Arbitrary amplitude and linear phase approximations for non-prototype ladder and lattice wave digital filters

Abstract

Two different approaches are introduced for the design of non‐prototype ladder and lattice wave digital filters (WDFs) exhibiting arbitrary amplitude in the baseband (passband, transition band and stopband) and linear phase in the passband. The first approach is based on the phase correction of a minimum phase lattice (or ladder) WDF designed to satisfy the amplitude specifications in the three bands. In the second approach the amplitude and phase requirements are approximated simultaneously. It is devoted to the design of a lattice WDF that is constructed from the parallel arrangement of two allpass subfilters. The design procedure relies on preconstructing one of the subfilters to have exact linear phase at all frequencies and constructing the other to interpolate an arbitrary phase at a set of frequencies distributed all over the baseband. The hidden relationship between the amplitude and phase functions of the filter is utilized to approximate both of them. The approximation problem is solved by applying an interpolation technique combined with the Remez exchange algorithm. Prototype filters with amplitude specifications in the passband and stopband and phase specifications in the passband are also considered as special cases. Design examples are presented to show the efficiency of the two methods.

New Simultaneous Approximation for Wave Digital Lattice Filters Based on the Alternative and Iterative Generation of the Two Branch Polynomials

“…Now, we will compare the delivered method with previous approximation methods [4][5][6][7] 1-The methods of references [4][5][6] are relying on the restriction of one of the two branch polynomials to exhibit exact linear phase. The other polynomial is employed to control both the amplitude and phase.…”

“…Consequently, Equations (5,6) express the amplitude and phase functions A() and B() in terms of the two branch polynomials phase functions  1 () and  2 (). On the other hand, Equations (7,8) express the two branch polynomials phase functions  1 () and  2 () in terms of the amplitude and phase functions A() and B().…”

“…In literature [1][2][3], several design methods are available for wave digital lattice filters with only amplitude restrictions. For simultaneously specified amplitude and phase, some approximation methods have been established [4][5][6]. The common feature of these methods is the restriction of one of the two branch allpass functions to exhibit exact linear phase.…”

On the design of multiband transmission functions synthesized by one wave digital lattice structure

“…Determine also if the polynomial h(c) will be odd or even and consequently, f(c) will be even or odd, respectively. Accordingly, determine which one of Equations (20) and (22) will be applied. 2.…”

On the simultaneous amplitude and phase approximations of wave digital lattice filters

“…For lowpass cases, which are considered in this contribution, optimal and e cient amplitude approximation methods are available in the literature [4][5][6]. For simultaneously approximating the amplitude and phase functions, some design methods have been delivered [7][8][9] for wave digital lattice structures. In fact, any of these methods cannot be considered the optimal solution for the problem.…”