### Abstract

An exact formula for the output noise spectrum of a double-loop sigma-delta modulator 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. All results assume that the quantizer does not overload, and hence apply only to multi-bit coders. 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}(w/2) envelope.

Original language | English (US) |
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Title of host publication | Midwest Symposium on Circuits and Systems |

Publisher | Publ by IEEE |

Pages | 216-219 |

Number of pages | 4 |

ISBN (Print) | 0780317610 |

State | Published - 1993 |

Event | Proceedings of the 36th Midwest Symposium on Circuits and Systems - Detroit, MI, USA Duration: Aug 16 1993 → Aug 18 1993 |

### Publication series

Name | Midwest Symposium on Circuits and Systems |
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Volume | 1 |

### Other

Other | Proceedings of the 36th Midwest Symposium on Circuits and Systems |
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City | Detroit, MI, USA |

Period | 8/16/93 → 8/18/93 |

### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Electrical and Electronic Engineering

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## Cite this

*Midwest Symposium on Circuits and Systems*(pp. 216-219). (Midwest Symposium on Circuits and Systems; Vol. 1). Publ by IEEE.