and identification of nonlinear systems whose input-output. (1-0) relationship can be summarized as the Volterra or. Wiener functional expansions of the system ...
Modeling and Identification of Parallel Nonlinear Systems: Structural Classification and Parameter Estimation Methods HAI-WEN CHEN, MEMBER, IEEE
Structural class$cation and purameter estimation (SCPE) methods are used for stud.ying single-input single-output (SISO) parallel linear-nonlinear-linear (LNL), linear-nonlinear (LN), and nonlinear-linear (NL) system models from input-output ( I - 0 ) measurements. The uniqueness of the 1-0 mappings (see the definition of the 1-0 mapping in Section III-A) of some model structures is discussed. The uniqueness of 1-0 mappings of different models Tells us in what conditions direrent model structures can be differentiated from one another. Parameter uniqueness of the I - 0 mapping of a given structural model is also discussed, which tells us in what conditions a given model’s parameters can be uniquely estimated from 1-0 measurements. These methods are then generalized so that they can be used to study single-input multi-output (SIMO), multi-input single-output (MISO), as well as multi-input multi-output (MIMO) nonlinear system models. Parameter estimation of the iwo-input singleoutput nonlinear system model (denoted as the 2f-structure in [ l ] and [2]), which was left unsolved previously, can now be obtained using the newly derived algorithms. Applications of SCPE methods for modeling visual cortical neurons, system fault detection, modeling and identification of communication networks, biological systems, and natural and artificial neural networks are also discussed. The feasibility of these methods is demonstrated using simulated examples. SCPE methods presented in this paper can be further developed to study more complicated block-structured models, and will therefore have future potential for modeling and identibing highlj complex multi-input multi-output nonlinear systems.
I. INTRODUCT~ON
A. Background System modeling and identification is important for systems analysis, control, and automation as well as for scientific research. Though linear system identification methods have been well developed, nonlinear system identification methods are still in a relatively early stage of development [3]-[6]. Many systems we encounter are more or less nonlinear. Under certain conditions, some of them may be linearized. In many cases, they can only be represented Manuscript received November 17, 1993; revised August 29, 1994. This work was supported in part by a grant from NE1 (EY08610). The author is with the Biophysics Group M715, Los Alamoq National Laboratory, Los Alamos, NM 87545 USA. IEEE Log Number 9406487.
by nonlinear models. In this paper, we consider modeling and identification of nonlinear systems whose input-output (1-0) relationship can be summarized as the Volterra or Wiener functional expansions of the system outputs in terms of their inputs [7]-[17], [103], 11221, [146]-[1511.’ Such expansions are fully specified by a set of Volterra (or Wiener) kernels which can be estimated from the 1-0 measurements of the system under study. Such a system is generally time-invariant (stationary), causal, and continuous [7]-[1 I]. The system kernels provide a characterization of the system 1-0 relationship, and thus allow one to predict the response of a physical system to an arbitrary s t i m u h 2 However, the calculation of higher-order kernels is time-consuming, and interpretations of kernels based on structural or physical mechanisms is generally not readily apparent. Alternatively, one can develop the system’s blockstructured network models for characterizing the system’s 1-0 relationship. Block-structured network models may be easier to realize and implement and may be more directly related to the system’s inner structures or physical mechanisms. Linear time-invariant (LTI. or L ) operators and static (or zero memory) nonlinear ( N ) operators are two basic and important types of operators used in nonlinear system representation and modeling [6], [SI, [ 1 I], [21]-[23], [44]. There are various types of interconnections between L and N operators, such as cascades, parallel, multiplication, additive and shunting feedback, etc. Structures containing these interconnections are referred to as block-structured network models. Such models are often used to describe the input-output (1-0) relationship of practical nonlinear systems. Indeed, as proved by Korenberg and others [24]-[26], a discrete-time, finite-memory nonlinear system having a finite-degree M ( M < x)Volterra series (also referred to as the polynomial system of degree M by Rugh [ 111)
’
The mathematical foundation of the Volterra series has been recently advanced by Sandberg [ 141, [ 151, who proved that a Volterra-like inputoutput expansion exists for an important large class of nonlinear systems. ‘Identifying nonlinear systems by measuring all of the system kernels is referred to as the nonpurainetric trpprocrch [ 18]-[20] or the “black box” approach [ 1201. See Section X-C for further discussion on this issue.
