CONSTANT RESISTANCE NETWORKS 185 



peaks are to occur at real frequencies, F(\) must have poles at 

 X^ = — 1/5^^ and zeros at X^ = — Pm^- Moreover, since 1/[1 + F{\)'] 

 and l/l^l + (1/F(X)^ must always be positive for real frequencies, the 

 expression for F(\) when all its zeros and poles occur at real frequencies 

 must be a perfect square. It may then be written 



(1 + Si^x^y •••(! + s\n-i)/2\'y 



In order to get an idea of the significance of the expression, let 

 X = i(J/fo) and restrict the P's and the S's to values less than unity. 

 The first network will then have zero loss points at/ = and/ = Pmfc 

 and infinite loss points at/ = fo/Sm and/ = oo. The second network 

 will have infinite loss points when the loss of the first is zero, and zero 

 loss points when the loss of the first is infinite. The first network is 

 therefore a low-pass filter and the second a high-pass filter. 



The following work is considerably simplified if Sm = Pm- This 

 implies that the characteristic of the second filter is the same function 

 of 1/X that the first is of X. If the cross-over point is fixed at X^ = — 1, 

 the value of .<4o is — 1 and in order to write equation (6) or (7), it is 

 necessary to find those zeros with a negative real part of 



1 ,, (Pi' + ^'y • • • (P^n-n/2 + X^)^ 



"^ (1 + P{'\'y •••(!+ P\n-l)/2\'y 

 (Pl2 + X2) . • • 



1 +x 



(1 + Pl'\') 



J , (P,2 + X2) 



(1 + Pi'X') 



Now since the zeros of the second factor on the right are the negatives 

 of the zeros of the first factor, it will be sufficient to find all of the 

 zeros of the first factor and reverse the signs when necessary to secure 

 negativ^e real parts. Consider, then, the equation 



1 , . (^l'' + X'') • • • (P^n_l);2 + X^) __ .. 



' '^''(1 -f Pi2x2) ... (1 +P2^„ .1)^2X2) ''• 



One root is X = — 1. It may be shown further that the magnitude 

 of all of the roots is unity. Writing X = pe^^ as a root, the magnitude 

 of the typical product term (P^ -f X2)/(l + P^X') may be written 



1 ^„\ / „ 1 



P2 -f X2 p ^ J , \ -P' 



-^-P'Hp'-^^ 



1 -f P2X2 /o , 1 \' . . ,. 



' I Pp + -p- j - 4 sm2 d 



