By Rayleigh's energy theorem,
(as
shown in §2.3.8). Therefore,
From this ``ratio error'' expression in the
frequency domain, we can
see that contributions to the error are smallest when
![$ \vert{\hat Y}(e^{j\omega})\vert>\vert Y(e^{j\omega})\vert$](http://www.dsprelated.com/josimages_new/sasp2/img1754.png)
. Therefore, LP tends to
overestimate
peaks. LP cannot make
![$ \vert{\hat Y}\vert$](http://www.dsprelated.com/josimages_new/sasp2/img1755.png)
arbitrarily large because
![$ A(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1756.png)
is
constrained to be monic and
minimum-phase. It can be shown that the
log-magnitude
frequency response of every minimum-phase monic
polynomial
![$ A(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1756.png)
is
zero-mean [
162]. Therefore, for each
peak overestimation, there must be an equal-area ``valley
underestimation'' (in a log-magnitude plot over the unit circle).
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