Unified LMI-based design of ΔΣmodulators
- M. Rizwan Tariq^{1}Email author and
- Shuichi Ohno^{1}
DOI: 10.1186/s13634-016-0326-2
© Tariq and Ohno. 2016
Received: 16 June 2015
Accepted: 18 February 2016
Published: 29 February 2016
Abstract
Optimal finite impulse response (FIR) error feedback filters for noise shaping in Δ Σ modulators are designed by using weighting functions based on the system norms. We minimize the weighted norms of the quantization error in the output of a Δ Σ modulator, which corresponds to the minimization of the system norm. Three norms, the H _{2} system norm, the H _{ ∞ } system norm, and the l _{1} norm of the impulse response of the system, are adopted. The optimization problem for three types of FIR filters are evaluated by using linear matrix inequalities (LMIs) and then solved numerically via semi-definite programming. Design examples are provided to demonstrate the effectiveness of our proposed methods.
Keywords
Delta-sigma modulation Noise shaping Quantization Semi-definite programming1 Introduction
Analog-to-digital (A/D) and digital-to-analog (D/A) data converters are some of the most important parts of the electronic systems which act as the interface between the digital signal world and the real analog world. In A/D converters, the continuous-valued signals are discretized and quantized for transmission in wireline or wireless systems [1]. The process of quantization maps the continuous-valued signal to the discrete-valued signal. This usually introduces an undesirable effect, which is known as quantization noise. The important aspect of these converters is their ability to determine whether and how much the conversion can correctly keep the important information of signals, while suppressing undesirable noises.
Currently, the delta-sigma (Δ Σ) modulation is a popular technique for making high-resolution A/D and D/A converters [2, 3]. Modern Δ Σ converters offer several benefits including high resolution, low power consumption, and low cost, making them a reasonable choice for the A/D converter for many signal processing applications such as audio devices [4, 5]. These Δ Σ A/D converters are effective for converting analog signals over a wide range of frequencies, from DC to several megahertz.
The Δ Σ modulator mainly consists of a static uniform quantizer and an error feedback filter to shape quantization noise [6], which is called noise shaping filter. The input to the modulator is an oversampled signal which is to be digitized. In oversampling, the signal is sampled at a frequency much higher than the Nyquist frequency (two times the input bandwidth) which reduces the effect of the quantization noise in the frequency band carrying the information signal, while the total noise remains the same.
The high-rate digital output of the modulator has two components, one is the signal which is located in the low-frequency region and the other is the noise which has to be reduced.
In the design of a Δ Σ modulator, the objective is to minimize the in-band quantization noise which as a result improves the signal-to-quantization-noise ratio (SQNR) of a Δ Σ modulator. It has been observed that the technique of oversampling alone may not be enough to improve the SQNR in the band of interest, and we need to exploit the noise shaping properties of the Δ Σ modulator to further reduce the in-band quantization noise. This can be achieved by using a feedback filter which employs the noise shaping to obtain a high SQNR while keeping the oversampling ratio (OSR) not too high. Although the overall quantization noise may not be changed by the noise shaping, the SQNR is increased in the information signal frequency band of the frequency spectrum. Our objective is to design the finite impulse response (FIR) noise shaping filter of the Δ Σ modulator so that we can minimize the noise in the frequency region which constitutes our signal bandwidth.
Several designs for feedback filters have been proposed which also use the noise spectrum shaping technique [7, 8]. The FIR error spectrum shaping filters have been proposed for recursive digital filters composed of cascaded second order section in [9]. In [10], the noise transfer function (NTF) is assumed to have an infinite impulse response which is converted to a minimization problem by virtue of generalized Kalman-Yakubovich-Popov (GKYP) lemma. Then, an iterative algorithm is developed to solve this minimization problem subject to quadratic matrix inequalities. The method in [11] is a min-max design to optimize the NTF via GKYP lemma. This approach minimizes the worst case gain of the NTF over the signal frequency band and is shown to be able to improve the overall SNR of Δ Σ modulators as well. However, the method in [11] cannot incorporate the system connected to the quantizer into its design, while we consider a non-ideal output filter to minimize the quantization noise. In [12, 13], the optimization problem based on H _{2} norm is formulated as a convex quadratic optimization problem where the weighting function (output filter) impulse response in truncated to finite number of samples.
