456 BELL SYSTEM TECHNICAL JOURNAL 



to/s/2, the actual received noise power is multiplied by k. Then the signal- 

 to-noise ratio in db for a full load test tone is 



D = 10 1ogio/-db (1.4) 



In practical applications the value of k is about 3/4 which gives the con- 

 venient rule: 



D = 201ogior + 3db (1.5) 



In other words, we add 3 db to the ratio expressed in db of peak-to-peak 

 quantizing range to the range occupied by one step. For various numbers 

 of binary digits the values of D are: 



Table I 



From Table I we can make a quick estimate of the number of digits re- 

 quired for a particular signal transmission system provided that we have 

 some idea of the required signal-to-noise ratio for a full load test tone. The 

 latter ratio may be expressed in terms of the full load test tone which the 

 system is required to handle and the maximum permissible unweighted 

 noise power at the same level point. Since quantizing noise is uniformly 

 distributed throughout the signal band, its interfering effect on speech or 

 other program material is probably similar to that of thermal noise with the 

 same mean power. Requirements given in terms of noise meter readings 

 must be corrected by the proper weighting factor before applying the table. 

 If the signal transmitted is itself a multiplex signal with channels allotted 

 on a frequency division basis, the noise power falling in each channel is the 

 same fraction of the total noise power as the band width occupied by the 

 signal is of the total band width of the system. 



We have thus far considered only the case in which the quantized steps are 

 equal. In actual systems designed for transmission of speech it is found ad- 

 vantageous to taper the steps in such a way that finer divisions are available 

 for weak signals. For a given number of total steps this means that coarser 

 quantization applies near the peaks of large signals, but the larger absolute 

 errors are tolerable here because they are small relative to the bigger signal 

 values. Tapered quantizing is equivalent to inserting complementary non- 

 linear transducers in the signal branch before and after the quantizer. In 



