652 BELL SYSTEM TECHNICAL JOURNAL 



Theorem 22: The rate for a white noise source of power Q and band W\ 

 relative to an R.M.S. measure of fideUty is 



where N is the allowed mean square error between original and recovered 

 messages. 



More generally with any message source we can obtain inequalities bound- 

 ing the rate relative to a mean square error criterion. 



Theorem 23: The rate for any source of band Wi is bounded by 



W',log|<J?<irilog| 



where Q is the average power of the source, Qi its entropy power and TV the 

 allowed mean square error. 



The lower bound follows from the fact that the max Hy{x) for a given 

 {x — y) = N occurs in the white noise case. The upper bound results if we 

 place the points (used in the proof of Theorem 21) not in the best way but 

 at random in a sphere of radius ■\/Q — N. 



Acknowledgments 



The writer is indebted to his colleagues at the Laboratories, particularly 

 to Dr. H. W. Bode, Dr. J. R. Pierce, Dr. B. McMillan, and Dr. B. M. Oliver 

 for many helpful suggestions and criticisms during the course of this work. 

 Credit should also be given to Professor N. Wiener, whose elegant solution 

 of the problems of filtering and prediction of stationary ensembles has con- 

 siderably influenced the writer's thinking in this field. 



APPENDIX 5 



Let 5i be any measurable subset of the g ensemble, and ^2 the subset of 

 the/ ensemble which gives Si under the operation T. Then 



Si = TS2. 



Let H^ be the operator which shifts all functions in a set by the time X. 

 Then 



H% = H^'TSi = TH^'Si 



since T is invariant and therefore commutes with H . Hence if m[S\ is the 

 probability measure of the set S 



m[H%] = mlTH^Si] = mlH^Si] 



= miSi] = m[Si] 



