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Fig. 9-8. Response of Butterworth (Dotted Line), Least-Squares Quadratic (Dashed Line), and Averaging (Solid Line) Filters to Gaussian Noise (Means I V, Standard Deviation=0.5 V)

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Fig. 9-8. Response of Butterworth (Dotted Line), Least-Squares Quadratic (Dashed Line), and Averaging (Solid Line) Filters to Gaussian Noise (Means I V, Standard Deviation=0.5 V)

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Fig. 9-9. Response of Butterworth (Dotted Line), Least-Squares Quadratic (Dashed Line), and Averaging (Solid Line) Filters. (K(f)~ I +coswf + r, where «„=0.13 sec"1, E(v)=02S, o(r)=0.25 and wc = 0.13 sec"'.)

attenuate the signal. The predicted attenuation factor is 0.S for the Butterworth for which ue=u. Note the phase lag in the response of the Butterworth filter.

Figures 9-10 and 9-11 illustrate the use of the frequency response of the Butterworth and LSQ filters to obtain a desired output frequency spectrum. In Fig. 9-10, the Butterworth filter cutoff, «,=2^/100, is chosen to attenuate the input frequency, whereas in Fig. 9-11 the cutoff, <or = 2tt/ 12.5, is chosen to pass the input frequency and only attenuate the noise. The frequency dependence of the Butterworth filter's phase lag is apparent by comparing Figs. 9-9 and 9-11.

The frequency response of the LSQ filter is not as easily controlled as that of the Butterworth. In Fig. 9-10, with m = 50, there is some attenuation of the input frequency, whereas in Fig. 9-11, with m = 5, the signal attenuation is negligible but the noise is not removed completely.

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