Minimax and WLS Designs of Digital FIR Filters Using SOCP for Aliasing Errors Reduction in BI-DAC

Abstract

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.