Addendum to “The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems”

This addendum contains clarifications and a sharpening of some of the results on the VIT transform framework developed in [1]. The focus is on the right-coefficient and left- coefficient forms of the transform, the extraction of a first-order term from a left polynomial fraction, and the application to linear time-varying systems.


Definition of the VIT transform
Given a real or complex-valued discrete-time signal ( ), where is the integer-valued time variable, in [1] the VIT transform ( , ) of ( ) is defined to be the formal power series in −1 given by the right-coefficient form where is the integer-valued initial time variable. Here is a symbol or indeterminate. The transform ( , ) is an element of the set [[ −1 ]] consisting of all formal power series in −1 with coefficients in , where is the ring of all functions from the integers into the field of complex numbers with the usual pointwise operations.
It follows from Equation (1) that the VIT transform ( , ) depends only on the values of the signal ( ) for ≥ . Hence the VIT transform is a one-sided transform. Note that the VIT transform of ( ) is equal to the VIT transform of ( )ℋ( − ), where ℋ( − ) is the Heaviside step function defined by ℋ( − ) = 1 for ≥ and ℋ( − ) = 0 for < .
In [1], the set [[ −1 ]] is given the structure of a noncommutative ring by defining the usual addition of power series and with multiplication defined by −( + ) = − − and ( ) − = − ( + ), ( ) ∈ , (2) sometimes referred to as skew power series, and polynomials in with coefficients in and with the multiplication ( ) = ( − ) are referred to as skew polynomials.

Right-Coefficient and Left-Coefficient Forms of the VIT Transform
Now consider the signal ( ) = ( )ℋ( − ), where ( ) is a real or complex-valued function of , and the values of ( ) do not depend on the initial time variable . Then using the definition of multiplication (2), we can write the transform ( , ) in the left-coefficient form Since ∑ − ∞ =0 = ( − 1) −1 , the left-coefficient form of ( , ) reduces to ( ) multiplied on the right by the fraction ( − 1) −1 ; that is, To check this result, first observe that the inverse VIT transform of ( − 1) −1 is equal to ℋ( − ). Then by the multiplication by a time function property given in [1], the inverse VIT transform of ( )( − 1) −1 is equal to ( )ℋ( − ). This verifies the form ( )( − 1) −1 for the VIT transform of the signal ( ) = ( )ℋ( − ).
Note that setting = 0 in the right-coefficient form in Equation (3) which is equal to the formal z-transform of the function ( ). However, setting = 0 in the left-coefficient form in Equation (4) results in (0)( − 1) −1 , which is not equal to the ztransform of ( ). To eliminate this inconsistency, we define the evaluation of the VIT transform ( , ) at a particular value of to be the evaluation of the right-coefficient form at that value of .
It is important to note that if the values of ( ) depend on the initial time variable , then the form ( )( − 1) −1 for the VIT transform of ( )ℋ( − ) is not valid. If To verify Equation (7), divide − ( ) into using left long division.
The left-coefficient form of the VIT transform of the signal given by Equation (6) is The right side of Equation (8) can be written in the right polynomial fraction form From Equation (10), it is seen that moving ( + 1) to the left through ( − ( )) −1 changes the coefficient ( ) in the denominator polynomial to . This noncommutativity is a fundamental aspect of the VIT transform framework.
In this example, it can be shown that the right polynomial fraction form of the transform ( , ) can be derived directly from the left polynomial fraction form. This turns out to be true in the general case when ( , ) = ( , ) −1 ( , ), where ( , ) and ( , ) are skew polynomials in with coefficients in : Similar to the discussion of the extended right Euclidean algorithm given in [1], the extended left Euclidean algorithm can be used to determine polynomials ( , ) and ( , ) such that ( , ) ( , ) = ( , ) ( , ). In general, the coefficients of ( , ) and ( , ) belong to the quotient field ( ) of . Then

Extraction of a First-Order Term
Given a skew polynomial ( , ) with coefficients in and a function ( However, it is not necessary to express ( − ( )) ( , ) in the form given by Equation (13) in order to extract a first-order term. A sufficient condition for extracting a first-order term is that there exist a polynomial ( , ) with coefficients in ( ) and ( ) ∈ ( ) such that To prove sufficiency, multiple both sides of Equation (14) on the left by ( , ) −1 and on the right by [ − ( )] −1 . This results in the decomposition and thus, the first-order term ( )[ − ( )] −1 is extracted from the fraction. Also note that the denominators of the terms in the decomposition (15)  The computation of ( ) that satisfies Equation (14) can be carried out by evaluating both sides of Equation (14) at = ( ), where 0 ( ) = 1, ( ) = ( ) ( + 1) ⋯ ( + − 1), ≥ 1. First, it follows from the results in [3] that the evaluation at = ( ) of a skew polynomial ( , ), with coefficients written on the left, is equal to the remainder after dividing ( , ) on the right by − ( ). If ( , ) has − ( ) as a right factor, the remainder after dividing by − ( ) on the right is equal to zero. Hence, in this case, the evaluation of ( , ) at = ( ) is equal to zero. Finally, evaluation is an additive operation; that is, the evaluation of the sum of two skew polynomials is equal to the sum of the evaluations of the two polynomials.
Inserting Equation (17)  given in terms of a sum of first-order terms.

