3.2 Frequency-Domain Techniques for Small-Signal Distortion
3.2.1 Volterra Series Model of Weakly Nonlinear Systems
In the following sections we will discuss two powerful analysis techniques that are of paramount importance to intermodulation problems:power series analysisand Volterra series analysis.Their merit comes from the fact that they can be conceived as a direct extension of the widely known linear techniques. They have a precise mathematical foundation, provide closed-form solutions for nonlinear system responses, and can be directly used in frequency-domain. These properties are consequences of the restriction imposed on the system’s nonlinearities: they must be approximated by a power series (a polynomial) if they are memoryless, or by a generalization of this (a Volterra series) if they are dynamic.
3.2.1.1 Memoryless Systems’ Representation
Like any other model, or scientific theory, power series is a representation of nature, in the sense that it cannot be said to be true or false, but only that it may be simply useful or not. A power series is a useful model for, at least, two different orders of reasons.
In the first place, a power series is a simple mathematical representation that has the benefit of allowing the direct response computation of a nonlinear device, circuit or system, in the frequency-domain. That is, contrary to all other natural
‘‘time-domain’’ models, we need not convert our frequency-domain excitations to their time-domain representation (usually by an appropriate inverse Fourier transform), calculate the model response, and then go back to the frequency- domain. With a power series, we simply have to make multiple convolutions of signals’ spectra. In fact, since a power series is nothing more than the addition of several time-domain product terms, one can directly compute them by the spectral addition of the correspondent frequency-domain convolutions.
For the system defined by yO(t) ≡ S[xI(t)], if in a limited range of xI(t) amplitude, yO(t) can be approximated by
yO(t)≈a1xI(t)+a2xI(t)2+a3xI(t)3+. . . (3.16) then, in the frequency-domain,
YO()≈a1Xi()+a2Xi() *Xi()+a3Xi() *Xi() *Xi() +. . . (3.17) Because most analog, RF and microwave circuit designers generally deal with signals represented in the frequency-domain as a sum of a small number of discrete tones—and repeated convolutions of those signals are very easily calculated—this property of a power series model becomes a very attractive advantage.
In the second place, there is a rigorous mathematical foundation that provides certain power series models with two other important advantages. If we restrict our power series to be a Taylor series expansion around a predetermined quiescent point (usually the dc bias point), we immediately gain a systematic parameter extraction procedureand model consistency.
The former refers to the fact that each of the model coefficients can be easily extracted from the nth-order device’s derivatives:
a1≡dS(xI) dxI |xI=XI
; a2≡1 2
d2S(xI) dx2I |xI=XI
; . . . ;an≡ 1 n!
dnS(xI) dxnI |xI=XI
(3.18) whereXIis the referred quiescent point.
The other Taylor series intrinsic property we mentioned is consistency. This means that, even though our power series model was derived to predict moderate signal level nonlinear effects, it inherently represents the device’s small-signal behav- ior. In electronic device terms, this corresponds to saying the model is able to accurately predict the circuits’ weakly nonlinear behavior, while it nicely converges to the small-signal [Y], [Z], [S], etc., parameters, if input excitation level is decreased.
This is a consequence of the fact that the Taylor series representation ofS[.] around the bias point (XI, YO), with an input signal xi(t) ≡xI(t) − XI (defined as the dynamic deviation of the control variablexI(t) from its quiescent valueXI), is
yo(t)≡yO(t) −YO=dS(xI) dxI |xI=XI
[xI(t)−XI]+. . . (3.19) + 1
n!
dnS[xI] dxnI |xI=XI
[xI(t)−XI]n+. . . or, in our power series model form,
yo(t)=a1xi(t) +a2xi(t)2+a3xi(t)3+. . .+anxi(t)n+. . . (3.20) Thus, if xi(t) is very small [xI(t) tends to XI], the higher nth-order terms rapidly become negligible compared to a1xi(t), and the model automatically
behaves as a linear one. In this sense, the Taylor series is what one could ever think as the simplest nonlinear extension of a linear memoryless, or algebraic, model.
By the way, this explanation also gives an insight onto the Taylor series model validity. It gets useless (or, in other words, hopelessly inaccurate) whenever the device excitation is so hard that other higher order terms we have not initially considered become important.
This rather small validity domain, which restricts power series analysis to small- signal nonlinear distortion studies (or weak nonlinearities), is one of its two major disadvantages. The other is the absence of memory.
Although it is not possible to represent a general dynamic system by a power series model, this kind of representation can still be used in cases where the system can be described by several noninteracting subsystems, and where the nonlinearities are memoryless. An illustrative example is depicted in Figure 3.3.
