Rca1948FrequencyModu.. - The New Jersey Antique Radio Club

.TRANSMISSION THROUGH A NETWORK 433and imaginary parts is the amplification—frequency characteristicThe arc tangent of the quotient of the imaginary and real parts offM/gmX is the phase-shift—frequency characteristic (d).C. Approximate the network characteristic by a finite Fourierseries over an interval equal to or greater than the essential(c)intervaldetermined in step A.Example : Unless the network-characteristic function and its derivatives at one end point of the essential interval chosen in step A happento be equal to the function and its derivatives, respectively, at theopposite end point of the interval, it is necessary to take the followingsteps to avoid extrinsic discontinuities which would needlessly burdenand reduce the accuracy of the Fourier approximation. First, theperiod for the Fourier approximation should be chosen slightly largerthan the interval over which the approximation must accurately representthe network characteristic.Second, the function should be alteredin the two margins where the chosen period overlaps the essentialinterval of approximation in such a way that the new function has nodiscontinuities either within the chosen period or at its end points.The artificial function constructed within the margins should be -assmooth as possible to avoid its contributing to the higher-order termsof the Fourier approximation.Recall that the essential interval of approximation was 2tt X 0.965X 10 6 g w ^ 2tt X 1.035 X 10 e ; let CI, the period of approximation, beslightly greater, say, 2tt X 0.960 X 10 G § «) ^ 2tt X 1.040 X 10 6 . Thenconstruct an artificial function in the margins of overlap of theintervals (2tt X 0.960 X 10 6 g o> s 2tt X 0.965 X 10 6 and 2-n X 1.035X 10 e ^ a ^ 2tt X 1.040 X 10 6 ) that smoothly joins the ends of theperiod. This artificial function has been dotted into (b) of Figure 4.In Figure 4, (a) is so nearly continuous at the end points that no alterationis necessary. The essential interval of approximation and theperiod of approximation are also indicated-A 24-point numerical analysis 4 of this improved function in the indicatedperiod yields its Fourier coefficients. When using numericalmethods to obtain these coefficients, one must be careful that the signsof the coefficients are correct to place the approximation above o> asrequired by (11).^0