In this paper, to keep Δ Σ modulators versatile, we utilize the weighting function to design Δ Σ modulators. We minimize the weighted quantization noise in the output of the Δ Σ modulator. Three norms are adopted to measure the quantity of the weighted quantization noise. One is the variance of the weighted quantization noise when the quantization errors at different time are assumed to be independent of each other. The others are the l _{2} and the l _{ ∞ } norms of the weighted quantization noise. They correspond to the minimization of the H _{2} system norm, the H _{ ∞ } system norm, and the l _{1} norm of the impulse response of a system, respectively, and can be formulated as convex optimization problems with linear matrix inequalities (LMIs), which can be solved efficiently. The three norms considered in this paper are the most commonly used norms for quantifying signals. Our proposed method based on LMIs is termed unified for these three most commonly used norms only. If one imposes a constraint on the filter, then there are nine combinations for the design, three types of objectives, and three types of constraints, which can be handled by LMIs. One of these nine combinations, H _{2} norm subjected to the Lee criterion, is similar to the design criteria of the method in [12, 13]. However, for our proposed H _{2} norm design, we provide an alternate approach based on expressing the H _{2} norm by using LMIs. The similarity lies in the fact that our proposed design and the method in [12, 13] use the idea of incorporating the non-ideal output filter for the minimization of the quantization noise.
The stability condition of Δ Σ modulators is also described by an LMI, which is incorporated into our design. Simulations with our designed noise shaping filters are performed, and comparisons with existing methods are made to demonstrate the effectiveness of our proposed design.
This paper is organized as follows: Section 2 gives the input/output relation of a linearized Δ Σ modulator with a weighting function. Then, we formulate our design problem which minimizes the weighted quantization noise under the stability condition. Section 3 is the main section of this paper, and we propose the design of the FIR feedback filter using LMIs for H _{2}, H _{ ∞ }, and l _{1} system norms. Section 4 gives design examples to show advantages of our method using simulation results. Section 5 provides us with the conclusion of our study.
Notation: Throughout this paper, \(\mathbb {R}\) and \(\mathbb {Z}\) denote a set of real numbers and a set of integer respectively. S denotes the set of all stable, proper, and rational transfer functions with real coefficients. The subscript (·)_{+} is used to indicate a subset restricted to non-negative numbers.
2 ΔΣ modulator and output weighting filter
Let us consider a general linearized model of a Δ Σ modulator for analyzing the noise shaping characteristics and designing the optimal noise shaping filter. We only consider the discretized single-input/single-output system with discrete-time signals. Let us denote the z transform of a sequence \(f=\left \{f_{k}\right \}_{k=0}^{\infty }\) as \(F[\!z]=\sum _{k=0}^{\infty }f_{k} \mathrm z^{-k}\) and express the output (sequence) b of the linear time invariant (LTI) system F[ z] to the input \(a=\{a_{k}\}_{k=0}^{\infty }\) as b=F[ z]a.
where d is the quantization interval and i is an integer. We assume that the saturation level is sufficiently large to avoid the saturation.
The gain from the input y to the output of the modulator u is known as as signal transfer function (STF), while the gain between the quantization error w and the modulator output u is commonly known as noise transfer function (NTF). In our setting, the STF and NTF for the Δ Σ modulator are P[ z] and R[ z], respectively.
The feedback loop acts in such a way that the quantization noise is shifted away from a certain frequency band. If the input to the modulator lies within this certain frequency band, then most of the noise due to quantization lies outside the frequency band of interest.
where H _{ W }[ z]∈S. Without loss of generality, we normalize the maximum magnitude of H _{ W }[ z] to be in unity. The weighting function is selected to reduce the effect of the quantization noise in the passband of the y. For example, when the passband of y is [−ω _{ p },ω _{ p }], we will use the weighting filter that meets H _{ W }[e ^{ j ω }]≈1 for ω∈[−ω _{ p },ω _{ p }] and |H _{ W }[e ^{ j ω }]| is small enough outside the passband to let most of the noise be outside the passband.
In [12], the noise H _{ S }[ z]R[ z]w is minimized based on the H _{2} system norm to reduce the in-band quantization noise. When H _{ W }[ z]=H _{ S }[ z], our minimization based on the H _{2} system norm is equivalent to the minimization of [12]. Then, the difference between the proposed method and [12] lies in the usage of different optimization procedures for solving the H _{2} norm objective function. As pointed out in [12], if one knows the system H _{ S }[ z] connected to the Δ Σ modulator, one should set H _{ W }[ z]=H _{ S }[ z]. If not, we could design more general Δ Σ modulators using weighting functions. Thus, our objective is to obtain the optimal filter R[ z] in (8) for a given H _{ W }[ z] that minimizes ε=H _{ W }[ z]R[ z]w in a sense.
