Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

'344 FREQUENCY MODULATION, Volume I[ is a complete elliptic integral of the second kind with1 + x)modulus2Vx~1 + x'Root-Mean-Square Voltage of Rectified EnvelopeThe Root-Mean-Square voltage of the rectified carrier envelope is:Root-Mean-Square voltage = e ilK j- I (l + a; 2 + 2x cos 8) dfl1*= eiV^L (1 + x 2 ) 8 + 2x sin 6Fourier-Series Analysis of Envelope= e,\/l + x 2 (35)The envelope of the earrier isgiven by:Envelope = ei Vl + x 2 + 2x cos 2iry.t where x ^1. (36)Consider the expression:Vl + x 2 + 2x cos (3 = (1 + xe^)i(l + xer^-U+l w « _ W x *e** mJSl xieM _ KD (3 ) (5 ) ^s, I"r + 2 :Ce 2(4) a;e + Xe +2(4)(6) 2(4) (6)-"\(8)X r + 2 W 2(4) X + 2(4)(6)^ 2(4) (6) (8)% & +'"Jby the usual binomial series expansion. 18 (37)where:Multiply these factors together, term by term, then:Vl + z 2 + 2x cos /3 = a + a i cos + a 2 cos 2/J + .. . (38)x 2 x 4 z 6 25za- 81 + 4- + 64+ 256+ 16384+ -'- ^(„ x 2 x i 5x 6 S5x s )ai = xv--i-6i-im-iMi- •• \(4018Edwin P. Adams, SMITHSONIAN MATHEMATICAL FORMULAEAND TABLES OF ELLIPTIC FUNCTIONS, Smithsonian Institution, Washington,D. C, 1939, p. 117.