Communicated by J. BanasiakFourier analysis plays a vital role in the analysis of continuous-time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous-time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform.
The classical Discrete Fourier Transform (DFT) satisfies a duality property that transforms a discrete time signal to the frequency domain and back to the original domain. In doing so, the original signal is reversed to within a multiplicative factor, namely the dimension of the transformation matrix. In this paper, we prove that the DFT based on Simpson's method satisfies a similar property and illustrate its effect on a real discrete signal. The duality property is particularly useful in determining the components of the transformation matrix as well as components of its positive integral powers. Copyright
Mutagenesis had no effect on number of stalks/plot, stalk height, fibre and sucrose content of mutants. Imazapyr tolerance is likely due to a S622N mutation in the acetolactate synthase gene. The herbicidal compound imazapyr is effective against weeds such as Cynodon and Rottboellia species that constrain sugarcane production. This study aimed to compare agronomic characteristics of three imazapyr tolerant mutants (Mut 1, Mut 6 and Mut 7) with the non-mutated N12 control after 18 months of growth, and to sequence the acetolactate synthase (ALS) gene to identify any point mutations conferring imazapyr tolerance. There were no significant differences in the number of stalks/plot, stalk height, fibre and sucrose contents of the mutants compared with the N12 control. However, Mut 1 genotype was more susceptible to the Lepidopteran stalk borer, Eldana saccharina when compared with the non-mutated N12 (11.14 ± 1.37 and 3.89 ± 0.52% internodes bored, respectively), making Mut 1 less desirable for commercial cultivation. Molecular characterisation of the ALS gene revealed non-synonymous mutations in Mut 6. An A to G change at nucleotide position 1857 resulted in a N513D mutation, while a G to A change at nucleotide position 2184 imposed a S622N mutation. Molecular dynamics simulations revealed that the S622N mutation renders an asparagine side chain clash with imazapyr, hence this mutation is effective in conferring imazapyr tolerance.
Communicated by J. BanasiakThe Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform.
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