Daniel J. Alfsmann

A novel approach to the design of oversampling low-delay complex-modulated filter bank pairs

In this contribution we present a method to design prototype filters of oversampling uniform complex-modulated FIR filter bank pairs. Especially, we present a noniterative two-step procedure: (i) design of analysis prototype filter with minimum group delay and approximately linear-phase frequency response in the passband and the transition band and (ii) Design of synthesis prototype filter such that the filter bank pairs distortion function approximates a linear-phase allpass function. Both aliasing and imaging are controlled by introducing sophisticated stopband constraints in both steps. Moreover, we investigate the delay properties of oversampling uniform complex-modulated FIR filter bank pairs in order to achieve the lowest possible filter bank delay. An illustrative design example demonstrates the potential of the design approach.

Determination of the Biquaternion Divisors of Zero, Including the Idempotents and Nilpotents

Abstract.The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.

Efficient linear-phase directional filters with selectable centre frequencies

We consider complex-valued digital directional filters that extract two freely selectable user signals from an FDM signal, which comprises up to four independent user spectra uniformly allocated along the frequency axis from zero frequency up to the sampling rate. Two different approaches, both applying linear-phase FIR halfband filters, are investigated and compared with each other in terms of computational efficiency. The first approach is based on a complex-modulated frequency demultiplexer filter bank (FDMUX); it requires the least number of delay elements. The second approach combines two complex offset halfband filters (COHBF) that allow for the exploitation of coefficient symmetry; hence, the computation required is minimal. Finally, the dual directional filter effectuating frequency multiplexing is derived by transposition of the FDMUX and COHBF approaches.