DESIGN OF COMPLEX DIGITAL FIR FILTERS IN THE CHEBYSHEV SENSE

Lina  J. KARAM

Algorithms are developed for the efficient design and realization of finite‑duration impulse response (FIR) filters with arbitrary magnitude and phase responses. The emphasis is on minimizing and controlling the Chebyshev error norm. For that purpose, new practical characterization theorems and optimality criteria for complex Chebyshev approximation are presented and proved. In particular, the alternation theorem of Chebyshev approximation is extended from the real‑only to the complex case.

In the case of one‑dimensional filters, it was shown how to reformulate the complex filter design problem so that the second Remez exchange algorithm can be used in computing the optimal Chebyshev approximation to the desired complex‑valued frequency response. As a result, a very flexible and efficient multiple‑exchange algorithm is presented for designing optimal one‑dimensional FIR filters with arbitrary magnitude and phase specifications and, more generally, for computing the best Chebyshev approximation of a complex‑valued function. The new algorithm can design both causal and non‑causal FIR filters with complex‑ or real‑valued impulse responses. It was also shown how to modify the proposed algorithm for the design of optimal multi‑dimensional FIR filters with complex or real frequency responses.

The applicability and limitations of the McClellan transformation method are discussed. It is shown how new types of one‑dimensional filters can be transformed into two‑dimensional filters, and how all the possible types of symmetries can be designed by the transformation method. The result is that very efficient transformation procedures are developed for the design of complex or real, positive‑ or negative‑ symmetric, two‑dimensional filters with a rectangular region of support having either odd‑ or even‑length sides.

Using transformation functions, it is shown how a complete set of filters can be designed by regarding it as a single multi‑dimensional filter. Consequently, a very efficient technique is presented for designing a complete set of one‑ and multi‑dimensional filters by transforming a single one‑dimensional prototype filter. The new design technique is very fast since it reduces the design of a large set of one‑ and multi‑dimensional filters to the design of a single one‑dimensional filter and, possibly, a low‑order two‑dimensional transformation function. It is shown how the designed filters can be realized efficiently without having to explicitly compute the coefficients of the separate filters in the set. Among the types of filters, which can be designed by this transformation, techniques are depth migration filters and linear‑phase filters.