mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, σ-approximation adjusts a Fourier summation to greatly reduce the

Gibbs phenomenon In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The Nth partial Fourier series of the function (formed by summing ...

, which would otherwise occur at discontinuities. An ''m-1''-term, σ-approximated summation for a series of period ''T'' can be written as follows:

s(\theta) = \frac a_0 + \sum_^ \left(\operatorname \frac\right)^ \cdot \left_ \cos \left( \frac \theta \right) + b_k \sin \left( \frac \theta \right) \right

in terms of the normalized

sinc function In mathematics, physics and engineering, the sinc function ( ), denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatorname(x) = \frac. Alternatively, ...

\operatorname x = \frac.

a_

and

b_

are the typical Fourier Series coefficients, and ''p'', a non negative parameter, determines the amount of smoothening applied, where higher values of ''p'' further reduce the Gibbs phenomenon but can overly smoothen the representation of the function. The term

\left(\operatorname \frac\right)^

is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the

\operatorname

function to rolloff the higher frequency Fourier Series coefficients. As is known by the

Uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...

, having a sharp cutoff in the frequency domain (cutting off the Fourier Series abruptly without adjusting coefficients) causes a wide spread of information in the time domain (lots of ringing). This can also be understood as applying a

Window function In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval. Typically, window functions are symmetric around ...

to the Fourier series coefficients to balance maintaining a fast rise time (analogous to a narrow transition band) and small amounts of ringing (analogous to stopband attenuation).

References

Fourier series Numerical analysis {{mathanalysis-stub

See also

References