. The Bell System technical journal . re A and E are complete elliptic integrals of the first and second kindwith modulus k. Hence 37r (1 -t- 3X)(1 -f X) ^:vp = ^ = —I—^-^^ < where the modulus of A and E is \/(l — X)/2. This equation defines pas a function of X, and hence by inversion gives X as a function of p. Theresulting curve of X vs. p is plotted in Fig. 6 and may be designated as thefunction X = g (p). If we substitute X = V/P we then have V = P g{3Tr/Ra V2P) () This enables us to plot V as a function of P, for various values of Ra, Fig. P may represent the envelope of an
. The Bell System technical journal . re A and E are complete elliptic integrals of the first and second kindwith modulus k. Hence 37r (1 -t- 3X)(1 -f X) ^:vp = ^ = —I—^-^^ < where the modulus of A and E is \/(l — X)/2. This equation defines pas a function of X, and hence by inversion gives X as a function of p. Theresulting curve of X vs. p is plotted in Fig. 6 and may be designated as thefunction X = g (p). If we substitute X = V/P we then have V = P g{3Tr/Ra V2P) () This enables us to plot V as a function of P, for various values of Ra, Fig. P may represent the envelope of an amplitude-modulated (or diflf-erentiated FM) wave, and V the corresponding recovered signal outputvoltage, the curves of Fig. 7 give the complete performance of the circuitas an envelope detector. In general the envelope would be of form P —Po[l -f c s{l)\, where s{t) is the signal. We may substitute this value of Pdirectly in () provided the absolute value of c s{t) never exceeds unity. 152 BELL SYSTEM TECHNICAL JOURNAL. Fig. 6.—The Function X = g{f>) defined by Eq. ().
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