The Art of Doing Science and Engineering by Richard W. Hamming

Author:Richard W. Hamming [Hamming, Richard W.]
Language: rus
Format: epub
Tags: Engineering, Computing & Technology
Publisher: Taylor and Francis CRC ebook account
Published: 2007-03-29T21:00:00+00:00

Figure 16.V

carefully worked out all the algebraic details to convince myself what I thought had to be true from theory was indeed true in practice.

Then we use the FFT, which is only a cute, accurate, way of getting the coefficients of a finite Fourier series. But when we assume the finite Fourier series representation we are making the function periodic—and the period is exactly the sampling interval size times the number of samples we take! This period has generally nothing to do with the periods in the original signal. We force all nonharmonic frequencies into harmonic ones—we force a continuous spectrum to be a line spectrum! This forcing is not a local effect, but as you can easily compute, a nonharmonic frequency goes into all the other frequencies, most strongly into the adjacent ones of course, but nontrivially into more remote frequencies.

I have glossed over the standard statistical trick of removing the mean, either for convenience, or because of calibration reasons. This reduces the amount of the zero frequency in the spectrum to 0, and produces a significant discontinuity in the spectrum. If you later use a window, you merely smear this around to adjacent frequencies. In processing data for Tukey I regularly removed linear trend lines and even trend parabolas from some data on the flight of an airplane or a missile, and then analyzed the remainder. But the spectrum of a sum of two signals is not the sum of the spectra—not by a long shot! When you add two functions the individual frequencies are added algebraically, and they may happen to reinforce or cancel each other, and hence give entirely false results! No one I know has any reasonable reply to my objections here—we still do it partly because we do not know what else to do—but the trend line has a big discontinuity at the end (remember we are assuming that the functions are all periodic) and hence its coefficients fall off like 1/k, which is not rapid at all!

Let us turn to theory. Every spectrum of real noise falls off reasonably rapidly as you go to infinite frequencies, or else it would have infinite energy. Figure 16.VI. But the sampling process aliases the higher frequencies in lower ones, and the folding as indicated, tends to produce a flat spectrum—remember the frequencies when aliased are algebraically added. Hence we tend to see a flat spectrum for noise, and if it is flat then we call it white noise. The signal, usually, is mainly in the lower frequencies. This is true for several reasons, including the reason "over sampling" (sampling more often than is required from the Nyquist theorem), means we can get some averaging to reduce the instrumental errors. Thus the typical spectrum will look as shown in the Figure 16.VI. Hence the prevalence of low pass filters to remove the noise. No linear method can separate the signal from the noise at the same frequencies, but those beyond the signal can be so removed by a low pass filter.

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