Explain Gibbs phenomenon and States its configuration in FIR filter design.
1 Answer

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.

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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 $H_{d}(\omega)$

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