Quantization Noise Shaping on Arbitrary Frame Expansions

  • PDF / 689,875 Bytes
  • 12 Pages / 600.03 x 792 pts Page_size
  • 97 Downloads / 179 Views

DOWNLOAD

REPORT


Quantization Noise Shaping on Arbitrary Frame Expansions Petros T. Boufounos and Alan V. Oppenheim Digital Signal Processing Group, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 36-615, Cambridge, MA 02139, USA Received 2 October 2004; Revised 10 June 2005; Accepted 12 July 2005 Quantization noise shaping is commonly used in oversampled A/D and D/A converters with uniform sampling. This paper considers quantization noise shaping for arbitrary finite frame expansions based on generalizing the view of first-order classical oversampled noise shaping as a compensation of the quantization error through projections. Two levels of generalization are developed, one a special case of the other, and two different cost models are proposed to evaluate the quantizer structures. Within our framework, the synthesis frame vectors are assumed given, and the computational complexity is in the initial determination of frame vector ordering, carried out off-line as part of the quantizer design. We consider the extension of the results to infinite shift-invariant frames and consider in particular filtering and oversampled filter banks. Copyright © 2006 P. T. Boufounos and A. V. Oppenheim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Quantization methods for frame expansions have received considerable attention in the last few years. Simple scalar quantization applied independently on each frame expansion coefficient, followed by linear reconstruction is well known to be suboptimal [1, 2]. Several algorithms have been proposed that improve performance although with significant complexity either at the quantizer [3] or in the reconstruction method [3, 4]. More recently, frame quantization methods inspired by uniform oversampled noise shaping (referred to generically as Sigma-Delta noise shaping) have been proposed for finite uniform frames [5, 6] and for frames generated by oversampled filterbanks [7]. In [5, 6] the error due to the quantization of each expansion coefficient is subtracted from the next coefficient. The method is algorithmically similar to classical first-order noise shaping and uses a quantity called frame variation to determine the optimal ordering of frame vectors such that the quantization error is reduced. In [7] higher-order noise shaping is extended to oversampled filterbanks using a predictive approach. That solution performs higher-order noise shaping, where the error is filtered and subtracted from the subsequent frame coefficients. In this paper we view noise shaping as compensation of the error resulting from quantizing each frame expansion

coefficient through a projection onto the space defined by another synthesis frame vector. This requires only knowledge of the synthesis frame set and a prespecified ordering and pairing for the frame vectors. Instead of attempting a purely algorithmic generalization, we incorpo