BAND WIDTH AND TRANSMISSION PERFORMANCE 595 



A direct sampling process which avoids shifting the band may therefore be 

 preferred. A useful theorem for uniformly spaced samples is that the 

 minimum sampling frequency is not in general twice the highest frequency 

 in the band but is given by the formula: 



/. = 2.r(i + ^J, (1) 



where: 



fr — minimum sampling frequency 



11' = width of band 



/> = highest frequency in band 



m = largest integer not exceeding /2/l'r 



The value of k in (1) varies between zero and unity. When the band is 

 located between adjacent multiples of W, we have ^ = and it follows that 

 fr = 2TF no matter how high the frequency range of the signal may be. As 

 k increases from zero to unity the sampling rate increases from 2IF to 2W 



(1 + -). The curve of minimum sampling rate versus the highest frequency 



in a band of constant width thus becomes a series of sawteeth of successively 

 decreasing height as shown in Fig. 37. The highest samplmg rate is re- 

 quired when m = I and k approaches unity. This is the case of a signal 

 band lying between W — A/ and 2 IF — A/ with A/ small. The sampling 

 rate needed is 2(2IF — A/) which approaches the value 4TF as A/ approaches 

 zero. Actually when A/ = 0, we change to the case oi m = 2, k = 0, and 

 fr = 2W. The next maxhnum on the curve is 3 IF, which is approached when 

 fj nears 3W. The successive maxima decrease toward the limit 2TF as fi 

 increases. The sampling theorem contained in Eq. (1) may be verified 

 from steady state modulation theory by noting that the first order sidebands 

 on harmonics of 2W do not overlap the signal when the equation is satisfied. 



