594 



BELL SYSTEM TECHNICAL JOURNAL 



For large values of x, the following asymptotic formula has been derived by 

 Mr. S. O. Rice by use of the Mellin-Barnes contour integral representation 

 and the method of steepest descents: 



TT xA, 



,(.v) ~ In .V + In 32 + 7-2 



( 



■-hi+ 



+ 



1 



2m — 



-,) 



-(-)' 



Tl/l^-^°^0-^-+i) 



+ 



(90) 



7 = Euler's constant = 0.5772 . . . 



As X approaches zero, S/I approaches 2Pj'Q, which is to be expected because 

 the frequency deviation of the unwanted carrier is represented by a pair of 

 first order sidebands P/Q times as great as those on the wanted carrier. 

 Averaging over the random carrier phase difference brings in a factor \/2, 

 and averaging over all frame phases accounts for another. 



4W 



as 



m — 

 ag 3W 



'^Q 



Z< 



HCD 



Q-u. 2W 

 < 



l^ w 



2Z 



W 2W 3W 4W 5W 6W 7W 



HIGHEST SIGNAL FREQUENCY IN TERMS OF BAND WIDTH, W 



Fig. 37 — Minimum sampling frequency for band of width W. 



APPENDIX V 



Sampling a band of frequencies displaced from zero 



It is often necessary to transmit a signal band which does not extend all 

 the way down to zero frequency. For example, a group of channels in FDM 

 may be based on a set of carrier frequencies remote from zero. When we 

 consider the application of pulse methods to transmit such a signal, the ques- 

 tion of what sampling rate is needed immediately arises. A band extending 

 from /i to/i + 11' could of course be translated to the range to W by stand- 

 ard modulation techniques, sampled at a rate 211', and restored to the 

 original range by an inverse translation at the receiver. The frequency 

 shifting apparatus required includes modulators, carrier generators, band 

 separating filters, and possibly amplifiers to make up the inevitable losses. 



