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Fourier Series Equations

Data which contains sinusoidal trends is not particularly well fitted by most polynomials or rationals. On the other hand, a Fourier series fit may be just what is needed. Like the Chebyshev equations, the Fourier Series models are scaled. Here the x is mapped from 0 to p.

The Fourier series equations are as follows:

Polynomials: 1x2 through 10x2, Equations 6841-6850

  • 6841 y=a+bcos(x)+csin(x)
  • 6842 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)
  • 6843 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)
  • 6844 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)
  • 6845 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)
  • 6846 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)+lcos(6x)+msin(6x)
  • 6847 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)+lcos(6x)+msin(6x)+ncos(7x)+osin(7x)
  • 6848 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)+lcos(6x)+msin(6x)+ncos(7x)+osin(7x)+pcos(8x)+qsin(8x)
  • 6849 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)+lcos(6x)+msin(6x)+ncos(7x)+osin(7x)+pcos(8x)+qsin(8x)+rcos(9x)+ssin(9x)
  • 6850 y=a+bcos(x)+csin(x)+dcos(2x)+esin(2x)+fcos(3x)+gsin(3x)+hcos(4x)+isin(4x)+jcos(5x)+ksin(5x)+lcos(6x)+msin(6x)+ncos(7x)+osin(7x)+pcos(8x)+qsin(8x)+rcos(9x)+ssin(9x)+tcos(10x)+usin(10x)

*Rationals: 1x2/1x2 through 5x2/5x2 order, Equations 7641-7645*

  • 7641 y=(a+ccos(x)+esin(x))/(1+bcos(x)+dsin(x))
  • 7642 y=(a+ccos(x)+esin(x)+gcos(2x)+isin(2x))/(1+bcos(x)+dsin(x)+fcos(2x)+hsin(2x))
  • 7643 y=(a+ccos(x)+esin(x)+gcos(2x)+isin(2x)+kcos(3x)+msin(3x))/(1+bcos(x)+dsin(x)+fcos(2x)+hsin(2x)+jcos(3x)+lsin(3x))
  • 7644 y=(a+ccos(x)+esin(x)+gcos(2x)+isin(2x)+kcos(3x)+msin(3x)+ocos(4x)+qsin(4x))/(1+bcos(x)+dsin(x)+fcos(2x)+hsin(2x)+jcos(3x)+lsin(3x)+ncos(4x)+psin(4x))
  • 7645 y=(a+ccos(x)+esin(x)+gcos(2x)+isin(2x)+kcos(3x)+msin(3x)+ocos(4x)+qsin(4x)+scos(5x)+usin(5x))/(1+bcos(x)+dsin(x)+fcos(2x)+hsin(2x)+jcos(3x)+lsin(3x)+ncos(4x)+psin(4x)+rcos(5x)+tsin(5x))

Note that in the case of a mixture of overlapping waveforms of different frequencies, these models are no more capable of zeroing in on the individuals frequencies than an FFT is. Such mixtures must be fitted to a non-linear sum of sine wave functions. For empirical data with a periodic character, however, these models may prove near-ideal approximating functions.

Although the Fourier series equations are scaled models like the Chebyshev, they do not appear to share this same near-immunity to ill-conditioning. As such, you may observe a breakdown in the computation of confidence or prediction intervals with the high-coefficient count models.

The rational models consist of a ratio of equal coefficient count Fourier series polynomials. Since the sine-cosine pairs are preserved, there are only 10 Fourier series polynomials and 5 Fourier series rationals.