D-QUANT - Delta Quantizer

By: Preethi Ramkumar

Meaning of D-QUANT – “Delta Quantizer”, Sigma-Delta quantizers are widely implemented to quantize oversampled bandlimited functions. When used to quantize oversampled bandlimited functions, firstorder 1-bit quantizers yield approximations where the pointwise approximation error is bounded by C1A-1 or better, and the MSE behaves like A-3, where A is the frame bound of the corresponding tight frame for the space of bandlimited functions.

In signal processing, one of the primary goals is to obtain a digital representation of the signal of interest that is suitable for storage, transmission, and recovery. Although in a discrete representation, it is certainly not “digital” since the coefficient sequence is real or complex valued. Therefore, a second step is needed to reduce the continuous range of this sequence to a discrete, preferably finite, set. This second step is called quantization.

Other Related Definitions:

“…One-bit Delta quantization is a method of representing bandlimited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density, convolving these one-bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization.” [Wiley Periodicals, Inc.]

“…The second order Sigma-Delta scheme with linear quantization rule is analyzed for quantizing finite unit-norm tight frame expansions for Rd. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N2, where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second order scheme only gives approximation error of order 1/N. For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second order scheme.” [Department of Mathematics]

“…Oversampled Sigma-Delta Quantizer (SDQ) is becoming a standard method for implementing high-resolution A/D and D/A converters in silicon [1]. Among the reasons of the increasing popularity of SDQ are: (a) high resolution quantization is attained with simple circuits, (b) these are robust to circuit imperfections, (c) allows for speed vs. resolution tradeoffs, and (d) lower cost of implementations. SigmaDelta Modulation extends the concepts of delta modulation by preemphasizing the low frequencies of the input by the use of an integrator. As a result, slope-overload and granular noise distortions, typical of delta modulation, are reduced. In addition, the de-emphasis (differentiator) required at the decoder, to account for the pre-emphasis placed at the input signal, cancels the integrator in the DM decoder, resulting in a much simplified decoder consisting of a low pass digital filter.” [Department of Electrical Engineering ]

Related Links:

Finite frames and Sigma-Delta quantization - Department of Mathematics University of Maryland
Quantization in Delta–Sigma ADCs - IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS

Technical Resources:

ERGODIC DYNAMICS IN SIGMA-DELTA QUANTIZATION - Courant Institute of Mathematical Sciences
First-Order Sigma Delta Modulator - Stanford

Products and Solutions:

DELTA-SIGMA DESIGN
Stereophile

Blogs, News, Feeds, Discussion Lists:

Analysis of the delta Modulator
Sigma-Delta Quantization

Books About:

Delta-Sigma Modulators: Modeling, Design and Applications - by George I. Bourdopoulos

See Also:

Other Delta Quantizer Related Resources

 

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