Exact formulation of the signal to noise ratio in continuous-time noise shaping A/D converters

Interest in multi-mode wireless system and software-defined radio has led to a need for high-speed high-resolution bandpass analog to digital (A/D) converters to digitalize signals near to the front end of a radio receiver. Such high-frequency applications require that the modulator be clocked at a high frequency, which in turn, requires the modulators´ loop filters to be built as continuous-time circuits (e.g. using LC resonators) rather than discrete-time circuits (e.g. switched capacitors). All the previous analysis of continuous-time bandpass ΣΔ modulators considered the assumption of having filter with an infinite quality factor (Q). This assumption is impractical especially with on-chip filters which leads to an approximate noise transfer function (an infinitely deep notch in the quantization noise) and approximate signal to noise ratio (SNR). In this paper, a more accurate z-domain loop transfer function is derived. This loop transfer function is used in finding the noise transfer function. The effect of the resonator Q in the depth of the notch on the noise transfer function is analyzed. Derivation of a mathematical expression for SNR, using the exact formula of the loop transfer function, is provided. Also, the dependence of the modulator´s SNR on the resonator Q is presented.