NAPI TR-li^-J^O 



V. CASE 2: THE ECKAET FILTER 



The previous section examined what might be termed a theoreti- 

 cally crude optimization scheme: the shaping filter was constrained 

 to be a noise pre-whitening filter only, and the optimization process 

 was performed only on bandwidth. In this section, the optimization is 

 more sophisticated, making use of signal spectral knowledge to deter- 

 mine the best form of shaping filter. The result is the well Eckart 

 filter, given by 



i^M 



To show that this is the optimum filter, equation 9 will be 

 maximized by a common procedure (8): assiune a general shaping filter 

 response given by 



r(f) = ro(f) + eY(f) C^^) 



where ro(f) is the optimum value of r(f) sought, Y(f) is an arbitrary 

 frequency function, and e is any scaler. Then the system output SKR 

 is, from equations 9, 13, and 20, ^ 



If To is truly the optimum filter response, then j- should be zero when 

 e is set equal to zero, i.e., 



is: 





where Do and No represent the denominator and numerator of equation 21 

 evaluated at e = 0. Further manipulation results in 



_ c«0 >■ 



18 