Since the transfer function from w to ε is linear, we can put d=2 without the loss of generality so that |w _{ k }|≤1 and hence |w _{ k }|^{2}≤1.
for a finite γ.
The constraint on the H _{ ∞ } norm of the NTF is known as Lee criterion [6, 14]. The peak value of the NTF magnitude response must be bounded to some constant value γ, where the value of γ depends on the number of saturation levels. For the case of binary quantizers, the value of γ is usually set as 1.5.
In summary, we would like to minimize \(\sigma _{\epsilon }^{2}\), ||ε||_{2}, or ||ε||_{ ∞ } under the stability condition (15), assuming that the variance of w _{ k }, ||w||_{2} or ||w||_{ ∞ } is finite. It should be noted that without the weighting function, we only have the trivial solution such that R[ z]=1, that is, the error feedback is not necessary.
3 Design of FIR noise shaping filters using linear matrix inequalities
The coefficient r _{0} of the impulse response of the FIR filter R[ z] is unitary to ensure R[ z]∈S, which makes the noise shaping filter strictly proper.
which defines C _{ R }(r) above.
The weighted quantization noise ε in (6) to be minimized is characterized by the the composite system H _{ W }[ z]R[ z], which has to be internally stable.
First of all, let us consider the minimization of the variance \(\sigma _{\epsilon }^{2}\) of the weighted quantization error under the white noise assumption. It is sufficient to minimize the H _{2} norm of H _{ W }[ z]R[ z] to minimize \(\sigma _{\epsilon }^{2}\) given by (11).
For FIR R[ z], \(||H_{W}[\!\mathrm {z}]R[\!\mathrm {z}]||_{2}^{2}\) can be expressed as a quadratic function of r=[r _{1},…,r _{ n }] by using inverse Fourier transform of |H _{ W }[e ^{ j ω }]|^{2} [8], which requires numerical integrations. On the other hand, a truncated impulse response of H _{ W }[ z]R[ z] is utilized in [12], where the order of some parameters is scaled by the length of the truncated impulse response. Here, we adopt LMIs to numerically evaluate the H _{2} norm based on the next lemma.
Lemma 1.
We can utilize the bounded real lemma that provides us an LMI to evaluate the gain.
Lemma 2.
We can reduce ||ε||_{ ∞ } by minimizing ||H _{ W }[ z]R[ z]||_{imp}.
Unlike the H _{2} norm and the H _{ ∞ } norm, only upper bounds of the H _{imp} norm are available. In [17, 18], an upper bound based on the invariant set of a discrete-time system has been utilized to design infinite impulse response (IIR) error feedback filters for dynamic quantizers. The invariant set of a discrete-time system is defined as follows [19]:
Definition 1.
where \(A \in \mathbb {R}^{n\times n}\), \(B \in \mathbb {R}^{n\times m}\) and \(w_{k} \in \mathbb {R}^{m}\). A set \({\mathcal {X}}\) that satisfies \(x_{k+1}\in {\mathcal {X}}\) if \(x_{k}\in {\mathcal {X}}\) and \({w_{k}^{T}}w_{k}\leq 1\) is called an invariant set of the system given by (37).
The following lemma describes how to obtain an ellipsoid which is an invariant set of the system (37).
Lemma 3.
([19]) Let \(\mathcal {E}(P)\) be the ellipsoid defined by an n×n real symmetric matrix P≻0 as \(\mathcal {E}(P)=\{x \in \mathbb {R}^{n}: x^{T}Px\leq 1 \}\).
where ρ(A) is the spectrum radius of A.
It should be noted that unlike H _{2} and H _{ ∞ }, H _{imp} depends on parameter α.
Thus, we can conclude that \(|CP^{-1}C^{T}|^{\frac {1}{2}}+|D|\) is an upper bound of the norm.
For a fixed α, the minimization of μ is a semidefinite program, which can be numerically solved by existing optimization packages, e.g., CVX [20]. Then, all we have to do is to find α which gives the minimum. Since A is our design parameter, a line search for α∈(0,1) is required to obtain the minimum. The optimal (A,B,C,D) is given by the arguments corresponding to the optimal α.
subject to (29), (30), and (44).
