### Abstract

An exact formula for the output noise spectrum of a double-loop sigma-delta modulator, under the no overloading assumption and with a sinusoidal input, is derived without the use of a white-noise model. In the case of a sinusoidal input with irrational input amplitude and digital frequency, the result agrees with the exact formula derived by ergodic theory for two-stage modulators. In addition, the present method also provides an exact formula for a sinusoidal inputs with rational frequency and amplitude. Furthermore, the period of the output with rational initial conditions and dc input is also calculated. The results are of primary interest to multibit sigma-delta modulators, which do not overload over the entire input amplitude range. The ergodic theory method for calculating the exact noise spectrum involves explicitly determining the autocorrelation of the internal quantization error with ergodic theory techniques, and then determining the noise spectrum from the correlation function. The present method, however, directly determines the quantization noise spectrum by using an open-loop model for the coder and applying a Fourier series representation of the quantization error function. The result of both of these methods is that the output noise spectrum for a sinusoidal input is composed of discrete spectral lines shaped by a sin^{4}(ω/2) envelope.

Original language | English (US) |
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Pages (from-to) | 168-173 |

Number of pages | 6 |

Journal | IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1994 |

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### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering