ADAPTIVE ENVELOPE-CONSTRAINED FILTERING Ba-Ngu Vo, Sumeetpal Singh and Vladislav Tadic bv@ee.mu.oz.au, ssss@ee.mu.oz.au, vtadic@ee.mu.oz.au Department of Electrical and Electronic Engineering, The University of Melbourne, Vic. 3010, Australia ABSTRACT In the discrete-time Envelope-Constrained filtering problem, the gain of the filter is minimised subject to the constraint that the filter output to a prescribed input fits into a given en- velope. In this paper, a novel adaptive algorithm for solving this problem based on stochastic optimisation is presented. The algorithm is simple to implement on-line and conver- gence is demonstrated in numerical examples. Under mild regularity assumptions convergence follows from standard stochastic approximation results. 1. INTRODUCTION The (discrete-time) Envelope-Constrained (EC) filtering prob- lem involves the design of a linear time-invariant (LTI) filter such that the response to a specified excitation fits into a prescribed template or envelope, , [1] as shown in Figure 1. The envelopes can arise either from practical con- siderations or from the standards set by certain regulatory bodies. For example, in pulse-compression for radar, the envelopes are selected to suppress side lobes while keep- ing the main lobe above a certain threshold [1]; in telecom- munications pulse shapes used in transmission systems are specified using templates issued by standard bodies such as CCITT or ANSI (see e.g. [2]-[4]). For a technically ori- ented survey of the subject the reader is referred to [5]. Filter s k ψ k ξ k + ε k + ε k - n k Fig. 1. Envelope Constrained Filter The EC problem was formulated for FIR filters in [1] and IIR filters in [6]. The resulting optimisation problem can be solved off-line by Quadratic Programming (QP) via active set strategy [5]. Direct use of this algorithm would be inappropriate in applications where the parameters of the underlying signal model are either not known or varying with time. In such cases, it may be necessary to employ an adaptive filter with parameters that can be adjusted to their optimum value (see Figure 2). In the most common procedure, a test or training signal corrupted by noise is periodically used as the filter input. The filter response (to the training signal ) is checked against the boundaries , of the template. The result of the comparison is then processed in some way and fed back to adjust the fil- ter coefficients. This process is repeated until, for practical purposes, convergence has occurred and the filter is ready to process data. Additional test pulses are then inserted into the data stream at regular intervals so that the filter can con- tinue to be adjusted. Adaptive algorithms for EC filters have been proposed in [7] and [8] based on the dual formulation. In [7] the prob- lem was converted to an unconstrained non-smooth dual problem, which was then solved using subgradients, while in [8] the smooth constrained dual problem was solved us- ing gradient flow. In [9], a modified penalty technique was used to approximate the primal problem as a smooth uncon- strained problem which can be solved using descent meth- ods. These algorithms converge under noise-free condition, however, for noisy input signals, no useful results on con- vergence has been established. In this paper, we propose an adaptive EC filtering al- gorithm based on standard stochastic approximation tech- niques where convergence can be established under mild regularity assumptions using standard stochastic approxi- mation results as in [10]. The proposed algorithm is based on solving the dual problem using stochastic gradient ascent with projection. Our algorithm is simple to implement and exhibits good convergence characteristic in our numerical examples.