This article provides a detailed explanation of a design method for delta-sigma data conversion systems that introduces a pre-filter to improve signal quality. Related articles, related papers, and MATLAB links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

Okajima, M. Honda, R. Yoshino and N. Matsunaga, A Design Method of Delta-Sigma Data Conversion System with Pre-Filter, SICE Journal of Control, Measurement, and System Integration, Vol. 8, No. 2, pp. 154–160 (2015) (Open Access)

This paper is co-work with Maho Honda, Rei Yoshino, and Prof. Nobutomo Matsunaga, all from Graduate School of Science and Technology, Kumamoto University.

Contents

Why Delta-Sigma Data Conversion Matters
Traditional Delta-Sigma Data Conversion System
Delta-Sigma Modulator
Proposed System with Pre-Filter
Error Evaluation System
Design Algorithm Using Particle Swarm Optimization
Numerical Simulations: Classical Piano
Numerical Simulations: J-POP Music
Connections to Related Research
MATLAB Code
Related Articles and Videos
Paper Information

Why Delta-Sigma Data Conversion Matters

AD/DA conversion is a core technology in digital signal processing. Signal compression is one of the most important components for building effective AD/DA conversion systems. In practice, applications such as:

Sound processing and music compression
Image processing
Networked control systems with communication channel constraints

all require high-quality data conversion that minimizes the effect of quantization noise while preserving the original signal.

The delta-sigma modulator (DSM) is well known as an effective method for encoding analog signals into digital signals. It is a noise-shaping type converter that can suppress low-frequency quantization noise. However, the traditional system — composed of a DSM and a post-filter — faces a fundamental trade-off between quantization noise reduction and signal distortion. If the post-filter's cut-off frequency is set too low, signal distortion increases; if set too high, quantization noise is not sufficiently removed.

This paper proposes a new approach: adding a pre-filter before the DSM, creating a two-degree-of-freedom structure that can separately handle quantization noise and signal distortion. The filters are designed using a particle swarm optimization (PSO) algorithm.

Traditional Delta-Sigma Data Conversion System

The traditional delta-sigma data conversion system consists of a DSM followed by a post-filter $F$ . The input signal $u _{\mathrm{in}}$ is fed into the DSM, which produces a coarse quantized output. The post-filter then removes high-frequency quantization noise to recover an approximation $u _{\mathrm{out}}$ of the original signal.

The design objective is to minimize the difference between $u _{\mathrm{in}}$ and $u _{\mathrm{out}}$ , but this requires balancing the trade-off between quantization noise and signal distortion through careful selection of the post-filter parameters.

Delta-Sigma Modulator

The DSM used in this paper is denoted as $Q _{\Delta\Sigma}$ and is given by the following two equations:

$\displaystyle \xi(k+1) = \xi(k) - (u(k) - v(k))$

$\displaystyle v(k) = Q_{st}[-\xi(k) + u(k)$ ]

where $\xi \in \mathbb{R}$ is the state, $u(k)$ is the input, and $v(k)$ is the output. The initial state is $\xi(0) = 0$ . The paper assumes a 3-level quantizer ( $M = 3$ , $d = 1$ ), so the permissible input range is:

$\displaystyle u(k) \in [-1, 1$ ]

The quantization noise $n _q(k)$ satisfies $n _q(k) \in \lbrack -d/2, d/2 \rbrack$ for all $k$ , and the input-output relationship can be described as:

$\displaystyle v = u + \left(1 - \frac{1}{z}\right) n_q$

where $z$ is the delay operator. The term $(1 - 1/z)$ represents the derivative transfer function, which is the key noise-shaping property of the DSM: quantization noise is reduced at low frequencies (where the input signal resides) and pushed to higher frequencies.

Proposed System with Pre-Filter

The proposed data conversion system adds a pre-filter $F _1$ before the DSM and modifies the post-filter to $F _2$ , creating a two-degree-of-freedom structure. The filters are parameterized as:

$\displaystyle F_1(z) = D(z)$

$\displaystyle F_2(z) = F(z) D(z)^{-1}$

where $D(z)$ is a stable, minimum-phase transfer function with relative degree zero, and $F(z)$ is a low-pass filter. Setting $D(z) = 1$ recovers the traditional system, so the proposed structure has the potential to improve performance beyond the traditional approach.

The key advantage is that the pre-filter $D(z)$ shapes the input signal before quantization, allowing the designer to improve the signal-to-noise ratio without affecting signal distortion, which depends only on $F(z)$ .

Error Evaluation System

The paper proposes an error evaluation system that separates the effects of quantization noise and signal distortion into two error signals.

The first error signal $e _1(k)$ evaluates the signal distortion caused by the filter $F(z)$ . Its z-transform is:

$\displaystyle e_1(z) = (z^{-m} - F(z)) u_{\mathrm{in}}(z)$

where $m = 0$ in this paper. To minimize signal distortion, the gain of $(z^{-m} - F(z))$ should be small in the dominant frequency range of $u _{\mathrm{in}}$ .

The second error signal $e _2(k)$ evaluates the effect of quantization noise. Its z-transform is:

$\displaystyle e_2(z) = F(z) D(z)^{-1} \left(1 - \frac{1}{z}\right) n_q(z)$

Since the quantization noise characteristics do not depend on the input signal, $D(z)$ can be designed independently to minimize $e _2$ under the signal range constraint.

The overall evaluation function combines both errors:

$\displaystyle J = f_1 |\sigma(e_1)| + f_2 |\sigma(e_2)|, \quad f_1, f_2 > 0$

where $\sigma(\cdot)^{2}$ denotes the variance, and $f _1$ , $f _2$ are weights that control the trade-off between signal distortion and quantization noise.

