DESIGN OF OVERSAMPLING ∆Σ DA CONVERTERS VIA H ∞ OPTIMIZATIONMasaaki Nagahara∗ , Toshihiro Wada† , Yutaka Yamamoto‡Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University,
Kyoto 606-8501, Japan
ABSTRACTIn this paper, we propose a new method for designing oversampling ∆Σ DA converters via H ∞ optimization. The design consists of two steps. One is that for ∆Σ modulators.
In ∆Σ modulators, the accumulator 1/(z −1) is conventionally used in a feedback loop to shape quantization noise. Incontrast, we give all stabilizing feedback filters for the modulator, and propose an H ∞ design to shape the frequency response of the noise transfer function (NTF). The other is a
design for interpolation filters in oversampling DA converters.
While conventional designs are executed in the discrete-time
domain, we take account of the characteristic of the original
analog signal by using sampled-data H ∞ optimization. A design example is presented to show that our design is superior
to conventional ones.
1. INTRODUCTION∆Σ modulators are widely used in high-resolution AD or DA
converters. They are applied to measurement, digital audio
processing and wireless communications (see [1, 2]). In combination with oversampling technique, ∆Σ AD or DA converters can have high resolution despite a coarse (by usualone-bit) quantizer.
∆Σ modulators reduce quantization noise by linear filtersin feedback loops, which are designed to shape the frequency
characteristic of the noise transfer function (NTF). The design
is commonly done by assuming that the quantization noiseis white, and independent of the input signal. Although theassumption is not strictly valid, the method leads to a linear
model. We can then adopt linear system theory, in particular,frequency domain approach. Noise shaping in the frequency
domain can be executed by the established H ∞ optimization,and hence such a linear model will be useful to design ∆Σmodulators. Moreover, attenuating the H ∞ norm of the NTF
leads the stability of binary (1-bit) (Lee criterion[3, 2]) andmulti-bit (see section 2.3) modulators.
∗ nagahara@i.kyoto-u.ac.jp† twada@acs.i.kyoto-u.ac.jp‡ yy@i.kyoto-u.ac.jpr−ΣψQyFig. 1. ∆Σ modulator
Then, we propose a new design method of ∆Σ DA converters. Since DA converters involve both continuous- anddiscrete-time signals, it is necessary for analysis and design
to take the characteristics of both of them into account. For
this purpose, the sampled-data control [4] is an optimal tool.
In the last few years, several studies have been made on digital signal processing via sampled-data control theory [5, 6].
Based on these studies, we propose sampled-data H ∞ optimization for designing ∆Σ converters. By sampled-data H ∞optimization, we can optimize intersample response of thesignals in ∆Σ converters, while only sampled values are optimized by conventional methods. We present design examples
to show that our design is superior to conventional ones.
2. ∆Σ MODULATORS
2.1. Conventional modulators
Fig. 1 shows the block diagram of a conventional ∆Σ modulator. In this figure, the difference between the input r andthe output y is fed back to Σ, which is conventionally an accumulator Σ(z) = 1/(z − 1) and outputs a signal ψ. Then thesignal ψ is quantized by a quantizer Q, which is a piecewise
constant function R → Q where Q is a finite subset of R.
The quantizer Q is a nonlinear system. To make the analysis easy, we introduce a linear model for Q. Define the quantization error n, that is, n := Q(ψ) − ψ. Assuming that theerror n is independent of the input ψ, we take the additivenoise model for the quantizer as shown in Fig. 2. By using
this model, we see that the input-output equation of the conventional modulator is obtained byy=1
Σr+
n = z −1 r + (1 − z −1 )n.
1+Σ
1+Σ

On the other hand, if Hdes (z) is given by an IIR filter,our problem is to approximate Hdes (z) by an FIR filter H(z).
Since desired NTFs are given by their frequency characteristics, approximation of H(z) should be done in the frequency
domain. Therefore, we formulate our problem as an H ∞ optimization:nrΣ−ψ+yProblem 1. Given a stable transfer function Hdes (z) (desiredNTF) and a stable weighting function W (z), find H ∈ S with
H(∞) = 1 which minimize k(H − Hdes )W k∞ .

Fig. 2. Linear model for ∆Σ modulatorSince 1/(1 + Σ) = 1 − z −1 is high-pass, the quantizationnoise is reduced at low frequencies and increased at high frequencies. If the input signal r contains few high frequency
components, we can separate the noise n from the output signal y by an appropriate lowpass filter.
Therefore, the accumulator Σ plays a noise-shaping rolein ∆Σ modulators. In the next section, we generalize thisproperty.
2.2. Quantization noise shaping
We here consider the linear model in Fig. 2. Let S denotethe family of all stable, causal, real-rational transfer functions
andS 0 := {G ∈ S : G is strictly causal}.

