Sigma approximation
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.
Animation of the additive synthesis of a square wave with an increasing number of harmonics by way of the σ-approximation with p=1
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, 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
(
θ
)
=
1
2
a
0
+
∑
k
=
1
m
−
1
(
sinc
k
m
)
p
⋅
[
a
k
cos
(
2
π
k
T
θ
)
+
b
k
sin
(
2
π
k
T
θ
)
]
,
{\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}\cdot \left[a_{k}\cos \left({\frac {2\pi k}{T}}\theta \right)+b_{k}\sin \left({\frac {2\pi k}{T}}\theta \right)\right],}
in terms of the normalized sinc function:
sinc
x
=
sin
π
x
π
x
.
{\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.}
a
k
{\displaystyle a_{k}}
and
b
k
{\displaystyle b_{k}}
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
(
sinc
k
m
)
p
{\displaystyle \left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}}
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
sinc
{\displaystyle \operatorname {sinc} }
function to rolloff the higher frequency Fourier Series coefficients.
As is known by the uncertainty principle, 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 (equivalent to large amounts of ringing).
This can also be understood as applying a window function 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).
- Lanczos resampling
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