3.3 Frequency-Domain Techniques for Large-Signal Distortion
3.3.3 Nonlinear Model Representation—Spectral Balance
When formulating the harmonic equation, we faced the problem of deter- mining the spectrum representation of a certain nonlinear response [e.g., iNL[vO(t)] =iNL冤∑k
Vokejk0t冥 in (3.201)]. At that time, the adopted procedure was to convert the spectral representation of Vo into the time-domain, vO(t), compute iNL[vO(t)] in a time-sample by time-sample basis, and then return to the frequency-domain with the help of the appropriate Fourier transformation.
However, that time-domain evaluation of the nonlinearities may not be viable for, at least, two important reasons.
First of all because the signals may not be periodic, thus obviating the use of the DFT (or its fast computation algorithm, the FFT) as the tool to jump between time- and frequency-domains. This is so important in the nonlinear distortion field of problems that it deserves a separate treatment in a special section. For now, just imagine the simple case of trying to simulate a two-tone test where the two frequencies, 1and 2, have no common divider (i.e., there are no simultaneous nonzero integers k1andk2such thatk11+k22=0).
The second possibility appears whenever the nonlinearity may not be expressed by the cascade of a linear dynamic operator and an algebraic nonlinear function.
For example, if the circuit includes one or more electron devices for which quasi- static approximation does not apply. In this case,iNL[vO(t)] would have memory,
Frequency-DomainTechniquesforLarge-SignalDistortionAnalysis149
Figure 3.25 Summarizing flow chart of the harmonic-Newton algorithm.
meaning that the time-sample by time-sample calculation is no longer possible [the response at any time also depends on the pastvO(t)], and some kind of transient integration is needed. But, this is exactly what we want to obviate when using a steady-state simulation technique like HB.
To cope with these problems, we would like to have the capability of determin- ing the output spectrum directly from the input spectrum. And, this is, in general, not possible, since there are no mathematical tools for analytically solving a nonlinear differential equation subject to any combination of sinusoids. Obviously, the power series or Volterra series are two very special methods to handle that situation, in which the particular format adopted for representing the nonlinearities allowed the desired spectral manipulation. Indeed, the output spectral calculation could be easily done by transforming the time-domain additions and products into spectral additions and pseudoconvolutions, respectively. That is, ifx(t),X();y(t),Y();
z(t), Z() are the time-domain and frequency-domain representations of three signals such that
x(t) = ∑K
k=−K
Xkejkt;y(t)= ∑K
k=−K
Ykejkt; z(t)= ∑R
r=−R
Zkejrt (3.215) then, the spectrum of the addition
z(t) =a1x(t)+a2y(t) (3.216) will be
Z(r)=a1X(k)+a2Y(k);r=k= −K, . . . , 0, . . . , K
(3.217) and the spectrum of the product
z(t)=x(t)y(t) (3.218)
will be
Z(r) =X(k1)Y(k2);r=k1+k2= −2K, . . . , 0, . . . , 2K
(3.219a) or
Z() =X() * Y()=Y() *X() (3.219b)
whererspans through all possible linear combinations ofk1andk2, in a way that Z() = X() * Y() represents true spectral convolution if the various k are multiples of a fundamental0:k=k0, and another frequency transformation (herein called pseudoconvolution) otherwise. Again, from all mixing products, only the ones falling in the original frequency set are considered, in order to limit the spectral regrowth handled by the simulator. Therefore, it is assumed that many of theX冠k1冡and Y冠k2冡 are null, and anyr< −Korr>Kwill be discarded.
In this way, that pseudoconvolution corresponds to (2K + 1)2 complex entity products that may be put in matrix form as
冤ZZZ⯗⯗−KK0 冥=冤TxTx(2K+⯗111)1 …… TxTx(2K+1(2K⯗1)(2K++1) 1)冥 冤YYY⯗⯗−KK0 冥 (3.220a)
=冤TyTy(2K+⯗111)1 …… TyTy(2K+1(2K+⯗1)(2K+1) 1)冥 冤XXX⯗⯗−KK0 冥
or
Z=TxY =TyX (3.220b)
in whichTx andTy are the spectrum transform matrices of the signalsx(t) and y(t), respectively.
At this point, it should be noted that if x(t), y(t) and z(t) are periodic (k = k0), these transform matrices have a regular Toeplitz form such that Txij=X[(i−j)0]. The matrix-vector products of (3.220) can be implemented in a much more efficient way in time-domain, if the convolution properties of the FFT are used.7Indeed, these orderO[(2K +1)2] frequency-domain convolutions can be done with only two timesO[(2K+1) log2(2K+1)] for FFT inversion ofX() andY(),O[2K+1] forz(t)=x(t)y(t) multiplication andO[(2K+1)log2(2K+ 1)] for FFT transformation of z(t) to Z(), and we end up in the conventional mixed time-domain frequency-domain HB algorithm.
