The transversal filter - Design in the time domain

T-filter belongs to the sampling filters ? it is indicated by that we can specify only the T-interval samples of the output signal, and this output values are compiled from the T-interval samples of the input signal. We can write the y(t) = x(t) * k(t) convolution connection written during the time domain analysis of the T-filter only defined to t = kt + t0 times:

Since the T-filter has 2N+1 free cn parameters, we can specify only 2N+1 from the output patterns independently of each other. Let they be the yk, (k=0,1 ... 2N) patterns. Then the 2N+1 linear equation in the above mentioned coherence would be the following written to a matrix format:

X . c = y

where the X2N+1 squared matrix contains the xl (|l|<=2N) samples of the input signal, at the i,j place samples of xi-j. It is practical to be chosen the maximum of the input signal for the main transversal element xi,i = x0, of the X matrix. Column vectors c and y contain the cn and yt values.

So generally we have to solve a 2N+1 linear equation system. However, if the other elements are small compared to the main transversal element of the matrix, approximative solution of the equation system can be determined easily. In case of this kind of input signal the samples specified by T-intervals can be set easily with an iteration method ? without solving the equation system. Setting algorithm is the following: we can set the a y N values with the c0 coefficient, and then yN-1 and y N+1 with c-1 and c1, respectivelly. After it we can set y N-2 and yN+2 with c-2 and c2, respectivelly, etc. Then we must go back to c0 and after that to c-1 and c1, etc. We have to repeat the setting series a few times because of the impact to each other.

This algorithm is followed by the manual settings, equalizing processes, and the principle of the automatic T-filtered data transmission equalizers is similar, too. If we connect the T-filter to the output of the network (line section) to be equalized, the response function of the network to be equalized to the elementary signal will be the x(t) input signal of the T-filter. With the help of the T-filter x(t) will be modified in critical moments of nT from the viewpoint of the signal lapstreak. /I. criterion of Nyquist/.