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Fig. 9-2 Ideal of Fig. 9-1 and its approximation by one single-tuned circuit. (Passband centered about zero frequency.)

Fig. 9-2 Ideal of Fig. 9-1 and its approximation by one single-tuned circuit. (Passband centered about zero frequency.)

realized almost as readily with cascaded single-tuned or other interstage networks.

Another approximation problem of interest is the attainment of moderately effective filtering, but with the principal interest a linear-phase characteristic in the passband. Such a characteristic can be obtained still using only simple interstage networks.

9-1 Maximally Flat Gain Function.1 This function, which is also known as the "Butterworth" function 2 and as "approximation in the Taylor sense," has for its normalized magnitude, or amplitude response, the following form:

The shape of the response curve for various values of n (the number of stages) is shown in Fig. 9-3. As n increases, the shape becomes more nearly the rectangle of Fig. 9-2 and the 3-db bandwidth remains constant. The function is always monotonic, i.e., decreases uniformly toward zero on

1 V. D. Landon, Cascade Amplifiers with Maximal Flatness, RCA Rev., vol. 5, pp. 347-362, January, 1941 (first introduced term "maximal flatness"); W. A. Lynch, The Role Played by Derivative Adjustment in Broadband Amplifier Design, Proc. Symposium on Modern Network Synthesis, Polytechnic Institute of Brooklyn, N.Y., 1952, pp. 193-201.

2 S. Butterworth, On the Theory of Filter Amplifiers, Wireless Engr., vol. 7, pp. 536-541, October, 1930.

either side of band center, and for the n-pole function it represents "maximal flatness" in that the maximum number of derivatives (2n — 1) are zero at band center. This feature can be demonstrated as follows:

Compare Eq. (9-3) with the corresponding Taylor (Maclaurin) series for fix).

X2 f2n-1i0)x2n~l f2n(0)x2n fix) = /(0) + /'(0)x + /"(0) - + •••+ — + —ttt" +'

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