Indeed, if the input and output subsystems are both linear, defined by yi(t)
= Si[xi(t)], yo(t) = So[xo(t)], and characterized by frequency-domain transfer functions Hi(), Ho(), while the inner one is a memoryless nonlinear system represented by a power series like (3.20), the output to any frequency-domain excitation can be easily computed using the simple relations of (3.17). For example, ifxi(t) is given by a sum ofQ complex exponentials,
xi(t)= ∑Q
q=1
Xiqejqt (3.21)
the linear output can be calculated by yo1(t) = ∑Q
q=1
Ho(q)a1Hi(q)Xiqejqt (3.22) the second-order components’ response by
yo2(t)= ∑Q
q1=1 ∑Q
q2=1
Ho(q1+q2)a2Hi(q1)Hi(q2)Xiq1Xiq2ej冠q1+q2冡t (3.23)
Figure 3.3 Power series model system’s representation.
the third-order ones by yo3(t) = ∑Q
q1=1 ∑Q
q2=1 ∑Q
q3=1
Ho(q1+q2+q3) (3.24)
⭈a3Hi(q1)Hi(q2)Hi(q3)Xiq1Xiq2Xiq3ej冠q1+q2+q3冡t and so on.
Unfortunately, if the blocks interact with each other, or the system cannot be described by the simple cascade connection of Figure 3.3, as is the situation presented in Figure 3.4, then the straightforward calculation just performed is no longer possible, and the analysis demands for the true nonlinear dynamic representa- tion of Volterra series.
3.2.1.2 Dynamic Systems’ Representation
The main difference between power series and Volterra series models is the ability of the latter to represent true nonlinear dynamic systems. A Volterra series is, in fact, nothing more than a Taylor series with memory. Hence, it can also be interpreted as the extension of linear, or first-order, dynamic systems. And so, to introduce its foundations, we will begin by recalling the derivation of the convolutive response of a time-invariant linear dynamic system. The explanation follows the one pre- sented by [4].
Let us begin by considering again our general single-input single-output system S[.] whose signal responseyo(t) to an input signalxi(t) can be expressed asyo(t)
≡S[xi(t)]. As is seen in Figure 3.5,xi(t) may be approximated by an appropriate
Figure 3.4 Example of a nonlinear dynamic system for which the power series model is no longer valid.
Figure 3.5 Ladder function approximation of the system’s excitation.
ladder function in the domain−T<t<Tcomposed by a sum of 2Q+1 rectangular pulses, p(t), of⌬duration and 1/⌬ amplitude:
xi(t) ≈ ∑Q
q=−Q
xi(q⌬)p(t−q⌬) ⌬ (3.25)
AssumingS(t,q⌬) is the response ofS[.] to the rectangular pulse located at q⌬,
S(t,q⌬)≡S[p(t−q⌬)] (3.26) yo(t) may be also approximated by
yo(t)≈S冤q=−Q∑Q xi(q⌬)p(t−q⌬)⌬冥 (3.27)
If S[.] were a linear dynamic system, superposition would apply, and thus, yo(t)≈ ∑Q
q=−Q
S[xi(q⌬)p(t −q⌬)⌬] = ∑Q
q=−Q
xi(q⌬)S(t,q⌬)⌬
(3.28) In a time-invariant systemS(t,q⌬) =S(t−q⌬), andyo(t) can be approxi- mated by
yo(t) ≈ ∑Q
q=−Qxi(q⌬)S(t−q⌬)⌬ (3.29) as is depicted in Figure 3.6.
The approximations assumed for bothxi(t), (3.25), andyo(t), (3.29), improve their accuracy when the pulses’ duration, ⌬, is reduced. In the limit where ⌬
tends to zero, (3.25) tends to an infinite sum of Dirac delta functions, ␦(t −), and xi(t) can be represented in its whole domain ]−∞,+∞[ by
xi(t) = 冕∞
−∞
xi()␦(t −) d (3.30)
If h(t−) is now defined as the response ofS[.] to the Dirac impulse located at , (3.29) turns into the convolution integral
yo(t)= 冕∞
−∞
xi()h(t −) d = 冕∞
−∞
h()xi(t−)d (3.31)
Equation (3.31) is the usual time-domain response representation of a general linear dynamic time-invariant system.
Any direct attempt of extending that theory to nonlinear systems would fail, because superposition assumed in (3.28) no longer applies.
A convenient way of circumventing that difficulty consists of expanding a wide class of nonlinearities into Taylor series around some quiescent point (null
Figure 3.6 Linear system’s output approximation as the response of the ladder function.
excitation). In that way, the response ofS[.] to the generic rectangular pulse centered at q1⌬may be expressed as
S[xi(q1⌬)p(t −q1⌬)⌬] =S0+S1(t −q1⌬)xi1⌬
+S2(t−q1⌬)x2i1⌬2 (3.32) +S3(t−q1⌬)x3i1⌬3+. . .
