Effects Of Coefficient Quantization Research Articles

Coefficient quantization effects on new filters based on Chebyshev fourth-kind polynomials

The aim of this paper is to construct non-recursive filters, extensively used type of digital filters in digital signal processing applications, based on Chebyshev orthogonal polynomials. The paper proposes the use of the fourth-kind Chebyshev polynomials as functions in generating new filters. In this kind, low-pass filters with linear phase responses are obtained. Comprenhansive study of the frequency response characteristics of the generated filter functions is presented. The effects of coefficient quantization as one type of quantization that influences a filter characteristic are investigated here also. The quantized-coefficient errors are considered based on the number of bits and the implementation algorithms.

The Finite-Precision Numerical Effects Analysis of Infinite Impulse Response

This paper mainly uses the MATLAB software to design IIR digital filter with the amplitude frequency characteristic method and zero-pole method, and then analyzes the effect of coefficient quantization and quantization effects in IIR digital filter arithmetic. It is that we can use the software to draw some pictures. These pictures can show coefficient system function of IIR digital filter with a certain length to quantify the zero pole after the system distribution, frequency response, after quantizing the coefficients under the direct, parallel and cascade structures, as well as the limit cycle oscillation and the overflow oscillation of fixed-point operation in the quantization effect of operation are drawn.

Optimal Coefficient Quantization in Optimal-NTF <inline-formula> <tex-math notation="LaTeX">$\Delta \!\Sigma $ </tex-math> </inline-formula> Modulators

Strategies for the adaptation of $\Delta \!\Sigma $ modulators ( $\Delta \!\Sigma $ Ms) to the embedding environment with respect to the ability of the latter to remove quantization noise have recently been introduced. They rely on formal optimization techniques to deliver finely tuned values for the coefficients of the $\Delta \!\Sigma \text{M}$ internal filters. However, in digital $\Delta \!\Sigma $ Ms, coefficients will necessarily be quantized to short wordlengths, the shorter the better, cost-wise. Unfortunately, this can severely hinder optimality. In this brief, a strategy is proposed to re-use the very optimization framework to minimize the effects of coefficient quantization. The technique is discussed by a practical design example based on a $\Delta \!\Sigma \text{M}$ for a wideband fractional- ${N}$ phase locked loop (PLL).

Exploring DCT Coefficient Quantization Effects for Local Tampering Detection

In this paper, we focus on local image tampering detection. For a JPEG image, the probability distributions of its DCT coefficients will be disturbed by tampering operation. The tampered region and the unchanged region have different distributions, which is an important clue for locating tampering. Based on the assumption of Laplacian distribution of unquantized ac DCT coefficients, these two distributions as well as the size of tampered region can be estimated so that the probability of each DCT block being tampered is obtained. More accurate localization results could be got when we consider the prior knowledge of common tampered regions. We also design three kinds of features that can distinguish truly tampered regions from the false ones to reduce false alarm. For a tampered image which is saved in lossless compressed format, we also propose the specialized approach, which employs the quantization noise of high-frequency DCT coefficient, to improve the tampering localization performance. Extensive experiments on large scale databases prove the effectiveness of our proposed method and demonstrate that our method is suitable for locating tampered regions with different scales.

On the coefficient quantization of Multiplicative FIR filters

This paper analyzes the effects of coefficient quantization of Multiplicative Finite Impulse Response (MFIR) filters used to approximate the behavior of pole filters. Statistical analysis, zero displacement sensitivity and frequency domain analysis are used as measures of the filter performance for different coefficient lengths. A practical expression for determining the required number of bits for the coefficient quantization as a function of a predefined maximum deviation in the magnitude response is proposed in combination with an alternative method based on a time domain analysis. The time domain analysis allows, for a specific pole approximation, to investigate the sensitivity of the MFIR structure to coefficient variations. The paper concludes that, statistically, the MFIR pole approximation filter does not require a larger number of quantization bits for its coefficients than the corresponding Infinite Impulse Response (IIR) filter.

