Noise gain expressions for low-noise second-order digital filter structures

J.H.F. Ritzerfeld

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Abstract

The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power q2/12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.
Original languageEnglish
Pages (from-to)223-227
Number of pages5
JournalIEEE Transactions on Circuits and Systems. II, Analog and Digital Signal Processing
Volume52
Issue number4
DOIs
Publication statusPublished - 2005

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Digital filters
Transfer functions
Poles
Polynomials

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abstract = "The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power q2/12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.",
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Noise gain expressions for low-noise second-order digital filter structures. / Ritzerfeld, J.H.F.

In: IEEE Transactions on Circuits and Systems. II, Analog and Digital Signal Processing, Vol. 52, No. 4, 2005, p. 223-227.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Noise gain expressions for low-noise second-order digital filter structures

AU - Ritzerfeld, J.H.F.

PY - 2005

Y1 - 2005

N2 - The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power q2/12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.

AB - The quantization noise of a fixed-point digital filter is commonly expressed in terms of its noise gain, i.e., the factor by which the noise power q2/12 of a single quantizer is amplified to the output of the filter. In this brief, first a closed-form expression for the optimal second-order noise gain in terms of the coefficients of the numerator and denominator polynomials of the transfer function is derived. It is then shown, by deriving a similar expression for its noise gain, that the second-order direct form structure has an arbitrarily larger noise gain the closer the filter poles are to the unit circle. The main result, however, is that the wave digital form and the normal form structures have noise gains which are only marginally larger than the minimum gain. For these forms, the expressions for their noise gain in terms of the transfer function are given as well. The importance of these forms lies in the fact that they use less multipliers than the optimal structure and that they are much easier to design: properly scaled forms are given requiring no design tools.

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JO - IEEE Transactions on Circuits and Systems. II, Analog and Digital Signal Processing

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