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39
can be represented by a finite number of parallel LN or LNL cascades of alternating L and N operators. Therefore, all dynamical discrete-time polynomial systems can be represented by the interconnection of the two basic L and N operators. The L and N operators are mathematical representations, and they do not necessarily represent real components of actual physical systems. However, under certain conditions, along with access to some structural information, their interconnections may reflect the structure of the underlying physical network. For example, the N operator may represent: 1) the operation of a nonlinear resistor [22], [23]; 2) the rectification of a diode in an electronic circuit; or 3) the threshold of a cell membrane potential in a biological system [55].
B. Structural Classijication and Parameter Estimation (SCPE) Methods 1) Structural Classijication: There are generally two steps in developing a block-structured network model for a system under study. One must first determine the topology (structure) of the network (structure classijication), and then parametrize each L and N element (parameter estimation). We emphasize that not only is the first step (structural classification) important but it is also necessary for the following reasons: To date, most parameter estimation algorithms have been developed for specific model structures, e.g., the algorithms for the LNL cascade [9], [191, [211, [441 and for the feedback with the LNL cascade in the forward path [44], [50],[131], [176], etc. We must know something about the model structure which can best represent the system under study in order to choose the most fitting algorithm. For example, one cannot use the algorithm for an LNL cascade model to estimate a feedback nonlinear system. In many cases, the major goal of scientific studies is to determine the inner structures of physical or physiological systems and the topology of neural networks. For example, in the visual system (natural or artificial), we may want to know how visual information is conducted along different pathways; how neurons are interconnected in different neural network layers inside the system; and what types of interconnections exist (i.e., cascade, parallel, feedback [centrifugal], divergence, and convergence). The fundamental question is how to accomplish the structural classification step from 1-0 measurements alone, i.e., how to identify a unique network model topology (structure) which best describes the system under study. Boyd and Chua [21], [22], and Chua [23] have proven the uniqueness (aside from uninteresting gain, delay, and loop transformations) of the input-output mapping of the LNL cascade and feedback nonlinear structures that contain only one static nonlinearity. Therefore, they have shown theoretically that 1-0 measurements alone are sufficient 40
to classify these structures because of their unique 1-0 mappings. System kernels can be obtained from 1-0 measurements and are directly related to the system’s 1-0 relationship, and can therefore be used to develop practical structural classification methods. Hung et al. [56] (also see [57], [58]) have developed a kernel identification technique to interpret the internal structure of systems by examining the shape of the higher-order kernels. Korenberg [34] first derived formulas that describe the relationship between 1st- and 2nd-order Wiener kernels that must be satisfied by specific classes of single-input LNL, LN, and NL cascade models. (Some methods for testing the structure of the LNL, LN, and NL cascade models, based on measurements of the 1st- and 2nd-order kernels, can also be found in [9], [44], [160], [161], and [164].) Chen et al. [l], [2], [491, [501, [5 11, [ 1761 have extended these results to include additional structural classes (longer cascades, feedback, multi-input parallel cascades), and have proposed systematic methods for structural classification that compare the 1st- and 2ndorder Wiener or Volterra kernels of single-input and multiinput nonlinear systems. The key point in these approaches is that the topological structure of a nonlinear system will be reflected in the interrelationship between the system kernels without regard to the specijic parametrization of the system model. If an unknown system does not satisfy the kernel relationships associated with a given structure, then that structure can be rejected as a model for the system-that is, the kernel relationship constitutes a necessary condition for a system to have a given structure. On the other hand, if an unknown system satisfies the kernel relationship for a given structure, one cannot conclude that the system has the given structure-rather, a sufficient condition must be available to positively identify the system structure. However, suppose that fundamental physical arguments can be used to establish that the structure of the system under study belongs to a broad structural superclass (see [l], [2], [ S O ] ) , then, in this situation the necessary condition may also be sufficient to assign the unknown system to a specific structural subclass of the superclass. This assignment task is referred to as structural classijication. In this paper, the uniqueness of the 1-0 mappings of different models is evaluated, which aids in defining the conditions under which different model structures may be differentiated from one another. 