. The Bell System technical journal . formly spaced frequencies of the transfer constant andimage impedance chains, and to contain only the cut-off factor, it iseasily shown that it must include a net change in phase of 37r/2 interval must therefore contain 3/2 uniform spaces if the averageslope is to be correct. Considerations of symmetry to be describedlater require that the cut-off be the center of the interval, which thuscomprises two three-quarter spaces. The behavior of the severalcomponents of the total insertion phase is exhibited by Fig. 19, inwhich B, Br, and Bi refer res
. The Bell System technical journal . formly spaced frequencies of the transfer constant andimage impedance chains, and to contain only the cut-off factor, it iseasily shown that it must include a net change in phase of 37r/2 interval must therefore contain 3/2 uniform spaces if the averageslope is to be correct. Considerations of symmetry to be describedlater require that the cut-off be the center of the interval, which thuscomprises two three-quarter spaces. The behavior of the severalcomponents of the total insertion phase is exhibited by Fig. 19, inwhich B, Br, and Bi refer respectively to the phase shifts contributedby the transfer constant, by reflection effects, and by the interactionfactor. The mutually annulling discontinuities of 7r/2 radians in B,and Br at the cut-off are noteworthy. The fact that this choice of parameters is sufficient as well asnecessary to obtain the desired linearity of phase shift is not easilyshown analytically. It can, however, be verified by direct computa- IDEAL FILTERS 247. FREQUENCY IN TERMS OF a Fig. 19—Transfer, reflection, and interaction phase in the transition interval. tion. For this purpose the customary resolution of the total insertionloss into transfer constant, reflections, and interaction is not very usefulbecause of the indeterminacies found at the cut-off. This difficultyis avoided by expressing Z i and 6 in terms of the lattice impedances,in which event gT = 1 + 7 7 R ^ R + Zy R z. R Zy R (26) where 7 is the total insertion loss. If iXx and iXy be written for Zx and Zy, the insertion loss and phaseshift are given by Ji. lA y R tan B^ = and R{Xx + Xy) e^y = yJ{R-\-Xx)iR + Xy^) R{X X — Xy) (27)(28) Equation (27) can be used to confirm our previous choice of thelocation of the cut-off. At this frequency one of the two reactances,Xx and Xy, will be either resonant or anti-resonant. It is evidentfrom (27) that if the phase shift is to have the desired value,{n -(- 3/4)7r, at the assumed cut-off the non
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