Fixed-Point Signal Processing

This book is intended to fill the gap between the ideal digital signal processing (DSP) that is widely taught, and the limited precision implementation skills that are commonly required in fixed-point processors and field programmable gate arrays (FPGAs). These skills are often neglected at the university level, particularly for undergraduates. We have attempted to create a resource both for a DSP elective course and for the practicing engineer with a need to understand fixed-point implementation. Although we assume a background in DSP, Chapter 2 contains a review of basic theory and Chapter 3 reviews random processes to support the noise model of quantization error. Chapter 4 details the binary arithmetic that underlies fixed-point processors and then introduces fractional format for binary numbers. Chapter 5 covers the noise model for quantization error and the effects of coefficient quantization in filters. Because of the numerical sensitivity of IIR filters, they are used extensively as an example system in both Chapters 5 and 6. Fortunately, the principles of dealing with limited precision can be applied to a wide variety of numerically sensitive systems, not just IIR filters. Chapter 6 discusses the problems of product roundoff error and various methods of scaling to avoid overflow. Chapter 7 discusses limit cycle effects and a few common methods for minimizing them. There are a number of simple exercises integrated into the text to allow you to test your understanding. Answers to the exercises are included in the footnotes.

Finite-precision Goertzel filters used for signal tone detection

The effect of coefficient quantization on the output of a second-order Goertzel filter tone-detector is analyzed in order that the Goertzel filter may be systematically designed for minimum complexity. Three alternative configurations are identified for the Goertzel filter's component second-order digital resonator, which are especially suited to VLSI implementation, and which complement each other on the zero-to-Nyquist frequency interval. The Goertzel filter's tone response is formulated using the Fourier summation transform (and shown to depend on both the frequency and phase of the incident tone). Hence analytic expressions are derived for the tone response of a fully finite-precision Goertzel filter when used with either a rectangular or half-end-point, input data window. These formulas are amenable to optimization, and are applied to improve a DTMF receiver design which is subject to the use of low-complexity second-order resonators.

A new approach to designing computationally efficient interpolated FIR filters

More efficient use of multipliers in FIR filters can be achieved at the expense of a slight increase in delay by designing sparse filter structures. We have developed a new, relatively simple approach to designing sparse cascaded filters, also described in the literature as interpolated FIR filters. Our method is heuristic in nature, but gives surprisingly good results without requiring iterative design or investigation of a large number of alternative parameterizations. The design uses the efficient and widely available Remez exchange algorithm along with some routines that we have written for Matlab. Although the resulting designs are not optimal in a minimax-error sense, they have reduced RMS error, which may be attractive for some applications. We give design examples, and study the effects of coefficient quantization.

Two-channel IIR QMF banks with approximately linear-phase analysis and synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Required coefficient word length in floating-point and logarithmic digital filters

The effects of coefficient quantization to the response of a digital filter are studied. Easy-to-use expressions for the required coefficient word length, with which the filter response deviations are within specified bounds, are derived for filters realized with floating-point or logarithmic arithmetic. The required coefficient word lengths of floating-point and fixed-point realizations of filters are compared with simulations. >

Synthesis of pipelineable complex wave digital filters

A decomposition algorithm is presented for the synthesis of pipelineable, modular, cascade, complex wave (unitary) digital filters (CWDFs) with emphasis on simplicity of implementation. The main features of the algorithm are the following: (1) the extraction step obviates coefficient-form polynomial manipulations and zero-finding operations by using an alternative representation of the canonic polynomials that describe the lossless two-port; (2) first-order sections that effect pipelineability are treated like any other section (they realize a at z/sup /spl minus/1/=0, thus eliminating the need for special treatment; (3) a fully general first-order complex section is derived that can realize a transmission zero anywhere in the z-plane and is the only section required for the cascade decomposition. The algorithm can also be used to synthesize real two-port networks; equivalences are given between a cascade of two complex sections with ke/sup /spl plusmn/jw(0/) zeros and a 2nd-order real section. Also, a method is presented for investigating coefficient quantization effects directly in the z-domain. >

Modified series and parallel adaptors that lead to reduced sensitivity in wave digital filters

By replacing multipliers in conventional series and parallel adaptors by digital subnetworks comprising multipliers whose constants are as far as possible machine representable, modified series and parallel adaptors are obtained whose operation is less dependent on multiplier constants that are not machine representable. Conventional and modified adaptors are then used to design several digital filters and the effects of coefficient quantization are investigated. The results obtained show that wave digital filters implemented with the modified adaptors are significantly less sensitive than filters implemented with conventional adaptors.