2 ) Paramete8 Estimation The SISO LNL model with cascaded L, N , L operators (also referred to as the sandwich model), the LN (Wiener) model and the NL (Hammerstein) model, are examples of the familiar and often used models in industrial control plants and systems, electrical circuits, and biological systems [9], [11], [18]-[21], [27]-[50], [72], [143], [171]. As a general approach to identifying the LNL system, the measured 1st- and 2ndorder Volterra or Wiener kernels are first used to derive the impulse responses (or transfer functions) of the linear See the definition of “parameter” in Section 111-A. PROCEEDINGS OF THE IEEE, VOL. 83. NO. 1, JANUARY 1995
prefilter and postfilter, and then a least squares algorithm or a curve-fitting algorithm is used to estimate the static nonlinearity [9], [ 111, [ 181-[20], [34], [40]-[49], [ 1431. Boyd and Chua [21] have alternatively developed a stable decomposition method using the measured 2nd-order Volterra kernel to identify an LNL system in the frequency domain. Furthermore, the measured 1 st- and 2nd-order Volterra kernels can be used to completely identify feedback systems with the LNL, LN, or NL cascade in the forward path [44), [SO], [ 1761. Marmarelis [ 13I] has recently presented important results concerning the relationship between the measured (1 st- and 2nd-order) Wiener kernels and a class of feedback systems (cubic and sigmoid feedback) with various parameters. In this paper, parameter uniqueness of the 1-0 mapping of a given structural model is evaluated for different structural models, which defines the conditions under which a given model parameter can be uniquely estimated from 1-0 measurements. The 1-0 mappings of some structural models have been proven to be parameter nonunique (see Sections IV-A and IV-B). Although the parameters of these models cannot be uniquely estimated from 1-0 measurements, the results may be useful in filter designs for selecting specific constraints or design criteria (see Section X-A). 3) Kernel Measurements: As discussed above, accurate kernel measurements are crucial to the applicability of the proposed SCPE methods. Although a firm mathematical foundation of Volterra-Wiener series has been established for a long time [7]-[17], [103], [122], [146]-[151], relatively few attempts have been made to measure Volterra or Wiener kernels for electronic or industrial systems [ 181, [401, [421-[441, [1051, [1091, [119l-[121l. On the other hand, for more than 20 years, many applications of the Volterra-Wiener series have been devoted to physiological and biological systems [I], [2l, [9], [19], [201, [241, 1261, [341-[391, [41l, 1451, [511-[541, [561-[.581, [671-[741, [76l, [811, [871, [88l, [104l, [1061-[108l, [ l l o l , [111l, [113l, [ 1231-[ 1441, [ 1.591, [ 1621-[ 1651, [ 16714 1691, [ 17.51. AS pioneers in this area, Lee and Schetzen [lo51 developed a practical method in 196.5 for measuring the Wiener kernels using Gaussian white noise as the input signal. One practical disadvantage of this method is the requirement of long record length of the input to approximate the true white noise (see [ 1071 for a discussion of artifacts in Wiener kernel estimation using the Lee-Schetzen method). Since their seminal work in 1965, many improved methods for measuring the Volterra and Wiener kernels have been developed using deterministic or pseudorandom probing signals P I , [IO], [201, [241, [411, 1451, [561, VOI, [721-[741, 11041, [106]-[111], [ I 19]-[121], [123]-[1251, [1291, [130], [132], [133], [136]-[144], [1.59], [162], [163], [167], [168], [171]. For example, the technique using a multi-tone probing signal [ 1091, [ 1201, [ 12 1 1 allows one to measure the 2nd- and 3rd-order Volterra kernels with very high accuracy. Other techniques such as the Laguerre expansion technique [8], [ 1321, [ 1331, [ 1361 and the exact orthogonalization technique [24l, [ 1 101 have obvious advantages (e.g., no strict requirements for whiteness of inputs, shorter experimental
h1.o(tj
y-(L;*tp ho (f)
Nz
hz.7 (f)
(d) Fig. 1. Single-input single-output parallel models with two pathways: (a) The PLNL2 model. (b) The PLN: model. (c) The PNL2 model. (d) The PLlNL2 model.
