﻿ Sinc filter

# Sinc filter

An ideal low-pass, as Si filter, sometimes referred to as Küpfmüller lowpass ( KTP) and in English often referred to as a sinc filter that is in the signal processing a low-pass filter having the ideal transfer function: Below a certain frequency limit of the bandwidth B all allowed to pass through the filter frequency components above the cutoff frequency B all frequency components are blocked. The ideal low-pass plays as a model in the filter theory an important role, for example in the Nyquist -Shannon sampling theorem.

An ideal low-pass filter is good in theory be written, in practice, not feasible. This is due to the non-causal and infinitely long impulse response. An ideal low-pass filter at the output of the filter is already a reaction before the triggering signal is applied to the filter input. Reliable low-pass filter therefore approach the transfer function of the ideal low-pass only, but never reach it, because they would have an infinite group delay otherwise.

## Transfer function

Mathematically, the transfer function of the ideal low-pass filter in the continuous time case, the square-wave function and the bandwidth B and the parameters of the frequency f can be described as:

In the time domain is obtained as the inverse Fourier transform of the transfer function of the impulse response h (t):

The thereby occurring function si is also called sinc function or sinc function, of which the term sinc filter is derived for continuous-time filter.

In the discrete-time case, for example, in digital signal processing, are changing over time and the spectrum no continuous gradients, but discrete episodes. The rectangular function shown above is replaced by a finite rectangle sequence with N spectral points. The inverse Fourier transform is replaced by the inverse discrete Fourier transform ( IDFT ). So that in the then time-discrete pulse response h ( x) passes through the periodic extension in the time domain does not si - function, but a Dirichlet kernel. This is also known as di - function in accordance with the sinc function, and defined as follows:

Occasionally ideal low-pass filter are referred to as sinc filter in the specialist literature and in most cases the discrete-time case.

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