In the field of FIR filter analysis, it is known that the frequency response of a FIR filter, as any complex number, can be expressed as
H(f) = |H(f)|.exp(j.A(H(f))),
where |.| is the modulus (a continuous function), and A(.) is the argument (a piecewise continuous function). On the other hand, it is known that this expression can be rewritten as
H(f) = H_r(f).exp(j.\phi(f)),
where H_r is a real continuous function, and \phi(f) is (at least) a C1 function. Which theorem is involved in this equivalence?
Dear Carlos,
thanks for your answer.
My question may be not clear, but I think the problem does not involve trivial trigonometry only. Let consider a more general expression of the problem:
Let f be a function R -> C.
It can be written f = |f| . exp(j.Arg(f))
where Arg(f) is piecewise continuous.
How to show that it can also be expressed as:
f = g . exp(j.p)
where g and p are two continuous functions?
In particular, what happends when real(f) = imag(f) = 0?
Vincent
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