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Signal Error

Module by: Don Johnson

Summary: This module discusses the accuracy of the Fourier Series approximation.

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 s K t to be the signal containing K K Fourier terms.

s K t= a 0 +∑k=1K a k cos2πktT+∑k=1K b k sin2πktT s K t a 0 k 1 K a k 2 k t T k 1 K b k 2 k t T (1)
Figure 1 shows how this sequence of signals increasingly portrays the signal accurately as more terms are added.

Figure 1: The Fourier series spectrum of a half-wave rectified sinusoid is shown in the upper portion. The index indicates the multiple of the fundamental frequency at which the signal has energy. The cumulative effect of adding terms to the Fourier series for the half-wave rectified sine wave is shown in the bottom portion. The dashed line is the actual signal, with the solid line showing the finite series approximation to the indicated number of terms, K K .
Fourier Series spectrum of a half-wave rectified sine wave
Subfigure 1.1
Fourier Series spectrum of a half-wave rectified sine wave, Subfigure 1.1 (spectrum2.png)
Subfigure 1.2
Fourier Series spectrum of a half-wave rectified sine wave, Subfigure 1.2 (fourier1.png)

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

ε K t=∑k=K+1∞ a k cos2πktT+∑k=K+1∞ b k sin2πktT ε K t k K 1 a k 2 k t T k K 1 b k 2 k t T (2)
To find the rms error, we must square this expression and integrate it over a period. Again, the integral of most cross-terms is zero, leaving
rms ε K =12∑k=K+1∞ a k 2+ b k 2 rms ε K 2 1 2 k K 1 a k 2 b k 2 (3)
Figure 2 shows how the error in the Fourier series decreases as more terms are incorporated. In particular, the use of four terms, as shown in the bottom plot of Figure 1, has a rms error (relative to the rms value of the signal) of about 3%. The Fourier series in this case converges quickly to the signal.

Figure 2: The rms error calculated according to Figure 1 is shown as a function of the number of terms in the series. The error has been normalized by the rms value of the signal.
rms error
rms error (fourier2.png)

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