IEEE Microwave Magazine - June 2015 - 105

the second degree term in (1) does not
yield a spectral term at the fundamental freqeuency that would be needed to
model any observed gain compression.
In its simplest formulation, admittedly, this has now raised an issue that
still promotes much debate in the lofty
ivory towers of behavioral modelists and
wannabe predistortion practitioners.
Do I have to accept that since the even
degree terms in the power series cannot
generate distortion products at the fundamental carrier frequency, my assertion
that Figure 1 is truly a real measured PA
response means that I am not allowed to
model it with a second-degree polynomial? It should also be noted that the second-degree polynomial fits (2) perfectly
to whatever level of precision I care to
check it. Are even degree terms just "off
limits" when modeling the amplitude
response of an amplifier?
There appears to be two schools of
thought on this issue; we have already
met a member of the Oddite faction,
but we can define the other faction as
"Evenites." Let us dig a little deeper.
In fact, the resolution is quite simple
in general terms. The basic polynomial power series deals with the entire
signal, represented by o in (t) . The plot
of Figure  1 is merely addressing the
amplitude of the signal, not the whole
signal. So, for example, we cannot simply plug in a two-carrier signal into (4)
and (5), as we did using the original
power series (1), because in (5), the "v"s
refer to envelope amplitude only, and
we only have measured information
about the behavior of the amplitude
of the envelopes of the input and output signals. To handle this, we need to
reformulate the expression for a twocarrier signal. This involves finding an
expression having the form
Vsig = A ^Xt h . sin " ~t + } ^X h,,

(6)

where A ^X h is an amplitude function that is only allowed to have positive values, since any change of sign
would mean that the phase of the signal was being affected, and we have
stipulated all along that our amplifier is
completely "blind" to the signal phase
variations. Amplitude is not the same
as magnitude! A key mistake, therefore,

June 2015

which I have to say I have seen made in
loftier circles than I care to mention, is to
assume the following for a two-carrier
signal that complies with the format
defined in (6):
o ^ t h = V sin ^ Xt h sin ^ ~t h .
)

(6)

At first sight, this may seem like
the logical way of representing a twocarrier signal, as a carrier at angular
frequency ~ whose amplitude is modulated sinusoidally at baseband frequency X. Well, clearly it does; even the
most reluctant trigonometrist would
recognize that (6) can be expanded
into the sum of two carriers having
equal amplitude of V/2 at frequencies
~ +/- X . But it does not comply with
the requirements of (4) and (5), since the
amplitude term, V sin Xt, still contains
phase information. In a two-carrier
signal, the phase of the carrier inverts
every amplitude cycle, and we need
some further mathematical trickery to
relocate this characteristic as a phasemodulated element in the expression
for the RF carrier. There are numerous
ways of doing this, but for the present
purposes, I use the following:
V ^ t h = V sin Xt.Sq ^Xt h . sin
^ ~t + ^ r ) Sw ^Xt hhh,

(7)

where Sq ^Xt h is a square-wave function switching between -1 and +1 with
a period of 2r/X, and Sw ^Xt h switches
at the same modulation frequency,
between zero and unity.
So the amplitude term now represents a "positive-definite" full-wave
rectified cosine wave, and the carrier
term is an RF carrier with a 180° phase
shift every half cycle of the modulation
frequency. We can now substitute the
amplitude function in (5) to obtain the
output voltage variation as defined in
the power series (1)
V0 ^ t h = " a 1 V sin ^Xt h .Sq ^Xt h
+ a 2 V 2 sin 2 ^Xt h .Sq 2 ^Xt h, .
sin $ ~t + r .Sq ^Xt h. .
2

Noting that
Sq 2 ^Xt h = 1,
we can expand the second-degree term

Vo2 = 1 a 2 V 2 ^1 - cos 2Xt h
2

.Sq ^Xt hj
$ sin ~t cos ` r
2

+ cos ~t sin ` r .Sq ^Xt hj.,
2
and noting further that
cos ` r .Sq ^Xt hj = 0, and
2
sin ` r .Sq ^Xt hj = Sq ^Xt h,
2
then
Vo2 = 1 a 2 V 2 ^1 - cos 2Xt h
2
" Sq ^ Xt h cos ~t , .
So given that Sq ^Xt h expands into
a series of fundamental and odd harmonics in cos ~t , there are spectral
components at the IM3 sideband frequencies, ~ ! 3X, despite the squarelaw amplitude response. In a sense,
this is not all that surprising since the
amplitude characteristic still results in
a symmetrically compressed envelope
around the peak values. More disturbing, however, for the Oddites is the
observation that the IM3 sidebands
are proportional to the square of the
amplitude parameter V. So my original
gut feeling that such an amplifier
would have a poor IM3 characteristic
was correct. The IM3s will not back off
as fast as one would expect from the
Oddite Creed.
So who wins the set, Oddite or
Evenite? I have to declare a tie. On the
one hand, it is certainly true that IM3s
can only be generated by odd degrees
of nonlinearity, but this applies to the
entire signal as a function of time, not
a constituent property such as its amplitude as a function of time. Indeed, even
degrees of nonlinearity in the "magnitude" function can generate IM3s and
gain compression. As such, there is
nothing wrong with using even-degree
terms to model the measured AM-AM
responses shown in Figure 1. But the
Oddites can simply argue that the same
measured curve can be modeled equally
well using only odd-degree terms, even
though it requires several of them.
However, all of the above information still raises questions in the mind
of an entrenched rebel for conventional

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