record lengths, and higher kernel estimation accuracy, etc.) over the traditional Lee-Schetzen crosscorrelation technique. Further discussion on kernel measurements and the impact of noise will be presented in Section X-C. C. Parallel Nonlinear Structural Models In this paper, SCPE methods for the more complicated nonlinear SISO, SIMO, MISO, and MIMO parallel LNL structural models are further developed using measurable low-order Volterra or Wiener kernels (1st-, 2nd-, andor 3rd-order). Some examples of SISO structural models with parallel pathways are shown in Figs. 1 and 2. Parallel structures are often encountered in complex nonlinear system models for control systems, electrical circuits, biological systems, as well as artificial neural networks. Indeed, parallel structures are fundamental in neural networks, which lead to fast and effective parallel neural computation. There is evidence that in the primate visual system, visual information is conducted in parallel pathways (e.g., luminance or magnocellular, and color/form or parvocellular pathways) [83], [140]. Recently, Sandberg [117] has proven that under certain conditions a complex nonlinear feedback control system can be approximated by a finite number of feedforward parallel (neural network) operations. A general SISO parallel LNL model (denoted as the PLNL, model in Fig. 2(a))4 and its substructures, SISO PLN, model (Fig. 2(b)), and SISO PNL,, model (Fig. 2(c), also referred to as the Uryson model [61], [62]) are
CHEN: MODELING AND IDENTIFICATION OF PARALLEL NONLINEAR SYSTEMS
4The PLNL,,, model is generalized from the well studied SI,model [44], [49], [59], [60],[ I 121. See Section Ill-B for further explanations. 41
examined in Sections I11 and IV. The uniqueness of the 1 - 0 mappings for these model structures is also discussed. These methods are then generalized, in Section V, to study simple single-input multi-output (SIMO) nonlinear system models, which can be used to determine the existence and configuration of interconnections between different outputs. These results may have potential applications in studying natural and artificial neural networks since the lateral interaction between neurons is a very important characteristic of the network [63]-[65], [ 1161. Parameter estimation of the two-input single-output nonlinear system model (a generalized motion detection model [77]-[79] shown in Fig. 4(a) and denoted as the 2f-structure in [I] and [2]), which was left unsolved previously, can now be obtained using the newly derived methods. This result is presented in Section VI-A. MISO and MIMO nonlinear system models are further studied in Section VI. Applications for modeling and identifying visual cortical neurons, communication networks, biological systems, and natural and artificial neural networks are considered in Section VIII. The feasibility of these methods is demonstrated using simulated examples in Section IX.
Fig. 2. Single-input single-output parallel models with m pathways. (a) The PLNLm model. (b) The PLNm model. (c) The PNL,, model.