Minimizing quantization effects in a practical second-order digital filter bank

Abstract The paper describes the solution to coefficient and product-quantization effects in a practical, real-time digital filter bank. The solution to the coefficient-quantization effects involves simple manipulation of the floating-point filter coefficients. A computer simulation of product-quantization errors enables the appropriate number of bits for the truncated products to be determined.

Coefficient quantization effects in cascade and parallel form realization of low-pass digital filters

Abstract This paper presents a comparison between cascade and parallel direct form canonic implementation of low-pass digital filters, when we regard the effects of coefficient quantization on the magnitude of the frequency response. A first-order sensitivity analysis is presented along with experimental results. Finally, a set of interesting observations for the four types of classical filter approximations are made.

Thinning digital filters: A piecewise-exponential approximation approach

An algorithm is presented for designing finite impulse response (FIR) recursive digital filters that require few multiplies to produce good frequency response, The process of reducing the number of multiplies needed to implement a digital filter is called thinning. This thinning algorithm uses dynamic programming techniques to find the best least-squares piecewise-exponential approximation to a desired impulse response of length P. Because these filters are implemented recursively, the number of arithmetic operations is independent of the model filter length P and is dependent only on the number of pieces or segments S used in the approximation ( S \ll P ). Examples of thinned narrow-band, broad-band, lowpass, and bandpass filters are given. Several of these thinned filters require fewer than one-third the number of multiplications required for the corresponding model filter, while still retaining desirable frequency response characteristics. The effects of coefficient quantization and finite precision arithmetic are also discussed.

The effects of quantization errors on rotated filters

A digital filter that has been designed by rotation of the frequency response of a one-dimensional continuous filter, and then bilinearly transformed into a two-dimensional digital filter is called a rotated filter. Other useful filters such as circularly symmetric low-pass, high-pass, or bandpass filters can be obtained by parallel or cascade connection of rotated filters. These filters can be used in image processing and geophysics. Rotated filters are marginally stable if the rotation angle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta</tex> satisfies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">270°&lt;\beta&lt;360°</tex> , when designed from a stable one-dimensional continuous filter. A slight change in the coefficients has a significant effect on the stability of rotated filters. In this paper, the effects of quantization errors on the stability of rotated filters are investigated. A method to predict the stability state of rotated filters after coefficient quantization is given. A coefficient perturbation technique is discussed and used to stabilize a filter if found to be unstable. For real coefficients, a region of guaranteed stability is defined for some forms of fixed-point and floating-point arithmetic. The effects of coefficient quantization and coefficient perturbation on the frequency response also are discussed.

On the performance of optimum linear phase low-pass FIR digital filters under impulse response coefficient quantization

The effects of coefficient quantization on the frequency response of FIR filters realized in direct form are analyzed in [1]. Applying such results to optimal low-pass filters, simple formulas are derived to predict, starting from the specifications on the frequency response, the filter order and the quantization step size which minimize the bit-multiplier product.

Error analysis in fixed-point digital filters using sign-magnitude truncation

The quantization error of a digital filter employing fixed-point arithmetic with sign-magnitude truncation is analyzed. The effect of coefficient quantization is also included. Exact analyses are presented first for Gaussian, sinusoidal and Gaussian plus sinusoidal inputs. Quasi-linearization is then employed to yield a simple computation method of the output noise variance. The resulting expression contains parameters that depend on the input signal statistics. An upper bound is given that is independent of these statistics. Explicit expressions are given for the second-order sections of the digital filter realized in the D1 and D2 forms together with a general expression for the cascade connection of the second-order sections.

On the zeros of polynomials used in digital filters

The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transform of the truncated impulse response of a digital filter with poles, is a polynomial that is important in the stability analysis of certain types of block recursion and inverse digital filters. This paper presents new results on the zeros of these polynomials, extending the earlier work of Read and Burrus and also studies the effect of coefficient quantization on the zeros.

Digital filters: a template method of design

It is shown that the frequency response of digital filters of the cascade biquadratic (ratio of two quadratic polynomials) form can be described by a set of templates. These templates enable the effects of coefficient quantization to be seen. Consequently, by using the templates to design filters, coefficient quantization can be taken into account in the initial stages of the designing process. Examples of filter designs are given.