calculate the magnitude of each of the five complex signals at each sample period, and compute the five average magnitudes over a time of one second. ...
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  • Re: Zero Padding In The Frequency Domain
    ... Real world signals are of finite duration, ... It seems very likely that the fft() of a finite duration signal would have a non-zero value at Nyquist. ... Fourier transform does not calculate frequency components outside [-Fs/ ...
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