Signal analysis of externally linear filters

Noise analysis of externally linear systems

“…Assume first that the ELIN systems examined in this paper are developed from a linear time-invariant prototype circuit, using a generalization [23], [24] of the state-space mapping method of [4]. Let the minimum dimension state-space equations of an th order prototype be (1) where and are the input and output signals, is the state vector with elements, and is its time derivative.…”

“…Using the chain rule of differentiation, it follows that (6) with denoting the Jacobian matrix 1 of mapping . Substituting (5) and (6) into (1) we obtain (7) In case , the Jacobian matrix is square and assuming that is a nonsingular matrix, premultiplying both sides of the first of (7) with the inverse of , we obtain the state-space equations of a minimum dimension externally linear system [23], [24]. Instantaneous companding filters are typical examples of minimum dimension systems.…”

“…Syllabic companding filters (e.g., those in [1], [10], and [27]) are typical examples of extended dimension systems. A detailed discussion of this material (excluding noise analysis) can be found elsewhere [23], [24]. Here only the material necessary for the development of noise analysis is reviewed (in condensed form, due to lack of space).…”

“…It should be noted, however, that the proposed noise analysis technique is not restricted only to circuits designed by the state-space mapping method. A straightforward technique has been developed in [23] and [24] for signal analysis of any externally linear system, whether it has been designed with the state-space mapping method or not. This signal analysis technique uniquely determines an observability canonical form [25] of a prototype linear system and a corresponding nonlinear mapping, such that the nonlinear system is derived from the linear system with the state-space mapping method.…”

“…The method is valid even if the input signal is not periodic. Incorporating the systematic method presented in [23], [24] the analysis is straightforward if the nonlinear circuit equations are given analytically.…”