The LMIs for other stability conditions ||R[ z]||_{2}<γ and ||R[ z]||_{imp}<γ can be obtained similarly, which are omitted to avoid the duplication.
In summary, our unified approach enables the design of the FIR noise shaping filter to minimize the H _{2}, the H _{ ∞ }, or the l _{1} system norm under the H _{2}, the H _{ ∞ }, or the l _{1} norm constraint. Moreover, since norms are described by LMIs, different types of problems can be solved numerically. For example, some signal processing applications may require us to design an error feedback filter for a Δ Σ modulator by adding a constraint that limits the magnitude of the weighted quantization noise to a certain value. Then, our design objective is to design the noise shaping filter that attains the optimal value of the stability threshold γ under the maximum weighted quantization noise constraint. If we adopt the Lee criterion, we can obtain the most stable error feedback filter by minimizing (43) subject to ||ε||_{ ∞ }≤c, where c is the maximum bound on the weighted quantization noise ε, by using LMIs in (41), (42), and (44).
4 Design examples
In this section, simulations for lowpass and bandpass Δ Σ modulators have been shown using the proposed design method based on H _{2}, H _{ ∞ }, and l _{1} system norms. For the design of a conventional Δ Σ modulator by NTF zero optimization method [6], the DELSIG toolbox [21] is utilized to obtain the frequency response of an IIR noise shaping filter with synthesizeNTF MATLAB function. The frequency response and the noise shaping characteristics of the FIR feedback filter proposed in [11] are also compared with our designed filters. As our proposed H _{2} norm minimization is mathematically equivalent to the method proposed in [12], the numerical results which we have performed also show that there is no significant difference between our proposed H _{2} norm method and [12]. To avoid redundancy in our simulations, we omit the comparison with [12].
All simulation results are obtained by using MATLAB programming, while semi-definite programming (SDP) problems are solved by using CVX tool [20], which is an effective solver for convex optimization problems.
4.1 Lowpass ΔΣ modulator with the first-order weighting function
Now, let us design a lowpass Δ Σ modulator by using a first-order lowpass Butterworth filter as our weighting function H _{ W }[ z]. The first-order Butterworth filter provides us the maximum flat response in the passband at the expense of a wide transition band as the filter changes from the passband to the stopband. The input signal y to the lowpass Δ Σ modulator is assumed to be oversampled with an oversampling ratio (OSR) of 512. Then, the cut-off frequency of the first-order Butterworth filter is set at π/OSR≈0.0061 in the normalized angular frequency interval [ 0,π].
For the stability of the Δ Σ modulator, we assume the value of the Lee coefficient γ to be 1.5 which is equivalent to 3.52 in decibels ; however, the value of γ can be increased further as long as the Δ Σ modulator remains stable.
The method in [11] designs the FIR noise shaping filter based on the weighted H _{ ∞ } norm of R[ z]. Near the cut-off frequency, the magnitude response of the FIR filter in [11] increases rapidly showing the high steepness in the transition band, while all of our proposed filters exhibit good performance, matching the steepness of the weighting function. Note that the maximum magnitude value of all filters are bounded to 3.52 dB approximately due to stability constraint which utilizes the Lee coefficient γ=1.5.
||H _{ W }[ z]R[ z]||_{2}, ||H _{ W }[ z]R[ z]||_{ ∞ }, and l _{1} norms of the impulse response of H _{ W }[ z]R[ z] for the first-order lowpass weighting function
H _{2} norm | H _{ ∞ } norm | l _{1} norm | |
---|---|---|---|
H _{2} norm design | 1.54×10^{−2} | 2.19×10^{−2} | 2.62×10^{−2} |
H _{ ∞ } norm design | 1.54×10^{−2} | 2.16×10^{−2} | 2.59×10^{−2} |
l _{1} norm design | 1.63×10^{−2} | 2.59×10^{−2} | 2.59×10^{−2} |
Nagahara design [11] | 1.92×10^{−2} | 3.82×10^{−2} | 4.89×10^{−2} |
Conventional design [6] | 2.61×10^{−2} | 6.92×10^{−2} | 11×10^{−2} |
The H _{ ∞ } and l _{1} norm designs exhibit an equivalent l _{1} norm, while the H _{2} and H _{ ∞ } norm designs have an equivalent H _{2} norm. This may be partially due to the implementation and the numerical errors in our numerical optimization. It should be noted that we minimize the upper bounds, which implies that we cannot guarantee that the quantizer designed based on a norm is optimal in the sense of the norm.