With that definition of the spectrum transform matrix, it is also possible to determine the spectrum mapping imposed by a time-domain division. Consider that the nonlinear function to be evaluated in the frequency-domain is
7. Strictly speaking, the use of the FFT algorithm requires that (2K+1) be a multiple of two. If it is not, the array should be padded with the necessary number of zeros before processing.
z(t)= y(t)
x(t) (3.221)
To determine Z() from X() and Y(), we should first recognize that if (3.221) holds, theny(t)=x(t)z(t) orY()=X() *Z() orY=Tx Z, and thus
Z =Tx−1Y (3.222)
whereTx−1is the inverse matrix of Tx.
In the same manner as the matrix-vector productTx Yrepresented a frequency- domain convolution in the periodic excitation case, now Tx−1Y stands for its inverse operation, or deconvolution.
It should be clear by now that since any algebraic function can be approximately expressed as a suitable combination of the four arithmetic operations (addition, subtraction, multiplication, and division), any memoryless nonlinearity can be directly evaluated in frequency-domain. For that, the nonlinearity should be approx- imated by a convenient power series or rational function (i.e., a ratio between two polynomials).8In this way, we can solve the HB equation in the frequency-domain obviating the need for the Fourier transformations. That special version of HB is usually known asfrequency-domain harmonic-balanceor simplyspectral-balance (SB), to distinguish it from the conventional mixed mode HB.
To solve the spectral-balance equation with the Newton-Raphson method, like in the previous harmonic-Newton, we need to compute the Jacobian matrix. That is straightforward if the spectrum transform matrix is again used for the time- domain products and divisions. If, for example,x(t),y(t), andz(t) are dependent on a signals(t), and, for example, the derivative ofX() with respect to thenth component ofS() is
X˙n≡冋∂X∂S−Kn . . .∂X∂Sn0. . .∂X∂SnK册T (3.223)
then the Jacobian of the addition (or subtraction)z(t)=x(t)±y(t) can be calculated by
Z˙n=X˙n±Y˙n (3.224)
the Jacobian of the product z(t) =x(t)y(t), by
Z˙n=TxY˙n+TyX˙ n (3.225)
8. This is only applicable to algebraic nonlinearities. If they include memory, alternative generalized power series could also be used [10].
and the Jacobian of the division z(t)=y(t)/x(t) by
Z˙n=Tx−1(Y˙n−Tx−1TyX˙n) =Tx−1(Y˙n−TzX˙n) (3.226) 3.3.3.1 Comparison of Frequency and Mixed-Mode Harmonic Balance
As previously stated, the main advantages of SB over HB are its ability to handle any type of input spectrum, and dynamic nonlinearities. Although the second argument does not have too strong a practical impact, as common nonlinear devices can generally be described by memoryless currents, charges, or magnetic fluxes, the first one plays an important role on nonlinear distortion simulation. Since no Fourier transformation is required, no additional requirement is imposed to the input spectrum other than that it must be composed of a finite number of discrete points.
However, as the number of different mixing products increases rapidly with the number of nonharmonically related tones, in practice, memory storage and computation time limit SB usage to inputs composed by a few tens of discrete uncommensurated tones.
Another important advantage of SB over HB, which is also very important to the distortion simulations filed, is its higher numerical range. Highly linear systems produce distortion components having amplitudes much lower than the linear components. Therefore, its accurate calculation may be compromised if any error associated to those linear components can be spread over the distortion components.
Because Fourier transformations compute any spectral point from all time-samples, and the inverse transformation compute the value of any time-sample from all spectral components, conventional mixed-mode HB is susceptible to the type of inaccuracies just referred, while SB is not. In fact, this is an advantage inherent to all frequency-domain methods, as they handle independently all spectral components.
Unfortunately, these three advantages of SB have a price, one high enough to obviate its widely use in commercial simulators: contrary to mixed-mode HB, which handles any SPICE-like model, SB requires a special device model format.
And this implies, really, a twofold problem.
First of all, this requirement imposes another step in circuit simulation: the model substitution by a convenient approximant. It may be a power series or a rational function. Although the second one is generally preferred because of its wider approximation range, it must be selected in a way that it osculates the first nderivatives of the original function, if small-signal distortion up to ordernis to be accurately predicted. A good example of that is the Hermite rational [11]. The second criteria that must be observed in selecting these ratios of polynomials is that it must be guaranteed that any possible zero of the denominator within the approximation range must be exactly canceled by a similar zero of the numerator.
And this may be some times difficult, due to the finite arithmetic precision used to compute the polynomial coefficients.
The second problem caused by the device model format is excessive computation time. Since we rely on spectral pseudoconvolutions and deconvolutions, it is first necessary to spend time calculating the frequency transform matrices and their inverses. Then, it is necessary to perform many matrix-vector products, which are comparatively much more expensive then evaluating the model in time-domain.