. The Bell System technical journal . itude in the neighborhood of coi, so that ai is zero. Then sin 5r fit) = 28M exp (-ai) --— cos (wit - (dx - wi^i)). 8t (6) Here the amplitude includes a constant factor which is proportional tothe bandwidth, 25, and to the magnitude, M exp( —aO, at the frequency,oji, and a function of time, a plot of which is shown in Fig. 1. This functionconsists of a sinusoidal wave of frequency, coi, the amplitude of which varieswith time, the envelope being symmetrical about the instant, Te =^ 61 , STEADY STATE DELAY A\D APERIODIC SIGXALS 111 at which it is a maximum.
. The Bell System technical journal . itude in the neighborhood of coi, so that ai is zero. Then sin 5r fit) = 28M exp (-ai) --— cos (wit - (dx - wi^i)). 8t (6) Here the amplitude includes a constant factor which is proportional tothe bandwidth, 25, and to the magnitude, M exp( —aO, at the frequency,oji, and a function of time, a plot of which is shown in Fig. 1. This functionconsists of a sinusoidal wave of frequency, coi, the amplitude of which varieswith time, the envelope being symmetrical about the instant, Te =^ 61 , STEADY STATE DELAY A\D APERIODIC SIGXALS 111 at which it is a maximum. 7,., the time of maximum envelope, is then aunique instant which is suitable for defining the time at which the dis-turbance occurs. It is determined solely by the slope of the phase frequencycurve for the spectrum. The instant, T^, may be interpreted, in accordance with the principleof stationary phase, as the one at which the sinusoidal components of (2)are most nearly in the same phase, and so have the least destructive inter-. Fig. 2—Graphical representation of the phase of an elementary disturbance ference. This condition will hold when the instantaneous phase angle ischanging least rapidly with frequency, that is, when 5co (a>/ - 0) = 0, from which / = The angle, Q\ — (s>\Q\, in (6), gives the phase of the wave at the instant,Te, when its envelope is a maximum. The interpretation of this angle willbe aided by the geometrical construction of Fig. 2 which is similar to that 228 BELL SYSTEM TECHNICAL JOURNAL employed for phase and group velocity . The abscissae are values of toand the ordinates are values of phase in radians. A portion of the function,6, in the neighborhood of coi is shown. The distance, OB, is di. The slopeof the tangent, CA, to the curve at A \?, d\. The distance, CB, is wi^, OC, or the intercept of this tangent on the phase axis, isdi — w\d[. If, as shown in the figure, the absolute value of this interceptis greater than it,
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Keywords: ., bookcentury1900, bookdecade1920, booksubjecttechnology, bookyear1