11. GENERAL THEORY
A. The Volterra Series for Single- and Multi-Input Nonlinear Systems The input-output relationship of a causal SISO system is a mapping of the past (and present) inputs to the present output of the system. When a system is linear and timeinvariant, the input-output relationship can be represented by the convolution integral. More generally, given an SISO nonlinear time-invariant system with certain restrictions [9]-[l I], the Volterra theory [7] enables one to express the relationship between the system input z ( t ) and output y(t) as a series of multiple convolution integrals:
where the function k n ( 7 1 , . . . , 7 , ) is the nth-order Volterra kernel which can be assumed to be symmetric without loss of generality [lo], [ 111. The Volterra functional expansion represents the linear system as a special case when all the kernels higher than the 1st-order are identically zero. In this case, the 1st-order kernel represents the impulse response function of the linear system. The Volterra functional expansion also represents the static nonlinearity as a special case where all the kernels in (1) are 6 functions. In this case, the Volterra series in (1) becomes the familiar Taylor series. Based on the Volterra series for a single-input system, the series for an MISO system can also be derived. Generally, for a time-invariant dynamic system with m inputs (stimuli) x l ( t ) , . . . , z m ( t )(1 I m < CO), we have the Volterra series 42
where l ~ ~ ~ , . . . (. , ~' ,.), , is the nth-order Volterra kernel of the MISO system. The kernel order is n = n1 + . ' . + n,. The Volterra kernels are called self-kernels when one of the m subscripts, ni(i = 1. . . , m), is nonzero and all other m - 1 subscripts are zero. Otherwise, they are called cross kernels. That is, in the Volterra series expansion (2a), each selfkernel is only convolved with one input, whereas each cross kernel is convolved with at least two different inputs. As with the SISO Volterra theory, the self-kernels of a multiinput system can be assumed to be symmetric without loss of generality. The cross kernels of an MISO system may, however, be either symmetric or asymmetric depending on the properties of the system. The cross kernels of a multiinput system reflect the nonlinear interaction between the different input pathways (see, for example, [l], [2], [9], and [67]). The sufficient condition for convergence of the m-input Volterra series is expressed as nl+:.:+n,
Lw
. . ' . , Tmn,)l . . d71nl . . dTm1 . dTmn_ 5 Anl,...,nm (2b)
l k n l , . . . , n m ( ~ l .l , > . 7 1 n 1 >.. ,7m1,
d711
'
*
'
PROCEEDINGS OF THE IEEE, VOL. 83, NO. 1, JANUARY 1995
and
where B1,. . . , B, are the absolute bounds on the m inputs x l ( t ) ,. . . , x m ( t )(i.e., Bi = maxoltLEL NONLINEAR SYSTEMS
‘The representation of the .\-operator can be extended to continuous, not necessarily analytic, nonlinear functions based on the Weierstrass polynomial approximation theorem [ 1 I ] , [20]. 6The term “system kernels” (or “kernels”) stands for Volterra kernels throughout this paper. See Section VI1 for the Wiener kernel representation. 43
inputs. This result can be extended to MIS0 and MIMO nonlinear systems using (2a), (2d), and (2e). Similarly, models with the same structure but different system parameters generally have different system kernels. In essence, each specific set of parameters is associated with one 1-0 relation. The set of 1-0 relations associated with all the possible sets of parameters for a given structural model is referred to as the 1-0 mapping of the given structural model. If two 1-0 mappings, associated with two different structural models, do not have any 1-0 relation (or system kernels) in common, we say that the two IO mappings associated with the two models are unique to one another. In this case, these two model structures can be distinguished from one another from their 1-0 measurements (i.e., structural classification). On the other hand, if these two 1-0 mappings overlap partially or totally, then they are considered nonunique. With regard to parameter estimation, if each 1-0relation in the 1-0 mapping is related to one and only one specific set of system parameters in a given model, then the IO mapping is considered parameter unique and one can uniquely identify the parameters of a given system from its 1-0 relation. However, there are cases, as discussed later, where the 1-0 mapping for a given model is considered parameter nonunique (i.e., different sets of parameters for a given model may be associated with the same 1-0 relation and thus the same system kernels).