The response to a sum of two rectangular pulses xi(q1⌬)p(t −q1⌬)⌬ + xi(q2⌬)p(t −q2⌬)⌬, would then be
S[xi(q1⌬)p(t−q1⌬)⌬+xi(q2⌬)p(t−q2⌬)⌬]
=S0+S1(t −q1⌬)xi1⌬+S1(t−q2⌬)xi2⌬
+S2(t−q1⌬,t −q1⌬)x2i1⌬2
+2S2(t−q1⌬, t−q2⌬)xi1xi2⌬2 (3.33) +S2(t−q2⌬,t −q2⌬)x2i2⌬2
+S3(t−q1⌬,t −q1⌬, t−q1⌬)x3i1⌬3+. . . +S3(t−q2⌬,t −q2⌬, t−q2⌬)x3i2⌬3+. . .
In general, when the input is a sum of 2Q+1 pulses, yo(t) may be given by
yo(t)= ∑Q
q1=−Q
S1(t−q1⌬)xi1⌬
+ ∑Q
q1=−Q ∑Q
q2=−Q
S2(t−q1⌬,t−q2⌬)xi1xi2⌬2 (3.34) + ∑Q
q1=−Q ∑Q
q2=−Q ∑Q
q3=−Q
S3(t−q1⌬,t−q2⌬,t−q3⌬)xi1xi2xi3⌬3+. . .
Again, in the limit where⌬tends to zero, and the rectangular pulses tend to Dirac impulses, we have
yo(t)=冕∞
−∞
h1(t−1)xi(1)d1
+冕∞
−∞ 冕∞
−∞
h2(t−1,t−2)xi(1)xi(2)d1d2 (3.35)
+冕∞
−∞ 冕∞
−∞ 冕∞
−∞
h3(t−1,t−2,t−3)xi(1)xi(2)xi(3)d1d2d3+. . .
which, rewritten in the compact form, yo(t)= ∑∞
n=1
yon(t) (3.36a)
with
yon(t)≡ 冕∞
−∞
. . . 冕∞
−∞
hn(1, . . . ,n)xi(t−1) . . .xi(t −n) d1. . . dn
(3.36b) results in the wanted Volterra series expansion of the nonlinear dynamic system’s response,yo(t), to a general inputxi(t).
As an extension to the linear case, hn(1, . . . ,n) is called the nth-order impulse response, ornth-order Volterra kernel.
The above deduction was made assuming the system, its input, and output were represented in their natural domain: time-domain. However, we usually have xi(t) described in the frequency-domain by some spectral representation Xi() and would like to directly computeyo(t) in the same domain [i.e.,Yo()]. To see how we can do that using the Volterra series model, we will assume that the input can be expressed as a finite sum of sinusoidal functions, or elementary complex exponentials:
xi(t)=1 2 ∑Q
q=−Q
Xiqejqt (3.37)
in which no dc component is expected (i.e.,q≠0). (The dc term is really already embedded in the Taylor series expansion of the nonlinearity, as its quiescent point.)
Substituting that input into the genericnth orderS[.] response, (3.36), we can obtain, after some algebraic manipulation,
yon(t) = 1 2n ∑Q
q1=−Q . . . ∑Q
qn=−QXiq1. . . Xiqnej冠q1+. . .+qn冡t (3.38)
⭈ 冕∞
−∞
. . . 冕∞
−∞
hn(1, . . . ,n)e−j冠q11+. . .+qnn冡d
1. . . dn
The integral part of (3.38) is a generalization of the conventional Fourier transform, known as the multidimensional Fourier transform:
Hn(q1, . . . ,qn)≡ 冕∞
−∞
. . . 冕∞
−∞
hn(1, . . . ,n)e−j冠q11+. . .+qnn冡d
1. . . dn
(3.39) and the Hn(q1, . . . ,qn) is called the nth-order nonlinear transfer function (NLTF). In that way, the system’s response yo(t) can be finally given by
yo(t)= ∑∞
n=1
1 2n ∑Q
q1=−Q. . . ∑Q
qn=−QXiq1. . . XiqnHn(q1, . . . ,qn)ej冠q1+. . .+qn冡t (3.40) from which it is possible to derive the variousYo() components. Therefore, (3.40) constitutes the basis for the frequency-domain analysis of the steady-state response of all mildly nonlinear systems.
Using this Volterra series formalism in real analog and RF circuits resumes to determining each of the nonlinear transfer functions. For that, theharmonic input method and thenonlinear currents method were proposed, which are the object of subsequent sections.