# A new set of orthogonal functions and its application to the analysis of dynamic systems

TL;DR: It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution.

Abstract: The present work proposes a complementary pair of orthogonal triangular function (TF) sets derived from the well-known block pulse function (BPF) set. The operational matrices for integration in TF domain have been computed and their relation with the BPF domain integral operational matrix is shown. It has been established with illustration that the TF domain technique is more accurate than the BPF domain technique as far as integration is concerned, and it provides with a piecewise linear solution. As a further study, the newly proposed sets have been applied to the analysis of dynamic systems to prove the fact that it introduces less mean integral squared error (MISE) than the staircase solution obtained from BPF domain analysis, without any extra computational burden. Finally, a detailed study of the representational error has been made to estimate the upper bound of the MISE for the TF approximation of a function f ( t ) of Lebesgue measure.

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##### Citations

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### Cites background or methods from "A new set of orthogonal functions a..."

...From the definition of TFs, it is clear that triangular functions are disjoint, orthogonal, and complete [9]....

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...[9] and studied and used by Babolian et al....

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...Operational matrix of integration Expressing ∫ s 0 T1(τ )dτ and ∫ s 0 T2(τ )dτ in terms of the triangular functions follows [9] ∫ s...

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64 citations

50 citations

### Cites background from "A new set of orthogonal functions a..."

...Orthogonality of 1D-TFs is shown in [15], that is, ∫ 1...

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...[15] integrate each element of T1(s) and T2(s), and express the results in terms of 1D-TF vector, ∫ s...

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...A review of one-dimensional triangular functions One-dimensional triangular functions were introduced by Deb et al. (2006) in [15]....

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41 citations

### Cites background from "A new set of orthogonal functions a..."

...For more details about triangular orthogonal functions, see [1]....

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...The main body of Section 2 is based on [1]....

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...The authors of [1] called T1ðtÞ the left-handed triangular functions (LHTF) and T2ðtÞ the right-handed triangular functions (RHTF)....

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