SPECTRA OF QUANTIZED SIGNALS 467 



From the defining equation (2.9) we note that ^ is a small quantity when 

 more than a very few steps are used in the quantizer so that exponentials 

 with exponent containing the factor —1/k are very small except when the 

 factor is multiplied by a number near zero. It will be seen that this can 

 only happen in the first series and then only when a approaches the value 

 unity. We recall that a lies in the range — 1 to +1 and it is apparent from 

 (2.25) that C{a) is an odd function of a. We thus need consider only posi- 

 tive values of a very slightly less than unity. Only the component of the 

 sin/? with positive exponent is then significant, and we write the very 

 accurate approximation for G{a): 



G{a) = r-5 2^ - exp — (2.26) 



ZTT^ n=l fl'^ k 



A typical curve of G(a) vs. a for a fixed value of k is shown in Fig. 1 1 . The 

 rapidity with which it falls away at the left of the point « = 1 is such that 

 the curve can only be plotted by greatly expanding the scale of a in this 

 region. The physical significance of the spike-shaped curve is that G{a) 

 is a measure of the correlation of the errors as a function of the correlation 

 of the applied signal. When there are many steps there is virtually no 

 correlation between errors in successive samples except when there is com- 

 plete correlation of successive signal values. 



Use of the approximation (2.26) enables us to derive a convenient formula 

 for the spectral density of the errors in a flat band input signal. Sub- 

 stituting (2.26) in (2.14) we obtain: 



«»« - i I i f -p [^ (^ - ¥)] - - ^^ (^-"^ 



The integrand is negligible except when s is near zero, and in this region we 

 may replace (sin z)/z by the first two terms of its power series expansion. 

 We then find 



2 



flo(T) - — 3 X) - / exp ( ^l"" ^ ) cos yz dz 

 ~ lir^y 2'whn^^''^\^n^l^^)' 



0n\y one set of calculations from the infinite series need be made since we 

 may define a function of one variable 



oc — 2/n2 



B{z) = Z ^ . (2.29) 



n=i rr 



Then _ 