Then the optimization is reducible to a linear matrix inequality (LMI) with respect to a matrix variable and the coefficients a1 , . . . , aN [8], and can be effectively solved by standard optimization software (e.g., MATLAB).
Moreover, the zeros of H(z) can be assigned by linearequations (linear constraints) of a1 , . . . , aN . Define nH (z) :=PNz N − k=1 ak z N −k . Then, H(z) has M zeros at z = z0 ifand only if¯dk nH (z) ¯¯= 0, k = 0, 1, . . . , M − 1,dz k ¯z=z00

where d ndzH0(z) := nH (z). The LMI with these linear constraints can be also effectively solved.

Then we characterize Σ for the linear system in Fig. 2.

2.3. Stability Constraints

Lemma 1. The linearized feedback system Fig. 2 is wellposed and internally stable if and only if
½
¾
R0Σ∈
:R∈S
1−RThe linearized model Fig. 2 is useful for analyzing the noise
shaping properties of ∆Σ modulators. The stability of ∆Σ
modulators, however, should be analyzed by considering theirnonlinear behaviors.
To analyze the stability, the H ∞ norm of H(z) (NTF)
is available. For the stability of binary ∆Σ modulators, thefollowing criterion (Lee criterion) is widely used [3, 2]:

Proof. See chapter 5 in [7].
This theorem gives all stabilizing feedback filters. By using the parameter R ∈ S 0 , the input/output relation of thesystem Fig. 2 is given by
y = Rr + (1 − R)n =: Rr + Hn,(1)

where H = 1 − R is the noise transfer function (NTF) to bedesigned. Note that the conventional first order ∆Σ modulator has R(z) = z −1 ∈ S 0 .
In implementation, finite-impulse response (FIR) filters
are often preferred, and hence we assume R is an FIR filter(so is the NTF H), that is,R(z) =NX
k=1

ak z −k ,H(z) = 1 −NX

ak z −kk=1Note that R(z) is always in S 0 . If a desired NTF Hdes (z)is given by an FIR filter, R(z) is obtained by R(z) = 1 −Hdes (z). Since R(z) must be strictly causal, we have to restrict Hdes (∞) = 1.

kHk∞ < 1.5.

(2)Note that this is not a sufficient nor necessary condition forthe stability. For multi-bit modulators with M -step quantizer,
the following is a sufficient condition for the stability [9, 2]:
khk1 ≤ M + 2 − krk∞ ,(3)

where khk1 is the `1 norm of the impulse response of H(z).
Let ν denote the order of H(z). Then, we have the following
relation [10]:khk1 ≤ (2ν + 1)kHk∞ .
By combining this with (3), we have another stability condition.
1(M + 2 − krk∞ ).
(4)
kHk∞ ≤
2ν + 1
From the conditions (2) and (4), attenuation of kHk∞ helps
the stability. Therefore, we add the following stability constraints to the design of modulators:
kHk∞ < C,u

↑LK(z)r∆ΣyHT /L

Frequency ResponseycP (s)30
2010Gain (dB)

0Fig. 3. Oversampling ∆Σ DA converter−10−20−30

wc

−40

e−ds
F (s)

uTyr
↑ L K(z) R(z) HT /L P (s)ec−50−60 −110−011010Frequency (rad/sec)ycFig. 4. Error system for designing DA converter

Fig. 5. Interpolation filters for DA converter: sampled-data
H ∞ design (solid) and equiripple design (dotted)

where C > 0 is a constant (e.g., by Lee criterion, C = 1.5).
This inequality is also reducible to LMI [8] and easily combined with the LMI optimization mentioned above.

Problem 2. Given a stable, strictly proper F (s), stable, proper
P (s), upsampling factor L, delay d, sampling period T , findK(z) which minimizes
kEk∞ := sup

3. DESIGN OF ∆Σ DA CONVERTERSwc

By using the feedback filter R(z) characterized in Lemma
1, we here design DA converters with ∆Σ modulators. To
take account of intersample response, we introduce sampleddata H ∞ optimization. From the input/output relation (1), thesystem from r to y is R(z). In this section, we assume thatR(z) is given.
3.1. Design problem of ∆Σ DA converters
Fig. 3 shows an oversampling ∆Σ DA (digital-to-analog) converter. Assume that the input signal u has sampling time Tand word length b [bits]. The digital signal u is first upsampled by an interpolator [11] K(z)(↑ L). By ↑ L, L − 1 zeros are introduced between two consecutive input values [11].
The following digital filter K(z), called interpolation filter,
operates on the L − 1 zero-valued samples inserted by ↑ L toyield nonzero values between the original samples.
Then the interpolated signal r goes through a ∆Σ modulator, and becomes a signal y whose word length is converted
to another one, by usual 1 [bit]. Then the discrete-time signal
y is converted to a continuous signal by the zero-order holdHT /L with hold time T /L, smoothed by a continuous-time
filter P (s), and finally becomes an analog signal yc .
Our objective here is to design the interpolation filter K(z)to interpolate samples taking account of the analog performance. If we a priori have the knowledge about the characteristic F (s) of the original analog signal (e.g., u is a sampled
data of an orchestral music), we can use it explicitly for design.
Therefore, we consider the error system Fig. 4 for designing the filter K(z). Let E denote the input/output operator
from wc to ec . Then, our design problem is then as follows:∈L2

