762 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



There will thus be pulse or signal echoes of amplitude a/2 at the time 

 T before and after the main pulse as illustrated in Fig. 26. 



With a cosine variation in the attenuation rather than in the amplitude 

 characteristic, the modified amplitude characteristic becomes 



A (a;) = ^o(co) e"*^"' "^ (7.03) 



and the modified impulse characteristic 



p(t) = ^ r ToMe'"'°"'V''' dw. (7.04) 



27r J- CO 



The expansion corresponding to (6.04) is in this case: 



a cos cor t / \ \ t / \ /' ^^t t — ia)T\ 



e = Io{a) + Ii{a){e + e ) 



+ /2(a) (e"'"^ + e~''"0 

 + /3(a)(e''"^ + e-^'"0,+ ••• (7.05) 

 where h , h • ■ ■ are Bessel functions for imaginary arguments in their 

 usual notation. 



The resultant modified impulse characteristic in this case becomes 



Pit) = /o(a)Po(0 + h(a)[Poit + r) + Po(t - r)] 



+ h(a)[Po(t + 2t) + Po(t - 2r)] (7.06) 



+ h(a)[Po(t + 3r) + Poit - St)] + • • • 



which can be interpreted in a similar way as discussed for (6.05). For 

 small values oi a, h (a) = 1, h (a) = a/2 and the remaining terms 

 in (7.06) negligible, so that (7.02) is obtained. 



As discussed in Section 1, when the amplitude characteristic is modi- 

 fied in accordance with (7.01), the resultant modification in the phase 

 characteristic is in accordance with (1.13) 



_i r sin wr , . 



4/1 = 2 tan -— — ; . (^.07) 



1 + r'' cos cor 



The modified transmission-frequency characteristic is in this case 



r(iw) = ro(2co)(l + a COS coT)e~''^S (7.08) 



which can be transformed into 



r(zco) = TS^) — 1-, (1 + re-n\ 

 \ -\- r- 



= Toiic) — ^, (1 + 2,-e-'"^ + r'e-'n. 

 1 -f r 



(7.09) 



