Sparse Representation for Sampled-Data $$H^\infty $$ H ∞ Filters

We consider the problem of discretization of analog filters and propose a novel method based on sampled-data $$H^\infty $$ H ∞ control theory with sparse representation. For optimal discretization, we adopt minimization of the $$H^\infty $$ H ∞ norm of the error system between a (delayed) target analog filter and a digital system consisting of an ideal sampler, the zero-order hold, and an FIR (finite impulse response) filter. Also, for digital implementation, we propose a sparse representation of the FIR filter to reduce the number of nonzero coefficients with the $$\ell ^1$$ ℓ 1 norm regularization. We show that this multi-objective optimization is reducible to a convex optimization problem, which can be solved efficiently by numerical computation. We then extend the design method to multi-rate filters, and show a design example. We also give an application to the feedback filter design of delta-sigma modulators.