Application to Linear Time-Varying Systems
Consider a causal linear time-varying discrete-time system with input ( ) and resulting output response ( ). It is assumed that the input is applied beginning at time , and is zero before time . Here is the initial time which is allowed to vary over the set of integers. Then the input can be expressed in the form ( ) = ( )ℋ( − ), which shows that ( ) depends on , so we shall write the input as ( , ). (23) This is the input/output relationship of the system when the input ( , )ℋ( − ) depends on both the current time and the initial time .
Here the output ( ) is the response of the system to the input ( ) applied beginning at the initial time = 0.
The transfer function ( , ) defined by Equation (24) is equal to the VIT transform of the unit-pulse response function ℎ( , ), and as proved in [1], the VIT transform ( , ) of the output ( , ) resulting from the input ( , ) is given by the product where ( , ) is the VIT transform of the input ( , ). Here the output ( , ) is the response of the system to the input ( , ) applied beginning at the initial time = , where varies over the set of integers. Hence, the VIT transform framework captures the dependency of the output response on the time when the input is applied, which is a key aspect of timevarying systems.
Note that the left-coefficient form of ( , ) is given by When is viewed as a complex variable, ( , ) defined by Equation (26) is equal to the ordinary -transform of ℎ( , − ), which is the definition of the transfer function given in [4]. Thus, the transfer function has the same form in both the -transform approach developed in [4] and the VIT transform approach developed in [1]. However, the two approaches differ significantly since the skew ring framework in [1] is based on the noncommutative multiplication ( ) = − ( − ), ( ) ∈ , whereas there is no noncommutative multiplication in the -transform framework. It is a consequence of the noncommutative multiplication in the ring framework that the VIT transform of the output is equal to the product of ( , ) with the VIT transform of the input; whereas the -transform of the output is not equal in general to the product of ( , ) with the -transform of the input.
From the results in [1] and [2], when the system is given by the input/output difference equation where ≤ , the transfer function ( , ) in the skew ring framework has the left polynomial fraction form given by ]. From the shifting property of the VIT transform given in [1], the inverse transform of −( − ) ( , ) is equal to ( − ( − ), ). Hence, the causal inverse system with transfer function [ ( , ) − ] −1 ( , ) reproduces a time delayed version of ( , ) from the output ( , ). Now suppose that the system defined by the difference equation (27) is asymptotically stable as defined in [1]. Using the VIT transform framework, we shall determine the steady-state output response of the system to the input ( , ) = − ( ) = ( ) ( + 1) ⋯ ( − 1), ˃ , with initial value ( , ) = 0 ( ) = 1 Here ( ) is an element of with the condition that − ( ) does not converge to zero as → ∞. Note that ( + 1, ) = ( ) ( , ), and the VIT transform of ( , ) is equal to ( − ( )) −1 . Then the VIT transform ( , ) of the output response is where ( , ) is a polynomial in with coefficients in ( ), and the remainder ( ) is equal to the evaluation of ( , ) at = ( ); that is, ( ) = ∑ ( ) ( ) =0 . Then inserting (31) into (30) results in The inverse VIT transform of the first term on the right side of Equation (32) decays to zero as → ∞ since the system is asymptotically stable.
As in the above constructions leading to the proof of Theorem 1, it is possible to determine a polynomial ( , ) with coefficients in ( ) and ( ) ∈ ( ) such that The inverse VIT transform of the first term on the right side of Equation (35) decays to zero as → ∞. Hence, the steady-state response to the input ( , ) = − ( ), ≥ , is equal to the inverse VIT transform of ( )( − ( )) −1 , which is equal to ( ) − ( ), ≥ . This proves the following result.
By Theorem 2, the steady-state output response of an asymptotically stable system to the input ( , ) = − ( ), ≥ , is equal to a scaling of the input by the time function ( ). It is interesting to note that the expression for ( ) given by Equation (34) can be generated directly from the input/output difference equation by inserting ( , ) = − ( ) and ( , ) = ( ) − ( ) into Equation (27) and solving for ( ). The VIT transform framework as utilized here verifies that this solution to the input/output difference equation is in fact the steady-state response in the case when the system is asymptotically stable.