SPECTRA OF QUANTIZED SIGNALS 457 



the usual case, the transducer ahead of the quantizer is of the "caftipf^ssing" 

 type in which the loss increases as the signal increases. If the full load sig- 

 nal just covers all the linear quantizing steps, a weak signal gets a bigger 

 share of the steps than it would if the transducer were hnear. The trans- 

 ducer after the quantizer must be of the "expanding" type which gives de- 

 creased loss to the large signals to make the overall combination linear. 



On the basis of the theory so far discussed, we can say that the error spec- 

 trum out of the linear quantizer is virtually the same whether or not the sig- 

 nal input is compressed. The operation of the expandor then magnifies the 

 errors produced when the signal is large. When weak signals are applied, 

 the mean square error is given by Eq. (1.1), as before, but when the signal 

 is increased an increment in noise occurs. The mean square value of noise 

 voltage under load may be computed from the probability density of the 

 signal values and the output-vs-input characteristic of the expandor, or its 

 inverse, the compressor. A first order approximation, valid when the steps 

 are not too far apart, replaces (1.1) by: 



12 h 



Qi 



pi{Ei) dEi , . 



where Qi and Q2 are the minimum and maximum values of the input signal 

 voltage El , pi (£1) is the probability density function of the input voltage, 

 and F'{Ei) is the slope of F{Ei), the compression characteristic. 



Some experimental results obtained with a laboratory model of a quan- 

 tizer are given in Figs. 6-9. Figs. 6-7 show measurements on the third 

 harmonic associated with 6-digit quantizing. As mentioned before, the 

 amplitude of any one harmonic oscillates with load. The calculated curves 

 shown were obtained by straightforward Fourier analysis. In the measure- 

 ments it was convenient to spot only the successive nulls and peaks. 



In Fig. 6 the bias was set to correspond to the stair-case curve of Fig. 1, 

 while in Fig. 7 the origin is moved to the point (£o/2, -Eo/2), i.e., to the mid- 

 dle of a riser instead of a tread. The peaks of ratio of harmonic to funda- 

 mental decrease steadily as the amphtude of the signal is increased to full 

 load, which is just opposite to the usual behavior of a communication sys- 

 tem. It is difficult to extrapolate experience with other systems to specify 

 quality in terms of this type of harmonic distortion. 



Figure 8 shows measurements of the total distortion power falling in the 

 signal band when the signal is itself a flat band of thermal noise. The 

 technique of making such measurements has been described in earlier ar- 

 ticles.^ -^"^ Measurements are shown for quantizing with both equal and 

 tapered steps. The particular taper used is indicated by the expandor 

 characteristic of Fig. 9. The compression curve is found by interchanging 



