By Mücahit Kozak
The research of the quantization noise in delta-sigma modulators isn't really a trivial job. state of the art research tools comprise modelling the quantization noise as a uniform dispensed white noise. in spite of the fact that, it is common to watch restrict cycle oscillations and tones on the output of a delta-sigma modulator. In many of the functions, those restrict cycles and tones are strictly objectionable. Such an program, for example, is a Fractional-N PLL frequency synthesizer, the place idle tones and restrict cycles generated from the delta-sigma modulator without delay seem within the synthesized RF waveform as spurious parts. The particularly small conversion bandwidth is one other vital hassle of delta-sigma modulators. because of their oversampling nature, delta-sigma modulators were utilized in low frequency functions. Oversampled Delta-Sigma Modulators: research, functions, and Novel Topologies offers theorems and their mathematical proofs for the precise research of the quantization noise in delta-sigma modulators. large mathematical equations are integrated through the publication to investigate either single-stage and multi-stage architectures. it's been proved that properly set preliminary stipulations generate tone unfastened output, only if the modulator order is at the least 3. those effects are utilized to the layout of a Fractional-N PLL frequency synthesizer to provide spurious unfastened RF waveforms. moreover, the publication additionally offers time-interleaved topologies to extend the conversion bandwidth of delta-sigma modulators. The topologies were generalized for any interleaving quantity and modulator order. Oversampled Delta-Sigma Modulators: research, purposes, and Novel Topologies is stuffed with layout and research innovations. The e-book includes adequate element that permits readers with little history within the topic to simply keep on with the cloth in it. Oversampled Delta-Sigma Modulators: research, purposes, and Novel Topologies should be of curiosity to graduate scholars, researchers, and practicing circuit designers within the parts of delta-sigma dependent info converters and Fractional-N PLL frequency synthesizer layout.
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Extra resources for Oversampled Delta-Sigma Modulators: Analysis, Applications and Novel Topologies
2 C + -y ? 2 ANALYSIS O F SINGLE-STAGE DELTA-SIGMA MODULATORS 94 ANALYSIS O F SINGLE-STAGE DELTA-SIGMA MODULATORS 95 We now define new labels for the accumulator outputs within the (L + -order modulator as Y-, = V, _, and b, _, = v,_, where I = 1,2,.... L - I , and 1)lh finally bL, = b L i l , . 23) and hence the proof is completed. 59), the output of the (L + 1)'" -order modulator is given by In this section, we determine the criterion that ensures the no-overload operation of the quantizer within an L'"-order modulator.
This suggests that le[tl]l< A/2 for all n < n,). 89). 84) satisfies the real solution for 17 = 0 and n = 1 . 84) is true for any n 2 2 . 66), the normalized quantizer error sequence ~ [ n at] n + 1 can be written as which proves the theorem. 6 employing L-bit quantization with output levels +b . , l x [ n ] lA/2 ~ for all n ), where A = 26/(2' - I ) . , lv, [n]l 5 2'-'A and Je[n]5 ~ / 2 )for all n ). 106) Using the induction hypothesis. 97), where L = N . 112) states that L'" -order differentiation of an ,v'"-order summation is zero, when the order of the differentiation is greater than that of the summation.
Additionally, the quantizer error sequence is uncorrelated with the dc input. To see this, we write the cross-correlation between the input and the quantizer error sequence in an L"' -order MASH modulator as I . E, ) . 90), it is concluded that v,,,. ( I ) = o for all lag indices I . To support our theory, several simulation results were also presented. Specifically, the first and second-order moments of the quantizer error sequence in first and higher-order AC modulators were simulated and compared against their corresponding analytic solutions.