Design Algorithm Using Particle Swarm Optimization

The filters $F$ and $D$ are designed using particle swarm optimization (PSO), a population-based optimization method inspired by swarm behavior. The PSO algorithm updates particle positions and velocities according to:

$\displaystyle x_{i}^{k+1} = x_{i}^{k} + v_{i}^{k+1}$

$\displaystyle v_{i}^{k+1} = c_1 v_{i}^{k} + c_2 \mathrm{rand}_{1,i}^{k} (x_{\mathrm{pbest},i}^{k} - x_{i}^{k}) + c_3 \mathrm{rand}_{2,i}^{k} (x_{\mathrm{gbest}}^{k} - x_{i}^{k})$

where $x _{\mathrm{pbest},i}^{k}$ is the personal best and $x _{\mathrm{gbest}}^{k}$ is the global best. The design follows a two-step procedure:

Design $F(z)$ : The post-filter is designed as a Butterworth low-pass filter with cut-off frequency $W _p$ and order $n$ as design variables, using PSO to minimize $J$ with $D = 1$ .
Design $D(z)$ : The pre-filter is designed as a first-order transfer function $D(z) = c(z - a)/(z - b)$ with parameters $a, b, c$ optimized by PSO, using the previously obtained $F(z)$ and enforcing the signal range constraint.

The two-step approach is advantageous because the number of design parameters for each PSO problem is smaller compared to simultaneous design of both filters.

Numerical Simulations: Classical Piano

The method is first demonstrated using a classical piano sound as the input signal, sampled at 44,100 Hz with a main frequency range of 0 to 2 kHz.

Post-Filter Design

The PSO algorithm ( $k _{\mathrm{max}} = 100$ , 50 particles) yields a Butterworth filter with:

Cut-off frequency: $W _p = 2163$ Hz
Order: $n = 4$

The evaluation function value is $J _{\mathrm{piano}} = 0.0315$ .

Pre-Filter Design

The PSO algorithm ( $k _{\mathrm{max}} = 100$ , 100 particles) yields:

$\displaystyle D_{\mathrm{piano}} = 10.71 \frac{z - 0.893}{z + 0.625}$

This filter has high-pass characteristics, which boosts the signal around 2 kHz before quantization, thereby improving the SN ratio. The resulting evaluation function value improves to $J _{\mathrm{piano}} = 0.0262$ , which is smaller than the case without the pre-filter (0.0315).

The maximum absolute error between $u _{\mathrm{in}}$ and $u _{\mathrm{out}}$ is also smaller with the pre-filter compared to the traditional method, confirming the effectiveness of the proposed approach.

Numerical Simulations: J-POP Music

To evaluate robustness to different input characteristics, J-POP music is used as a second test signal. J-POP music has a wider frequency range compared to the classical piano.

The PSO algorithm yields:

Post-filter $F _{\mathrm{JP}}$ : cut-off frequency $W _p = 3099$ Hz, order $n = 2$
Pre-filter: $D _{\mathrm{JP}}(z) = 1.79(z - 0.335)/(z + 0.229)$

The designed parameters are significantly different from the classical piano case. In particular, the effect of the pre-filter is smaller, and the cut-off frequency is higher due to the wider frequency range of J-POP music. The evaluation value is $J _{\mathrm{JP}} = 0.0727$ .

An important finding: when the filters designed for J-POP music are applied to the classical piano input, the evaluation value becomes $J _{\mathrm{piano}} = 0.0374$ , which is larger than the value obtained by the individually designed filters (0.0262). This demonstrates that individual filter design for each input signal type is important for minimizing quantization noise.

This work is part of a broader research program on quantized control and data conversion:

Dynamic Quantizer Design — H. Okajima, K. Sawada and N. Matsunaga, Dynamic Quantizer Design Under Communication Rate Constraints, IEEE Transactions on Automatic Control, Vol. 61, No. 10, pp. 3190–3196 (2016). Addresses optimal dynamic quantizer design for control systems with discrete-valued inputs under communication rate constraints.

Dynamic Quantizer for MIMO Systems — H. Okajima, K. Sawada, N. Matsunaga, and Y. Minami, Dynamic quantizer design for MIMO systems based on communication rate constraint, Electronics and Communications in Japan, Vol. 96, Issue 5, pp. 28–36 (2013). Extends quantizer design to MIMO systems.

Model Error Compensator (MEC) — The quantized control framework can be combined with the Model Error Compensator to achieve robust control in systems with discrete-valued signals.

MATLAB Code

GitHub (Dynamic Quantizer): MATLAB Dynamic Quantizer

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Dynamic Quantizer / Publications

Video

YouTube: Control Engineering Channel / Portal

Paper Information

Okajima, M. Honda, R. Yoshino and N. Matsunaga, "A Design Method of Delta-Sigma Data Conversion System with Pre-Filter", SICE Journal of Control, Measurement, and System Integration, Vol. 8, No. 2, pp. 154–160, 2015. DOI: 10.9746/jcmsi.8.154 (Open Access)

Co-authors: Maho Honda (Kumamoto University), Rei Yoshino (Kumamoto University), Nobutomo Matsunaga (Professor, Kumamoto University)

Earlier conference paper: H. Okajima, M. Honda, R. Yoshino and N. Matsunaga, "A design method of delta-sigma data conversion system with pre-filter," Proc. of SICE Annual Conference 2014, pp. 1388–1394 (2014)

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.