Suppose we have a signal as x(t) = exp(i(omega*t + phi)), after a simple calculation as y(t) = 1/x(t). We can clearly see that there is a phase shift of Pi=180deg. However, using directly calculation as y(t) = 1/x(t) = exp(-i(omega*t + phi)), the phase shift shall be:
delta phi = angle(x(t)*conj(y(t))) = angle(exp(i(omega*t + phi)*exp(i(omega*t + phi)))) = 2*omega*t + 2*phi, which does not equal to Pi.
This problem occur for a general combination of calculations for a signal. Suppose the calculations can be treated as an operator upon the signal function as L(f). Then after these operation, what phase shift will we get?
phase shift = angle(x(t)*conj(y(t))) = angle(x(t)*conj(L(x(t)))) = ?
Where conj() means to get the conjugate of the a complex value.
Argument of y(t) is equal -arg x(t)=arg y*(t). Therefore delta phi=0. The same result gives angle(x(t)*conj(y(t))) = 0 if calculated correctly.
Victor Palamodov
We have xy=1, hence argy=-argx /= arg(-x). Therefore argconj(y)=--argy=argx and angle(x(t),conj(y(t))) = argx-argconj(y)=0.
Victor Palamodov Yes, arg(conj(y)) = -arg(y) = arg(x), but what we need to get is the phase shift between x(t) and y(t). Here the conjecture applied on y(t) is just for the benefit to calculate phase shift during coding as:
delta phi = arg(x) - arg(y) = angle(x(t)*conj(y(t)))
Delta phi does not mean the phase shift between x(x) and conj(y(t)) which is 0.
Therefore, for y=1/x, as your derivation. The correct phase shift between x(t) and y(t) shall be:
arg(x) - arg(y) = arg(x) - (-arg(x)) = 2arg(x) = angle(x(t)*conj(y(t)))
And there is a correction to my first commend, it should be: -x(t) = exp(-i Pi)x(t) = exp(i(omega*t + phi - Pi)).
Well, in your grafical example x and y are posirive, hence argx=argy=0 and \phi=0.
Apparently we need to distinguish "argument" of a complex valued function and that of periodic function.
V.P.
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