4.2 Lowpass ΔΣ modulator with the fourth-order weighting function
Now let us introduce a higher order lowpass Butterworth filter of order 4 as our weighting function, where the OSR is 32. The maximum magnitude of NTF is limited to 3.52 dB by using the Lee coefficient γ=1.5. The fourth-order Butterworth filter with a cut-off frequency of π/OSR≈0.0098 has a better stopband attenuation than the first-order Butterworth filter by increasing the steepness of the passband to the stopband transition at the cost of reduced passband flatness.
||H _{ W }[ z]R[ z]||_{2}, ||H _{ W }[ z]R[ z]||_{ ∞ }, and l _{1} norms of the impulse response of H _{ W }[ z]R[ z] for the fourth-order lowpass weighting function
H _{2} norm | H _{ ∞ } norm | l _{1} norm | |
---|---|---|---|
H _{2} norm design | 3.95×10^{−2} | 9.71×10^{−2} | 1.40×10^{−1} |
H _{ ∞ } norm design | 4.07×10^{−2} | 9.09×10^{−2} | 1.24×10^{−1} |
l _{1} norm design | 4.43×10^{−2} | 1.22×10^{−1} | 1.23×10^{−1} |
Nagahara design [11] | 9.18×10^{−2} | 3.53×10^{−1} | 4.74×10^{−1} |
Conventional design [6] | 1.49×10^{−1} | 6.69×10^{−1} | 9.01×10^{−1} |
To assess the performance of the lowpass Δ Σ modulator with an error feedback filter obtained by our proposed H _{2} norm-based design, the MATLAB function simulateDSM in DELSIG toolbox [21] is used to simulate the Δ Σ modulator for obtaining the digital output. The input to the Δ Σ modulator is a sinusoidal wave with a frequency of 100 Hz and and amplitude of 0.5. We assume a uniform quantizer with saturation levels L=2 and quantization interval d=2.
4.3 Bandpass ΔΣ modulator with the sixth order weighting function
Finally, we adopt a sixth-order bandpass Butterworth filter as our weighting function, whose frequency response is found in Fig. 10.
The input to the modulator is assumed to have the center frequency ω _{∘}=π/2 and bandwidth parameter Ω=π/16. For the passband ω∈[π/2−π/16,π/2+π/16], we use the bandpass Butterworth filter that meets H _{ W }[e ^{ j ω }]≈1 for ω∈[ω _{∘}−Ω,ω _{∘}+Ω], and |H _{ W }[e ^{ j ω }]| is small enough outside the passband to let most of the noise be outside the passband. For the conventional design [6], OSR is set to be 16.
||H _{ W }[ z]R[ z]||_{2}, ||H _{ W }[ z]R[ z]||_{ ∞ }, and l _{1} norms of the impulse response of H _{ W }[ z]R[ z] for the sixth-order bandpass weighting function
H _{2} norm | H _{ ∞ } norm | l _{1} norm | |
---|---|---|---|
H _{2} norm design | 5.08×10^{−2} | 1.385×10^{−1} | 2.094×10^{−1} |
H _{ ∞ } norm design | 5.38×10^{−2} | 1.277×10^{−1} | 1.916×10^{−1} |
l _{1} norm design | 6.08×10^{−2} | 1.833×10^{−1} | 1.858×10^{−1} |
Nagahara design [11] | 5.45×10^{−2} | 1.408×10^{−1} | 2.222×10^{−1} |
Conventional design [6] | 10.19×10^{−2} | 4.253×10^{−1} | 5.461×10^{−1} |
4.4 Stability under the l _{ ∞ } norm constraint on the weighted quantization noise
Here, to obtain the most stable error feedback filter for a lowpass Δ Σ modulator, we minimize (43) under the l _{ ∞ } norm constraint on the weighted quantization noise such that ||ε||_{ ∞ }=1.96×10^{−2}. We use the same first-order Butterworth filter in Section 4.1.
5 Conclusions
We have proposed a design method of the FIR noise shaping filters of Δ Σ modulators based on H _{2}, H _{ ∞ }, and l _{1} norms. The minimization of the norm of the weighted quantization error is cast into a convex optimization problem by using LMIs, which can be efficiently and numerically solved. To ensure the stability of a Δ Σ modulator, we have also included LMI constraints which subsumes the Lee criterion. Our results show that the frequency response of our filters exhibits good performance throughout the low-frequency region providing uniform attenuation and matching the weighting function. Also, our proposed H _{2}, H _{ ∞ }, and l _{1} norm designed error feedback filters are shown to provide us with minimum H _{2}, H _{ ∞ }, and l _{1} norms of the weighted quantization error, respectively, which shows the effectiveness of our proposed design method.
