Top-hat filter

The Top-hat filter is several real-space or Fourier space filtering techniques. The name top-hat originates from the shape of the filter, which is a rectangle function, when viewed in the domain in which the filter is constructed.

Real space

In real-space the filter performs nearest-neighbour filtering, incorporating components from neighbouring y-function values. However, despite their ease of implementation their practical use is limited as the real-space representation of a top-hat filter is the sinc function, which has the often undesirable effect of incorporating non-local frequencies.

Fourier space

In Fourier space, a top hat filter selects a band of signal of desired frequency by the specification of a lower and upper bounding frequencies. Top-hat filters are particularly easy to implement digitally, and practically are only theoretically possible. Top-hat filters can be constructed by chaining theoretical low-band and high-band filters.

Related functions

The top hat function can be gnerated by differentiating a linear ramp function of width $epsilon$. The limit of $epsilon$ then becomes the delta function. It's real-space form is the same as the moving average, with the exception of not introducing a shift in the output function.

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