ANALOG ENVELOPE CONSTRAINED FILTER WITH INPUT UNCERTAINTY Ba-Ngu Vo, Alex S. Leong, Pok Iu Department of Electrical and Electronic Engineering University of Melbourne Parkville, Vic 3010, Australia ABSTRACT In an envelope constrained filtering problem with uncertain input (ECUI), we require the response of the filter to each signal in the input mask to lie within a prescribed output mask. The objective is to design the filter so as to minimize the noise enhancement whilst satisfying these constraints. The continuous-time ECUI problem was previously solved using filter structures which in practice were not physically realisable. This paper presents a new sub-optimal method which allows the use of more realistic filters, and which can converge to the optimal result provided a certain parameter is made small. 1. INTRODUCTION In signal processing the design of many filters can often be cast as a constrained optimization problem where the constraints are defined by the specifications of the filter. These specifications can arise either from practical considerations or from the standards set by certain regulatory bodies. For example, in telecommunication systems, pulse shapes used in transmission systems [7] are speci- fied using templates by recommendations issued by standards bod- ies (see e.g. [1] and [2]). The continuous-time envelope-constrained (EC) filtering prob- lem considers the design of a filter such that the noiseless response ψ to a specified excitation s fits into an envelope described by ε + and ε - , as shown in Figure 1. The EC with uncertain input (ECUI) filtering problem addresses the robustness to input disturbances by allowing for uncertainty in the input pulse. Here the input is not specified exactly, but is known to lie within an input envelope de- scribed by upper and lower boundaries s + and s - . The filter is required to fit the response of all excitations within the boundaries s + and s - into the output mask. The discrete-time ECUI problem was first addressed for FIR filters in [3]. In [10], the continuous-time ECUI problem was formulated as a quadratic program with non-differentiable con- straints. This problem was then solved by transforming it into a positive definite QP problem with affine (hence differentiable) constraints. For finite-dimensional filters, this transformation re- quires their impulse responses to possess very restrictive properties that are not physically realisable with analog or hybrid components [11]. In this paper, we present a new method for solving the ECUI problem by approximating the non-differentiable constraints with differentiable ones. It can be shown that if a solution satisfies these differentiable constraints, then it will also satisfy the original non- differentiable constraints. The advantage of this approach is that the approximation assumes no specific property on the filter im- pulse reponse, thus allowing solutions to the ECUI problem for Fig. 1. EC filter filter structures which are physically realisable. Numerical simu- lations are given for both an analog filter, and a hybrid filter which consists of both digital and analog components. 2. PROBLEM STATEMENT This section presents a general formulation of the ECUI filtering problem. The reader is referred to [10], [11] and [12] for deriva- tions and further details. Here x denotes the input, u denotes the impulse response of the filter, d≡0.5(ε + +ε - ) denotes the desired output, ε≡0.5(ε + − ε - ) denotes allowable deviation from the desired output, s ≡ 0.5(s + + s - ) denotes the nominal input and θ ≡ 0.5(s + − s - ) denotes the uncertainty on the nominal input. The general ECUI filtering problem can be stated as the following optimization prob- lem on a Hilbert space: min u∈H f (u) = 〈u, Lu〉, (1) subject to |Ξ x Ψu − d|  ε, ∀x : |x − s|θ (2) where L : H → H, Ψ: H → Y and Ξ x : Y → C are linear operators, H is the Hilbert space of filter impulse responses, Y is some vector space and C is the space of filter outputs. The partial ordering  on C is defined for all x, y ∈ C by: xy if and only if x(t)≤y(t) for all t ∈ Ω. d and ε are assumed to be bounded, and θ and s are finite energy signals with θ0. In this paper we consider 2 types of filters: 1) analog filters and 2) hybrid filters, which comprise both digital and analog com- ponents as shown in Figure 2. The ECUI problems for analog and hybrid filters are both special cases of problem (1-2). For analog filters, H = L 2 (R + ), the space of square inte- grable functions on R + ≡ [0, ∞), with inner product 〈x, y〉 =