B. Kernel Relationships for the SISO PLNL2 Model A general SISO parallel LNL model with m ( m = 1, 2, . . .) pathways, denoted as PLNL,, is shown in Fig. 2(a). Obviously the PLNL, model is a generalized case of the previously studied SM model [441, [491, [591, [601, [ 1121. The N operator in each pathway is represented by a general polynomial function for the PLNL, model versus a specific integer power nonlinearity for the S M model.' In this section, we will study the PLNL2 model (with two pathways) and its subclasses, as shown in Fig. 1. The more general PLNL, model will be discussed in Section 111-E. As shown in Fig. l(a), the output of the PLNL2 model is the sum of two LNL models and thus the system kernels of the PLNL2 model are the sum of the system kernels of each individual LNL model. The system kernels and their relations to the parameters of the PLNL2 model can therefore be readily obtained using the results of the well studied LNL model [9l, 1111, [181-[211, [34l, [411-[501. The 71th order kernel ( n = 1, 2, . . .) is expressed as
'As discussed in the Introduction section, a PLNL,,, model may represent an arbitrary polynomial nonlinear system [24]-[26]. On the other hand, an Shl model can only represent a subset of polynomial systems with a specific factoring kernel formation similar to the kernel formation of an LNL model. 44
where K,(wl,. . . ,U,) is the nth order kernel in frequency domain (the Fourier transforms of the time-domain kernels in (1)) for the PLNL2 model; al,, and a2,, are polynomial coefficients for the static nonlinear operators N1 and N2, respectively; and H I , o ( ~H) ,z , o ( ~ HI,I(w), ), and &,1(w) are transfer functions of the four linear subsystems in Fig. l(a), respectively. From (6), we have8 Kl(W) = a l , l H l , l ( ~ ) H l , o ( 4
+ a2,1H2,1(w)&,o(w), K2 (W , 0)
(7)
= U 1 ,2Hl,O(O)Hl,1( U )H 1,o ( U )
+ a 2 , 2 H z , o ( O ) ~ z , (w)Hz,o(w) l
(8)
and ~ 3 ( w0,o) ,
= al,3~12,0(0)~i,i(w)~~,o(w)
+ a2,3Hi,o(O) H2,1
( U )H2 ,O ( U ).
(9)
Let .:,2
= ~z,zHz,o(O), ah,3 = a2,3H:,,(O).
= al,zHl,o(O), = al,3H?,o(0), and
Then, for u ; , ~ u / # ~ ,U~ ~ matrix form as
, ~ U (8) ~ , and ~ ,
Hl,l(w)H1,o(w)) =
A-1
(
(9) can be solved in
K2(w,0)
K3(w, 0,O)
)
(10)
where A-' is the inverse matrix of A, and
Finally, one of the kernel relationships for the PLNL;! model can be obtained by substituting (10) into (7), as shown in (11).
+ plK3(w,O,o)
Kl(w) = alK2(w,O)
(11)
where a1
=
a1,1a:,3 - a2,la:,3 ':,2./2,3
-
'h,Za:,3
and
Equation ( 1 1 ) is a necessary condition for a nonlinear system to be represented by the PLNL:! model without regard to the specific parametrization of this model.' In (11), cy1 and are constants, which can be calculated by using two independent equations which are available and -Yzare neither even- nor odd-symmetric *Here, we suppose that functions. For cases where 4 1 and 2V2 are even- or odd-symmetric functions, only even- or odd-order kernels exist. Therefore, higher-order (e.g., the 1st-, 3rd-, and 5th-order) kernels must be used to obtain the kernel relationship. To avoid using higher-order kernels, some specific techniques [42], [44] may be used to obtain both even- and odd-order kernels where nonlinearity is even- or odd-symmetric. 'Equation (11) means that the 1st-order kernel is proportional to the linear combination of the one-dimensional (ID) slices of the 2nd- and 3rd-order kernels. PROCEEDINGS OF THE IEEE, VOL. 83, NO. I , JANUARY 1995
from (11) if one selects two different values for w.I0 Therefore, the structural testing procedure for the PLNL2 model includes two steps: first estimating the constants cy1 and PI, then substituting the estimated a 1 and p 1 into (1 1) to examine whether the kernel relationship of a system under study satisfies (1 1) for all frequencies w. Equation (1 1) can also be expressed by kernels in the time-domain using marginal integrals [50], [ 1641. Though the kernel relationship in (1 1) indicates only a partial relationship between kernels of a PLNL2 model," any nonlinear system (having the 1st-, 2nd-, and 3rd-order kernels) that does not satisfy (1 1) can not be a PLNL2 model. On the other hand, nonlinear systems having the same (partial) kernel relationship do not necessarily have the same 1-0 mappings. One example is the case of the SISO LN and LNL models. Their kernel relationships are shown in (13) and (141, respectively [91, [341.
