The Gerzon-Craven noise shaping theorem states that the ideal information capacity of a sigma delta modulator design is achieved if and only if the noise transfer function (NTF) is minimal phase. In this paper, it is found that there is a trade-off between the signal-to-noise ratio (SNR) and the information capacity of the noise shaped channel. In order to verify this result, loop filters satisfying and not satisfying the minimal phase condition of the NTF are designed via semi-infinite programming (SIP) techniques and solved using dual parameterization. Numerical simulation results show that the design with a minimal phase NTF achieves near the ideal information capacity of the noise shaped channel, but the SNR is low. On the other hand, the design with a non-minimal phase NTF achieves a positive value of the information capacity of the noise shaped channel, but the SNR is high. Results are also provided which compare the SIP design technique with Butterworth and Chebyshev structures and ideal theoretical SDMs, and evaluate the performance in terms of SNR and a variety of information theoretic measures which capture noise shaping qualities.
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