The tansversal filter - Design in frequency domain

Let our task be the approximation of the K(jω) transmission function specified in the 0...1/2T domain with a T-filter with 2N+1 pieces of weighting coefficients. Since the characteristic of the T-filter itself has a Fourier-line shape, too, the weighting coefficients of the T-filter approaching the specification with the smallest square error can be generated with a Fourier-line scaling. Permitting a constant difference in the running time of the specified K(jω):>

We can actually determine the weight function belonging to the specified characteristic with the Fourier-series method, then we can choose the patterns characterizing the appropriate weight function to be cn. Evaluation of the coherence mentioned above can be essentially simplified in numerous /e.g. clear real K(jω) /Figure T.4./.

During the Fourier-series approximation of torn functions (e.g. low-pass-like K(jω)) we may expect to important vibration because of the practically necessary line concision in the environment of tears and transitions. To reduce this so called Gibbs-oscillation, the characteristic should be modified in the places of tears, or in their environments.

We can do it with multiplying the chosen 2N+1 pattern of weight function with the appropriate chosen, the so called wn=w/nT/ smoothing weight function. Multiplication corresponds to K(jω) in the frequency domain, and to the convolution between narrow band functions W(jω)=F{w(t)}.

Relatively simple, but very effective smoothing function is the Hamming-window function:

Thus the coefficients to be set on the T-filter are products of the cn values determined by the coherence above, and the wk-s. Applying the smoothing functions we can reduce the waves of the realized characteristic, but - as it seems from the convolution - the gradient of the transition reduces, too. /Figure T.4./