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THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Then R{p) = R*ip), I(p) = I*{p), and both are real and symmetric. 

 If k is any vector, 



(Z(p)k, k) = (R(p)k, k) + i{I{p)k, k), 



and the seljf-ad joint property of R and / imply that each scalar product 

 on the right is real. Therefore 



(1) 



Re{Z(p)k, k) = {R(p)k, k), 



Im{Z{p)k, k) = (I(p)k, k). 

 We note that 



2il(p) = Z(p) - Z*(p) = Z*(p) - Zip) = -2il(p) 



so that, in particular, /(tco) is an odd function of co. 



4.31 Lemma: Let *S be a given real, constant, symmetric, and positive 

 definite matrix. Then there exists a unique number a > such that 

 (i) The matrix 



R(icci) — aS 



is semidefinite for every a;, 

 (ii) For some coo > 0, possibly + «> , 



Riiuo) — aS 



is singular. 

 Proof: We first show how the number a would be calculated, and then 

 reduce the claims of the lemma to a well-known and basic theorem in 



rz 



rz 



VI W Ni 



SCHEMATIC OF OPERATION ID 



W N, 



SCHEMATIC OF OPERATION AD 



Fig. 4 — Schematic of operation ID, left and AD, right. 



