The Filter Of Observation


The correspondence between continuous and discrete transfer functions in signal filtering involves many of the same issues that arise in the consideration of measurement in quantum mechanics.


Let x(t) signify a real physical variable regarded as a continuous function of time. We might attempt to construct a model dealing directly with x(t), but since any transfer of information must itself constitute a physical effect (i.e., we cannot observe x without physically interacting with it) we should base our model on a dynamic coupling to x(t) rather than on x(t) itself. A typical continuous-time dynamic coupling (i.e., filter) can be expressed schematically as a transfer function



where 's' denotes the differential operator. Here x(t) is the subject physical variable and y(t) is our observation of x(t) achieved via the coupling. This transfer function simply signifies that x(t) and y(t) are related according to the ordinary differential



It's worth noting that this coupling is symmetrical; neither x(t) nor y(t) must necessarily be regarded as the independent or dependent variable. The transfer function simply establishes a coupling between the two; y(t) is our observation of the system variable x(t), and conversely x(t) is the system's "observation" of our variable y(t).


Of course, this coupling by itself does not uniquely determine either x(t) or y(t). Each of these variables is subject to its own system constraints. By examining y(t) we hope to discern the constraints on x(t), from which we will infer something about the system of which x is a part. At the same time the constraints we impose on y(t) on our side of the coupling have an influence on x(t) and the system being observed.


The situation becomes more interesting when we attempt to translate such a coupling into the discrete-time domain (as, for example, when we model a continuous filter on a digital computer). In this case we can deal explicitly only with the values of the variables at discrete time intervals, i.e., we have only the values x(kT) and y(kT) where k = ... -1, 0, 1, 2 … and T is a constant time increment.


How do we translate the continuous transfer function into an equivalent coupling between the discrete-time values of x and y? There is no single discrete-time formula that will match the continuous relation for all possible signals x(t) and y(t) (because the discrete-time model has no knowledge of the behavior of the signals at frequencies greater than 2pi/T.) However, there are two specific translations that are, in different respects, optimum. One of these can be identified with "observed interactions" while the other can be identified with "unobserved interactions".


Suppose we intend to make a measurement of the variable x(t) associated with a particular system. For this purpose we design an interaction between x(t) and our local variable y(t) such that y(t) is as free of constraints (on our side of the coupling) as possible. In this context we can essentially treat x(t) as an independent variable and y(t) as the dependent variable (i.e., entirely dependent on x(t)). Now, given a sequence of discrete values x(0), x(T), x(2T),...x(nT) we still need to assume something about the form of the continuous variable x(t) in order to solve equation (1) for y. Of all the possible continuous functions the "most probable" is the unique nth degree polynomial that passes through the given values of x(kT). On that basis we can define the unique discrete-time recurrence relation corresponding to (1) with matched homogeneous response for y(t).


In contrast, consider an interaction in which x(t) and y(t) are symmetrical with respect to our state of knowledge, i.e., neither of them is regarded as “given”. In this context the optimum discrete- time recurrence is given by matching the homogeneous response of (1) "in both directions", i.e., for both x(t) and y(t). The resulting discrete-time recurrence is symmetrical and reversible.


Letting X and Y denote the column vectors with the components x(kT) and y(kT) respectively (for k = 0, 1, ..., n) the general form of the discrete- time recurrence is SaY = GX where Sa and G are constant row vectors. The components of Sa are the elementary symmetric functions of the exponentials of the roots of the characteristic equation of the right side of (1). Similarly we define the row vector Sb in terms of the characteristic roots of the left side of (1).


The components of G depend on which of the two contexts is assumed. For a "measurement coupling" when y(t) is treated as a purely dependent variable we have



where A, B, and M are square matrices defined by





On the other hand, if we treat x(t) and y(t) symmetrically (i.e., a non-measurement coupling) we have



The G vectors given by (2) and (3) converge to each other in the limit as T goes to zero. Combining these equations, we find that the vector given by



is invariant, i.e., it approaches the same vector as T goes to zero, regardless of the components of A. The most significant non-zero term of the components of this N+1 dimensional vector are of the form



where the coefficients ck are as shown below



These coefficients can be generated using Eulerian numbers, and are closely related to the generalized Bernoulli numbers.


What I find most interesting about these two forms of filters is that when we impose a "direction" on the transfer of information by determining one of the two sides of the equation, the resulting "most probable" transfer is irreversible, whereas if we allow the interaction to exist "unobserved" the most probable transfer is perfectly time-symmetric and reversible. This seemingly paradoxical situation arises only in the discrete-time case when the elementary increment of time T is non-zero.


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