Notes
Declarations
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- J Proakis, Digital Communications, 4th (McGraw-Hill, New York, 2001).MATHGoogle Scholar
- JC Candy, GC Temes, Oversampling Delta-Sigma Data Converters (IEEE Press, New York, 1991).View ArticleGoogle Scholar
- KCH Chao, S Nadeem, WL Lee, CG Sodini, A higher order topology for interpolative modulator for oversampling A/D conversion. IEEE Trans. Circuits Syst.37:, 309–318 (1990).View ArticleGoogle Scholar
- E Janssen, D Reefman, Super-audio CD: an introduction. IEEE Signal Process. Mag.20(4), 83–90 (2003).View ArticleGoogle Scholar
- U Zoler, Digital Audio Signal Processing, 2nd (Wiley, New York, 2008).Google Scholar
- R Schreier, GC Temes, Understanding Delta-Sigma Data Converters, 1st (Wiley-IEEE Press, US, 2004).View ArticleGoogle Scholar
- Tran-Thong, B Liu, Error spectrum shaping in narrow-band recursive filters. IEEE Trans. Acoustics, Speech Signal Process.25(2), 200–203 (1977).View ArticleGoogle Scholar
- T Laakso, I Hartimo, Noise reduction in recursive digital filters using high-order error feedback. IEEE Trans. Signal Process.40(5), 1096–1107 (1992).View ArticleMATHGoogle Scholar
- W Higgins, DC J Munson, Noise reduction strategies for digital filters: error spectrum shaping versus the optimal linear state-space formulation. IEEE Trans. Acoustics Speech Signal Process.30(6), 963–973 (1982).View ArticleGoogle Scholar
- X Li, C Yu, H Gao, Design of delta-sigma modulators via generalized Kalman-Yakubovich-Popov lemma. Automatica. 50:, 2700–2708 (2014).MathSciNetView ArticleMATHGoogle Scholar
- M Nagahara, Y Yamamoto, Frequency domain min-max optimization of noise-shaping delta-sigma modulators. IEEE Trans. Signal Process.60(6), 2828–2839 (2012).MathSciNetView ArticleGoogle Scholar
- S Callegari, F Bizzarri, Output filter aware optimization of the noise shaping properties of delta-sigma modulators via semi-definite programming. IEEE Trans. Circuits Syst. I. 60(9), 2352–2365 (2013).MathSciNetView ArticleGoogle Scholar
- S Callegari, F Bizzarri, Noise weighting in the design of delta-sigma modulators (with a psychoacoustic coder as an example). IEEE Trans. Circuits Syst. II: Express Briefs. 60(11), 756–760 (2013).View ArticleGoogle Scholar
- KCH Chao, WLL S. Nadeem, CG Sodini, A higher order topology for interpolative modulators for oversampling a/d converters. IEEE Trans. Circuits Syst.37(3), 309–318 (1997).View ArticleGoogle Scholar
- S Boyd, LE Ghaoul, E Feron, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, USA, 1997).Google Scholar
- P Gahinet, P Apkarian, A linear matrix inequality approach to h _{ ∞ } control. Int. J. Robust Nonlinear Control. 4:, 421–448 (1994).MathSciNetView ArticleMATHGoogle Scholar
- K Sawada, S Shin, Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework. Am. Control Conf. (ACC), 1662–1667 (2011).
- S Ohno, Y Wakasa, M Nagata, Optimal error feedback filters for uniform quantizers at remote sensors. IEEE Int. Conf. Acoustics, Speech Signal Process, 3866–3870 (2015).
- H Shingin, Y Ohta, in 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale systems: Theory and Applications (LSS). Optimal invariant sets for discrete-time systems: approximation of reachable sets for bounded inputs, (2004), pp. 401–406.
- M Grant, S Boyd, Cvx: Matlab software for disciplined convex programming, version 2.0 beta (2012). http://cvxr.com/cvx/. Accessed date Feb 2015.
- R Schreier, Delta sigma toolbox [online]. http://jp.mathworks.com/matlabcentral/fileexchange/19-delta-sigma-toolbox. Accessed date Feb 2015.