K1 ( W l ) K l ( W Z ) = P K z ( w 1 . w2)
(13)
uniqueness between these two 1-0 mappings. Previously, (1 3) was used to differentiate the LN model from the LNL model. In a similar manner, (14) for the LNL model can be used to derive conditions under which (14) does not satisfy its superclass, the PLNL2 model. Generally the kernel relationships for a structural subclass (e.g., (14)) may only be used to derive some sufficient conditions. Of course, the 1-0mappings of the LNL and PLNL2 models will be unique to one another when the subset under sufficient conditions12 is excluded from the 1-0 mapping of the PLNL2 model. Theorem 1 shows one such sufficient condition under which (14) can be used to distinguish the LNL model from the PLNL2 model. Theorem 1: One suflcient condition under which (14) does nor s a t i s - the kernel relarionship of the PLNL, model is
= (YKz(W.0)
(14)
= BN2.1(w) (15) where A and B are constants. Therefore, the 1-0 mapping of the LNL model is a subset of that of the PLNL2 model. In fact, the subset of the 1-0 mapping of the PLNL2 model under the conditions in (15) is equal to the 1-0 mapping of the LNL model. The 1-0 mappings of these two models will be unique to one another when this subset (i.e., the LNL model) is excluded from the 1-0 mapping of the PLNL2 model. Note that the subset under conditions in (15) is the minimum subset whose exclusion guarantees the = AHz.o(w)
and
u2.I
4.3 # --. U;,:,
4 . 2
-#
43 ~
a.;,:
0.30 is used as a criterion, the RLNindex can be used to classify simple cells versus complex cells. The results (see Fig. 6 in [53]) agree well with those obtained by the conventional method using modulation indices [ 1141. Complex cells are often modeled by a summation of parallel MISO LN channels whose linear spatiotemporal filters PROCEEDINGS OF THE IEEE, VOL. 83, NO. 1, JANUARY 1995
differ in spatial phase or position 1521-[54], [85], [86], 1881, 11721, [173]. One complex cell model that is currently popular is the "energy model" in which nonlinear simple cell-like subunits are summed to obtain a local estimate of spectral energy [52]-[541, [75],1761, [88]-[90], 11721, [ 1731. The MISO PLN2 model shown in Fig. 4(c) is such a model for complex cells where the filters h l ,i ( t )and h2,i ( t ) are quadrature phase pairs and the nonlinear operators N1 and N2 are pure square (full-squaring) functions (i.e., only the coefficients a1,2 and a 2 , 2 exist).22This "energy" model for complex cells has only a 2nd-order Therefore, (61) of the MISO PLN2 model can be used to test the model structure of complex cells. Similar to the case for simple cells, based on (61) of the MISO PLN2 model (a complex cell model), a R P L Nindex ~ may be introduced for: 1) characterizing the degree to which a cell's behavior deviates from the MISO PLN2 model prediction; and 2) classifying complex cells versus cells with more complex model structures. The results will be discussed at length in a separate paper. As stated in Theorem 3, the two linear spatiotemporal filters in an MISO PLN2 model can not be uniquely estimated from a 2nd-order spatiotemporal kernel only; one more independent equation is required. Nevertheless, as discussed in Remark 2 in Section IV, by virtue of the fact that the two linear spatiotemporal filters are a quadrature phase pair, one can still uniquely estimate these filters.
B. Some Potential Applications 1) Parallel Communication System: A two-channel communication system consisting of A4 (1
