226 BELL SYSTEM TECHNICAL JOURNAL 



simple matter. We need merely repeat, in each transmission or 

 transition interval, the rules for frequency spacing we have already 

 developed. 



The method can be understood from the study of a linear phase 

 shift high-pass filter. Since linear phase shift demands arithmetic 

 spacing of critical frequencies in the transmission band, it is clear that 

 the desired characteristic cannot be obtained over the complete 

 transmission band of the high-pass filter with a finite network. This 

 difiiculty will, however, be ignored for the moment. A method of 

 modifying the analysis to give a finite filter having a linear phase 

 characteristic in a finite interval above the cut-off will be described 

 later. 



We begin, then, by assigning to the transmission band of the 

 structure an infinite, evenly spaced chain of critical frequencies, as in 

 (13). The group of transition factors must evidently simulate the 

 reciprocal of this value in the attenuation range, if the condition of 

 high loss is to be realized; while in the transmission band, they must 

 simulate the P of our earlier analysis if we are to obtain a linear phase 

 characteristic. These conditions would be met by using for our 

 transition function the reciprocal of (19), using for the c's the same 

 values as before. Such a group, however, is not physically realizable 

 as part of a high-pass transfer constant, since the rational factors 

 would occur outside the theoretical transmission band. If, however, 

 we transfer the factor (1 — P) from (13) to (14), and seek a new Q 

 whose values will take the reciprocals of the old, thus altered, we 

 obtain a series identical with (17) except for a change of sign in every 

 term but | log (1 — P)- This change, however, reverses the sign of 

 the right-hand members of (21), and therefore changes the sign of 

 each c. The new solution then is the same as the original solution 

 except that the factors occur in reverse order on the frequency scale. 

 They can thus appropriately be combined with the remaining portion 

 of the high-pass transfer constant expression. 



A linear phase shift band-pass filter can be constructed similarly. 



The groups of transition factors associated with the upper and lower 



cut-offs should follow the arrangements prescribed, respectively, for 



low-pass and high-pass filters. An illustration will be found in 



Part II. 



The Impedance Property 



It will be recalled that the problem of approximating the ideal 

 transmission characteristics for each type of filter was solved only on 

 the assumption that the image impedance could be adjusted to a 

 nearly uniform value in the practical transmitting band. We can 



