The Design of Reactive Equalizers* 



By A. P. BROGLE, Jr. 



This paper describes a systematic method of approximating with a finite 

 number of network elements a transfer characteristic which is a prescribed func- 

 tion of frequency, rather than a constant, over the useful frequency band. Al- 

 though applied here only to input and output couphng networks as reactive 

 equalizers and where loss equalization to an extremely high degree of precision 

 over a wide frequency band is desired, the mathematical expressions which form 

 the basis for the design are applicable to any 4-terminal network whose transfer 

 characteristic is specified in a similar manner over the real frequency range. 



The selection of the appropriate form of the transfer function for equalization 

 purposes is the fundamental consideration. A squared Tchebycheff polynomial is 

 found to be particularly suitable to produce a desired cut-off characteristic with- 

 out impairing the precision of equalization in the useful band. 



A method of polynomial approximation based on the transformation co = 

 tan (p/i is used to obtain the coefficients of the in-band approximating function. 

 Predistorting the transfer specification and minimizing the mean-square error, 

 the coefficients become the Fourier cosine coefficients for an infinite frequency 

 range; and are the solutions of a linear set for a finite range, o < <p < w/o. 



1. Introduction 



IN MOST broad-band communication systems, the problems of loss 

 equalization and distortion correction are fundamental. Of the various 

 types of electrical networks which are found useful as equalizers and com- 

 pensators, the most frequently employed are the so-called constant re- 

 sistance networks. In particular, they are of three usual types, as indicated 

 in Fig. 1. 



In all cases, the relationship Z1Z2 = i?^ which is always possible to fultill 

 if Zi and Z2 are built up of resistive and reactive components in the well- 

 known manner, provides the means of altering the transmission properties 

 of the circuit without affecting its impedance. ' Methods are also available 

 which extend the problem to more complicated configurations having these 

 constant resistance properties. However, in some applications, where signal- 

 to-noise ratio considerations are of importance, the resistive elements in- 

 cluded as components of Zi and Z2 in these circuits place a limitation on the 

 final performance of the system. Hence, the satisfactory transmission and 

 impedance matching properties of these circuits are purchased at the expense 

 of a substantially increased noise level. As a consequence of this limitation 

 on the performance of standard constant resistance equalizers, recent work 



* The work presented in this paper is part of a thesis, "Design of Reactive Equalizers 

 with Prescribed Parasitic Capacitance," submitted by the author in partial fulfillment of 

 the requirements for the degree of Master of Science at the Massachusetts Institute of 

 Technology (Feb. 1949). 



1 Ref. 5, pp. 1-2. 



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