Elsevier

Signal Processing

Volume 144, March 2018, Pages 163-168
Signal Processing

An iterative LMI algorithm for quantization noise reduction in ΔΣ modulators

https://doi.org/10.1016/j.sigpro.2017.10.018Get rights and content

Highlights

  • The variance of the quantization noise is minimized in delta-sigma modulators.

  • The non-ideal output filter is incorporated into our minimization design problem.

  • An iterative LMI algorithm for the synthesis of noise shaping IIR filter is proposed.

  • The algorithm is initialized with the least-squares method and the hybrid design.

  • The effectiveness of our algorithm is demonstrated using the RF transmitter example.

Abstract

In this paper, we design noise shaping filters for quantization noise reduction in ΔΣ modulators. The design problem for the noise shaping infinite impulse response (IIR) filter in the feedback of the ΔΣ modulator is in non-convex form, which is hard to solve in general. We propose an iterative linear matrix inequality (LMI) algorithm to find a near optimal noise shaping IIR filter by solving the non-convex design problem. The least-squares and the hybrid design are two methods, either of which can serve as a starting point for our iterative LMI algorithm to obtain a near optimal solution. We also compare and analyze the performances of some other noise shaping techniques with our proposed algorithm. A design example for a bandpass ΔΣ modulator based radio frequency transmitter is provided which takes into account the non-ideal behavior of the output analog filter.

Introduction

The increasing demand for high signal bandwidth and dynamic range in various communication systems require highly efficient analog-to-digital (A/D) converters in the process of digitization. ΔΣ modulators are well known A/D converters which offer several benefits like high-resolution, low power consumption, and low cost, making them reasonable choice for the A/D converter for many signal processing applications. The use of a ΔΣ modulator as an A/D converter has shown to increase the overall efficiency of the RF transmitter [1], [2]. A switching audio amplifier [3], which is a widely used electronic component in many portable electronics and mobile phones, also makes use of a ΔΣ modulator for A/D conversion.

A discrete-time ΔΣ modulator utilizes a noise shaping digital filter in the feedback to shape as much quantization noise as possible away from the information signal band [4]. To minimize the quantization noise present in the information signal band, the design of the noise shaping filter in a ΔΣ modulator is of great importance. For a digital filter with finite impulse response (FIR), the optimal design can be obtained by solving the convex optimization design problem [5], [6], [7]. The method in [7] minimizes the worst case noise gain over the signal frequency band by obtaining H norm based optimal FIR filter via generalized Kalman–Yakubovich–Popov (GKYP) lemma [8]. In many applications, such as of a switching audio amplifier and a RF transmitter, a non-ideal filter is utilized at the output of a ΔΣ modulator to recover the baseband signal. This non-ideal filter may cause the quantization noise to leak into the signal band. The method in [7] cannot incorporate the system connected to the quantizer into its design, while the method in [5], [6] considers a non-ideal output filter to obtain FIR filters.

On the other hand, the design of an infinite impulse response (IIR) digital filter becomes non-convex, and the numerical solution for this non-convex problem is not guaranteed to be optimal. In the method proposed by [9], the noise shaping filter is assumed to have an IIR which is converted to a minimization problem by virtue of generalized GKYP lemma and solved by using an iterative algorithm. The method in [9] can only minimize the worst-case system gain for obtaining the IIR filter, and also, it does not incorporate the behavior of the non-ideal filter at the output. The IIR design problem proposed in [10] utilizes the non-ideal output filter, which is then solved by using the extended linear matrix inequality (LMI) technique. However, the order of the noise shaping filter in [10] is constrained to be identical to the non-ideal output filter. Recently, an indirect approach is used to obtain noise shaping IIR filters by using approximation techniques, where the non-convex IIR filter design problem is solved by first obtaining an optimal FIR filter which is then approximated by using an IIR filter [11], [12]. However, these approximation techniques cannot guarantee the stability of the ΔΣ modulator. Then, there is a method that proposes the idea of the hybrid design to obtain an IIR filter for a ΔΣ modulator [13]. The performance of the IIR filter obtained by using the hybrid design is dependent on the performance of the FIR filter.

In this paper, we propose an iterative LMI algorithm to solve the non-convex design problem for obtaining the near optimal noise shaping IIR filter. We minimize the variance of the quantization noise at the output of a ΔΣ modulator subject to the stability constraint. Since the non-ideal filter at the output of the ΔΣ modulator is an imperfect filter which may cause quantization noise leakage in the passband of the information signal, the non-ideal behavior of the output filter is also taken into consideration. Moreover, we also design and compare the performances of the noise shaping IIR filters by using the hybrid design and Schreier’s method [4]. The design example for the bandpass ΔΣ modulator in a RF transmitter is provided to show the effectiveness of the proposed technique.

The organization of this paper is as follows: Section 2 introduces the ΔΣ modulator and its approximated linearized model with error feedback noise shaping filter. Then, we formulate the design problem for the minimization of the quantization noise subject to the stability criterion. In Section 3, we propose an iterative LMI algorithm for solving the non-convex design problem. The initialization of the algorithm is also discussed in-detail. Section 4 provides a design example to demonstrate the effectiveness of the proposed algorithm.

Section snippets

ΔΣ modulator and noise shaping

A quantizer which utilizes a filter in its feedback, which is also known as error feedback or noise shaping filter, is developed to minimize the quantization errors with the technique called error spectrum shaping [14], [15], [16]. If we can utilize some prior knowledge about the input of the quantizer, it is possible to design effective quantizer. Since the effects of the quantization error cannot be ignored if significant number of bits are not available for the quantization, quantizers with

Proposed iterative algorithm

To minimize the variance of the quantization noise, we can evaluate the H2 norm of H[z]R[z] numerically. The H2 norm can be evaluated based on the Lemma 1 in [5], which gives us a bilinear matrix inequality (BMI) and an LMI of the function H[z]R[z]22<μ as follows: [PPAPBATPP0BTP01]0,[μCDCTP0DT01]0,where P is a positive definite matrix.

Also, the condition on the stability of ΔΣ modulator in (9) can be described by using the Lemma 2 in [5] as follow [PRPRARPRBR0ARTPRPR0CRTBRTPR0γ210CR11]0

Design example

Let us consider a design example of a bandpass ΔΣ modulator based RF transmitter. Fig. 3 shows a block diagram of a basic wireless transmitter which consists of an A/D converter, a frequency-up converter and a power amplifier (PA) with a bandpass filter (BPF) at its output. The input to the PA is usually a signal with varying envelope, and if the PA is driven to more than its maximum input saturating power, it will cause distortion. The peak power of the input signal with varying envelope

Conclusions

To minimize the variance of the quantization noise, we have obtained optimal noise shaping IIR filters for ΔΣ modulators. An iterative LMI algorithm has been proposed which converts the non-convex problem to convex form by converting BMIs into LMIs with alternation of unknown variables at each iteration. The hybrid design and the least-squares method are used to obtain the IIR filter whose denominator coefficients are used to initialize the proposed algorithm. Our design example utilizes the

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    This work was partially supported by JSPS KAKENHI Grant Number JP16H02921.

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