Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes $A$ and $B$ must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than Schur equality: that sA and sB have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.

Parallel-recusive filter structures for the computation of discrete transforms

A general approach is presented for implementing discrete transforms as a set of first-order or second-order recursive digital filters. Clenshaw's recurrence formulae are used to formulate the second-order filters. The resulting structure is suitable for efficient implementation of discrete transforms in VLSI or FPGA circuits. The general approach is applied to the discrete Legendre transform as an illustration.

Efficient computation of discrete polynomial transforms

This letter presents a new recursive method for computing discrete polynomial transforms. The method is shown for forward and inverse transforms of the Hermite, binomial, and Laguerre transforms. The recursive flow diagrams require only 2 additions, 2( +1) memory units, and +1multipliers for the +1-point Hermite and binomial transforms. The recursive flow diagram for the +1-point Laguerre transform requires 2 additions, 2( +1) memory units, and 2( +1) multipliers. The transform computation time for all of these transforms is ( )

Computation of discrete cosine transform using Clenshaw's recurence formula

Clenshaw’s recurrenee formula is used to derive recursive algorithms for the discrete cosine transform @CT) and the inverse discrete cosine transform (IDCT). The recursive DCT algorithm presented here requires one fewer delay element per coefficient and one fewer multiply operation per coeflident compared with two recently proposed methods. Clenshaw’s recurrence formula provides a unified development for the recursive DCT and IDCT algorithms. The M v e al gorithms apply to arbitrary lengtb algorithms and are appropriate for VLSI implementation.

Efficient implementation of discrete Pascal transform using difference operators

An alternative way is provided to define the discrete Pascal transform using difference operators to reveal the fundamental concept of the transform, in both one- and two-dimensional cases, which is extended to cover non-square two-dimensional applications. Efficient modularised implementations are proposed.

Digital signal processing and digital system design using discrete cosine transform [student course]

The discrete cosine transform (DCT) is an important functional block for image processing applications. The implementation of a DCT has been viewed as a specialized research task. We apply a micro-architecture based methodology to the hardware implementation of an efficient DCT algorithm in a digital design course. Several circuit optimization and design space exploration techniques at the register-transfer and logic levels are introduced in class for generating the final design. The students not only learn how the algorithm can be implemented, but also receive insights about how other signal processing algorithms can be translated into a hardware implementation. Since signal processing has very broad applications, the study and implementation of an extensively used signal processing algorithm in a digital design course significantly enhances the learning experience in both digital signal processing and digital design areas for the students.

Application of Discrete Fourier Inter-Coefficient Difference for Assessing Genetic Sequence Similarity

Digital signal processing (DSP) techniques for biological sequence analysis continue to grow in popularity due to the inherent digital nature of these sequences. DSP methods have demonstrated early success for detection of coding regions in a gene. Recently, these methods are being used to establish DNA gene similarity. We present the inter-coefficient difference (ICD) transformation, a novel extension of the discrete Fourier transformation, which can be applied to any DNA sequence. The ICD method is a mathematical, alignment-free DNA comparison method that generates a genetic signature for any DNA sequence that is used to generate relative measures of similarity among DNA sequences. We demonstrate our method on a set of insulin genes obtained from an evolutionarily wide range of species, and on a set of avian influenza viral sequences, which represents a set of highly similar sequences. We compare phylogenetic trees generated using our technique against trees generated using traditional alignment techniques for similarity and demonstrate that the ICD method produces a highly accurate tree without requiring an alignment prior to establishing sequence similarity.