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written 4.6 years ago by |

In FIR filter design, Desired Impulse Response $h_{d}(n)$ is generally infinite in length. It is made finite by truncating it with a window function. In the time domain, truncation is achieved by multiplying the impulse response $h_{d}(n)$ with a window function w(n).

$\therefore$ Finite Impulse Response $h(n)=h_{d}(n) \times w(n),$

where $w(n)$ is a finite length window that is equal to zero outside the interval $0 \leq n \leq M-1,$ where **M** is the filter length.

Truncating the impulse response introduces undesirable ripples and overshoots in the frequency response. This effect is known as the Gibb's phenomenon.

The Gibbs phenomenon effect manifests itself as a fixed percentage overshoot and ripple before and after an approximated discontinuity in the frequency response. Gibbs phenomenon occurs due to the non-uniform convergence of the Fourier series at a discontinuity. Thus, the frequency response so obtained contains ripples in the frequency domain.

With respect to Gibbs phenomenon following observation are made:

(i) As filter length M increases, the number of ripples increases but the ripple width decreases.

(ii) The height of the largest ripples remains constant, regardless of the filter length M.

(iii) As M increases, the height of all other ripples decreases. The main lobe gets narrower as M increases, that is, the drop-off becomes sharper.

Similar oscillatory behavior is seen in all types of truncated filters.

A rectangular window, which has abrupt drop from pass band to stop band, causes more ripple effect. In order to reduce the ripples, $h_{d}(n)$ should be multiplied with a window function that contains a taper and decays toward zero gradually.

As multiplication of sequences $h_{d}(n)$ and $w(n)$ in time domain is equivalent to convolution of $H_{d}(\omega)$ and $W(\omega)$ in the frequency domain, $W(\omega)$ has the smoothing effect on $H_{d}(\omega)$