To find the form of the SMF, note that the ratio, Equation 7.2-7, is maximized when the equality holds, which is when

Hm(ju) = K [S(ju) exp(juxo)]* = K S*(ju) exp(-juxo) 7.2.11

The asterisk denotes taking the complex conjugate of the vector (e.g., if S = p + jq, then S* = p - jq).

Now if s(x) is an even function, i.e., s(x) = s(-x), then clearly S(ju) = S(u) is real and even, and S*(ju) = S(u). Thus, Hm(ju) = K S(u) exp(-juxo). It follows that for the even s(x), hm(x) = F-1{Hm(ju)}, and hm(x) = K s(x - xo). If s(x) is not even, then in general, hm(x) = K s[-(x - xo)]. That is, s(x) is reversed in x and shifted some xo (see Figure 7.2-1).

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