Linear programming design of semi-digital FIR filter and ΣΔ modulator for VDSL2 transmitter

An oversampled digital-to-analog converter including digital ΣΔ modulator and semi-digital FIR filter can be employed in the transmitter of the VDSL2 technology. To select the optimum set of coefficients for the semi-digital FIR filter, an integer optimization problem is formulated in this work, where the model includes the FIR filter magnitude metrics as well as ΣΔ modulator noise transfer function. The semi-digital FIR filter is optimized with respect to magnitude constraints according to the International Telecommunication Union Power Spectral Density mask for VDSL2 technology and minimizing analog cost as the objective function. Utilizing the semi-digital FIR filter with one bit DACs, high linearity required in high-bandwidth profiles of VDSL2, can be achieved. The resolution of the conventional DACs are limited by the mismatch between DAC unit elements. By utilizing one-bit DACs in semi-digital FIR filter, there will be less degradation caused by mismatch between unit elements. The optimization problem is solved in two conditions; fixed passband gain and variable passband gain. It is shown in this paper that 38% saving in total number of unit elements can be achieved by employing variable passband gain in the optimization problem.