Sinc filter

In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase. The filter's impulse response is a sinc function in the time domain, and its frequency response is a rectangular function.

Realistic filters can only approximate this ideal, since an ideal sinc filter (aka "rectangular filter") has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the sampling theorem and the Whittaker–Shannon interpolation formula.

In mathematical terms, the desired frequency response is the rectangular function:

:$H(f)\; =\; mathrm\{rect\}\; left(\; frac\{f\}\{2B\}\; ight)$

where $B,$ is an arbitrary cutoff frequency (aka "bandwidth") (in Hz). The impulse response of such a filter is given by the inverse Fourier transform:

:

Interestingly enough, the name sinc filter is applied also to the filter shape that is rectangular in time and sinx/x (sinc) in frequency. How to tell which version is being used? As the sinc-in-time filter has infinite impulse response and must be approximated for real-world applications, it's often a "windowed sinc filter." Sinc-in-frequency filters, among many other applications, are almost universally used for decimating Sigma-Delta ADCs, as they are easy to implement and nearly optimum for this use. Papers on Sigma-Delta ADCs [cite journal | title=Time domain analysis of sigma delta modulation | author=Chou, W.; Meng, T.H.; Gray, R.M. | journal=Acoustics, Speech, and Signal Processing | year=1990 |pages=1751-1754 | volume=3 ] will use the second version.

References

See also

*Lanczos resampling
*Aliasing
*Anti-aliasing
*Whittaker–Shannon interpolation formula