kec kL2.
kwc kL2

Problem 2 is reducible to a finite-dimensional discretetime optimization [5], and hence the optimal filter K(z) canbe obtained by a standard numerical computation software(e.g., MATLAB).
4. DESIGN EXAMPLE
We here present a design example of ∆Σ DA converters. The
design parameters are as follows: the sampling period T = 1,the upsampling ratio L = 8, the reconstruction delay d =T /L = 1/8, and the analog filters are P (s) = 1/(Tc s + 1)2 ,Tc := T /π andF (s) =1, T = 22.05/π.
(T s + 1)(0.1T s + 1)

Note that the lowpass filter F (s) simulates the frequency energy distribution of a typical orchestral music, which are observed by FFT analysis of analog records of some orchestralmusic. We design the FIR filter R(z) (of order 7) in ∆Σmodulator by LMI [8], and the interpolation filter K(z) by
the sampled-data H ∞ optimization. For comparison, we takeR(z) = z −1 and the equiripple filter for K(z) of order 21 asa conventional design.
The obtained interpolation filters are shown in Fig. 5. The
gain around ω = 1/Tc = π of our filter is relatively large because the filter is designed by considering the lowpass characteristic of P (s).
Then, we simulate the oversampling ∆Σ DA converter
shown in Fig. 3 with a binary quantizer(1, ψ ≥ 0,Q(ψ) = sgn(ψ) =−1 ψ < 0.

ErrorTime Response
1
0.25
0.8
0.6

0.2
Magnitude

0.4

Magnitude

0.2
00.15

0.1
−0.2
−0.4

0.05

−0.6
00

−0.8
−1010203040Time (sec)50607010203040Time (sec)5060708080Fig. 7. Error: sampled-data H ∞ design (solid) and equiripple
design (dotted)(a) Sampled-data H ∞ design

Time Response
1tages of the present method.

0.8
0.6
0.4

6. REFERENCES

Magnitude

0.2
0[1] S. R. Norsworthy, R. Schreier, and G. C. Temes, DeltaSigma Data Converters, IEEE Press, 1997.

−0.2
−0.4
−0.6
−0.8
−1010203040Time (sec)50607080(b) Equiripple designFig. 6. Time response

kec k∞kec k2RMS(ec )Table 1. Comparison of error
Sampled-data design Conventional design
2.08 × 10−1
2.67 × 10−1−1
5.68 × 10
7.21 × 10−1
6.34 × 10−2
8.06 × 10−2

[2] R. Schreier and G. C. Temes, Understanding DeltaSigma Data Converters, Wiley Interscience, 2005.
[3] K. C. H. Chao, S. Nadeem, W. L. Lee, and C. G. Sodini,“A higher order topology for interpolative modulatorsfor oversampling A/D conversion,” IEEE Trans. on Circuits and Systems, vol. Vol. 37, pp. 309–318, 1990.
[4] T. Chen and B. Francis, Optimal Sampled-Data Control
Systems, Springer, 1995.
[5] M. Nagahara and Y. Yamamoto, “A new design forsample-rate converters,” Proc. of 39th IEEE Conf. onDecision and Control, pp. 4296–4301, 2000.
[6] M. Nagahara and Y. Yamamoto, “Optimal design offractional delay FIR filters without band-limiting assumption,” Proc. of IEEE ICASSP, 2005.

We take for the digital input u a sinusoidal wave u[k] =sin(0.1πk), k = 0, 1, 2, . . . , 80. The time responses areshown in Fig. 6. The conventional DA converter shows largeerrors around t = 10, 20, . . .. To see the difference, we show
the absolute error in Fig. 7, and some norms of the error
in Table 1. In the table, RMS is the root-mean-square values defined as follows: For fixed Tf > 0, RMS(ec ) :=o12n R
Tf
12. These comparisons show that our deTf 0 |ec (t)| dt
sign is superior to the conventional one.

[7] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory, Maxwell Macmillan, 1992.

5. CONCLUSIONS[10] M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems, Prentice Hall, 1995.

We have proposed a new design method for ∆Σ modulatorsand oversampling ∆Σ DA converters via H ∞ optimization.
We have presented design examples and shown the advan-[11] P. P. Vaidyanathan, Multirate Systems and Filter Banks,
Prentice Hall, 1993.

[8] Y. Yamamoto, B. D. O. Anderson, M. Nagahara, andY. Koyanagi, “Optimizing FIR approximation for
discrete-time IIR filters,” IEEE Signal Processing Letters, vol. Vol. 10, No. 9, 2003.
[9] J. G. Kenney and L. R. Carley, “Design of multibitnoise-shaping data converters,” Analog Int. Circuits Signal Processing Journal, vol. Vol. 3, pp. 259–272, 1993.