4)The closeness betwen the frequency response approach and the transient response approach may be explained simply as follows. To approximate the step responses is essent,ially the same as matching the frequency responses with emphasis in the lower frequency range.Comments on "&-Stability of Linear Time-Varying Systems-Conditions Involving Noncausal Multipliers" ROLIAIYO
AI. DE SANTISThe purpose of this correspondence is to comment briefly on the assumption of the above-ment.ioned paper' that every linear system in the Hilbert. space L2[0, = ) can be represented by the sum of a causal plus an ant,icausal component,. Another relevant comment, regards the cont.ent,ion that the transfer function N ( s ) , described by (6.2), defines a bounded syst,em.To justify the first assumpt.ion, t.he aut.hors invoke, in good faith, a causality decomposition theorem from [8] of t,heir paper. The att.ention is called, however, to the fact t.hat. this theorem has been recently recognized not t.o be correct., and examples of linear systems that cannot be decomposed into the sum of a causal plus an ant.icausal system have been given [ l ] , [3]. Reference [ l ] also gives necessary and sufficient conditions for a system to admit. a causalit.5decomposition. I n particular, these conditions are formulated in terms of integral t.ransformators of the type studied in [2], and are trivially satisfied for the class of linear convolution operat,ors considered by Zames and Falb. On t.he other hand, t.he quest.ion about whether a similar pr0pert.y might be enjoyed by more genera.1 convolution operators has no clear answer at. t.his time.In regard to the transfer function M ( s ) , it is observed that. M ( s ) hm poles on the right-hand side of the complex plane. 4 s a consequence, one can see that, in the contest of the space L?(m, Q ), either a noncausal hounded or a causal unbounded Pyat,em can be associated to M ( s ) , and therefore this transfer function ma)-indeed correspond to a bounded system. The sit.uation appears to be somewhat different, horvever, in the contest considered in the paper, namely, that. of t.he L? [O, 03 ) space. Here X ( s ) unambiguously defines a causal and unbounded system.We are unable t,o of'fer our views regarding the first comment. of De Santis [ l ] , since the necessary and sufficient. conditions for the additive decomposit.ion of a linear operator in L? [O, m ), as claimed to have been obtained by him, appear to have been derived very recently and are cont,ained in an unpublished report [ 11 to n-hich we do not. have access at present. In any case, it is interesting to note that. our r e d t s 1 do not necessitate any changes in view of these recent developments [ l ] , since the operators 31 for which we require such a decomposition (Section I : , Theorem 3) are elements of a Banach algebra CB of bounded, linear, time-invariant convolution hlanuscripr received .lugujt 24, 1972. tute of Science, 13angalore. India, The authors are with the Department of Electrical Engineering, Indian Insti-operat.ors ...
