A300727 Decimal expansion of the total harmonic distortion (THD) of the sawtooth signal filtered by a 2nd-order low-pass filter.

1, 8, 1, 1, 4, 1, 6, 1, 3, 7, 9, 3, 8, 2, 9, 0, 0, 4, 0, 8, 0, 2, 1, 8, 1, 0, 5, 5, 8, 1, 3, 0, 1, 6, 7, 8, 4, 4, 3, 8, 9, 2, 8, 3, 5, 1, 5, 9, 5, 6, 3, 5, 3, 8, 9, 1, 1, 5, 5, 6, 0, 6, 0, 8, 6, 2, 6, 4, 1, 4, 1, 9, 5, 6, 3, 6, 7, 9, 2, 4, 7, 3, 1, 6, 9, 8, 0, 7, 9, 1, 7, 9, 2, 7, 4, 4, 1, 6, 2, 1, 2, 2, 4

Offset

0,2

Comments

See formula (34) in Blagouchine & Moreau link.

Links

Table of n, a(n) for n=0..102.

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.

Formula

Equals sqrt(Pi*(cot(Pi/sqrt(2))*coth(Pi/sqrt(2))^2-cot(Pi/sqrt(2))^2*coth(Pi/sqrt(2))-cot(Pi/sqrt(2))-coth(Pi/sqrt(2)))/((cot(Pi/sqrt(2))^2+coth(Pi/sqrt(2))^2)*sqrt(2))+(1/3)*Pi^2-1).

Example

0.1811416137938290040802181055813016784438928351595635...

Maple

evalf(sqrt(Pi*(cot(Pi/sqrt(2))*coth(Pi/sqrt(2))^2-cot(Pi/sqrt(2))^2*coth(Pi/sqrt(2))-cot(Pi/sqrt(2))-coth(Pi/sqrt(2)))/((cot(Pi/sqrt(2))^2+coth(Pi/sqrt(2))^2)*sqrt(2))+(1/3)*Pi^2-1), 120)

Mathematica

RealDigits[Sqrt[Pi*(Cot[Pi/Sqrt[2]]*Coth[Pi/Sqrt[2]]^2-Cot[Pi/Sqrt[2]]^2*Coth[Pi/Sqrt[2]]-Cot[Pi/Sqrt[2]]-Coth[Pi/Sqrt[2]])/((Cot[Pi/Sqrt[2]]^2+Coth[Pi/Sqrt[2]]^2)*Sqrt[2])+(1/3)*Pi^2-1], 10, 120][[1]]

Prog

(MATLAB) format long; sqrt(sqrt(pi*(cot(pi/sqrt(2))*coth(pi/sqrt(2))^2-cot(pi/sqrt(2))^2*coth(pi/sqrt(2))-cot(pi/sqrt(2))-coth(pi/sqrt(2)))/((cot(pi/sqrt(2))^2+coth(pi/sqrt(2))^2)*sqrt(2))+(1/3)*pi^2-1))
(PARI) s2=sqrt(2);
A=Pi/s2;
B=1+2/(exp(2*A)-1)
C=1/tan(A);
sqrt(Pi*(B^2*C-B*C^2-C-B)/((C^2+B^2)*s2) + Pi^2/3 - 1) \\ Charles R Greathouse IV, Mar 11 2018

Crossrefs

Cf. A247719 (Pi/sqrt(2)), A300690, A300713, A300714.

Sequence in context: A133421 A128881 A154015 * A246722 A197492 A153621

Adjacent sequences: A300724 A300725 A300726 * A300728 A300729 A300730

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Mar 11 2018

Status

approved