648 BELL SYSTEM TECHNICAL JOURNAL 



The following are simple examples of evaluation functions: 

 1. R.M.S. Criterion. 



V = (x{t) - y{t)f 



In this very commonly used criterion of fidelity the distance function 

 p{x, y) is (apart from a constant factor) the square of the ordinary 

 euclidean distance between the points x and y in the associated function 

 space. 



p{x,y) = fjjw) -yiOfdt 



2. Frequency weighted R.M.S. criterion. More generally one can apply 

 different weights to the different frequency components before using an 

 R.M.S. measure of fidelity. This is equivalent to passing the difference 

 x{t) — y{l) through a shaping filter and then determining the average 

 power in the output. Thus let 



e{t) = x{t) - y{t) 



and 



fit) = f e{T)k{t - r) dt 



then 



1 r 



p{x, y) = fl M dt- 



3. Absolute error criterion. 



1 r^ . 



dt 



p(^, y) = ^l I ^(0 - ^(0 



4. The structure of the ear and brain determine implicitly an evaluation, or 



rather a number of evaluations, appropriate in the case of speech or music 

 transmission. There is, for example, an "intelligibility" criterion in 

 which p{x, y) is equal to the relative frequency of incorrectly interpreted 

 words when message x{l) is received as y{l). Although we cannot give 

 an explicit representation of p{x, y) in these cases it could, in principle, 

 be determined by sufficient experimentation. Some of its properties 

 follow from well-known experimental results in hearing, e.g., the ear is 

 relatively insensitive to phase and the sensitivity to amplitude and fre- 

 quency is roughly logarithmic. 



5. The discrete case can be considered as a specialization in which we have 



