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It is interesting to consider the sequence of signals that we obtain as we incorporate more terms into the Fourier series approximation of the half-wave rectified sine wave . Define s K t to be the signal containing K 1 Fourier terms.

s K t a 0 k 1 K a k 2 k t T k 1 K b k 2 k t T
[link] shows how this sequence of signals portrays the signal more accuratelyas more terms are added.

Fourier series spectrum of a half-wave rectified sine wave

The Fourier series spectrum of a half-wave rectified sinusoidis shown in the upper portion. The index indicates the multiple of the fundamental frequency at which the signal hasenergy. The cumulative effect of adding terms to the Fourier series for the half-wave rectified sine wave is shown in thebottom portion. The dashed line is the actual signal, with the solid line showing the finite series approximation to theindicated number of terms, K 1 .

We need to assess quantitatively the accuracy of theFourier series approximation so that we can judge how rapidly the series approaches the signal. When we use a K 1 -term series, the error—the difference between the signal and the K 1 -term series—corresponds to the unused terms from the series.

ε K t k K 1 a k 2 k t T k K 1 b k 2 k t T
To find the rms error, we must square this expression and integrate it over a period. Again, the integral of mostcross-terms is zero, leaving
rms ε K 1 2 k K 1 a k 2 b k 2
[link] shows how the error in the Fourier series for the half-wave rectified sinusoid decreases asmore terms are incorporated. In particular, the use of four terms, as shown in the bottom plot of [link] , has a rms error (relative to the rms value of the signal) of about 3%. The Fourier seriesin this case converges quickly to the signal.

Approximation error for a half-wave rectified sinusoid

The rms error calculated according to [link] is shown as a function of the number of terms in theseries for the half-wave rectified sinusoid. The error has been normalized by the rms value of thesignal.

We can look at [link] to see the power spectrum and the rms approximation error for thesquare wave.

Power spectrum and approximation error for a square wave

The upper plot shows the power spectrum of the square wave, and the lower plot the rms error of the finite-lengthFourier series approximation to the square wave. The asterisk denotes the rms error when the number of terms K in the Fourier series equals 99.
Because the Fourier coefficients decay more slowly here than for the half-wave rectified sinusoid, the rms error is notdecreasing quickly. Said another way, the square-wave's spectrum contains more power at higher frequencies than does thehalf-wave-rectified sinusoid. This difference between the two Fourier series results because the half-wave rectifiedsinusoid's Fourier coefficients are proportional to 1 k 2 while those of the square wave are proportional to 1 k . If fact, after 99 terms of the square wave's approximation, the error is bigger than 10 terms of theapproximation for the half-wave rectified sinusoid. Mathematicians have shown that no signal has an rmsapproximation error that decays more slowly than it does for the square wave.

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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