Design of ∆Σ Converters via Sampled-Data
H ∞ Optimization
Masaaki Nagahara∗ , Toshihiro Wada† and Yutaka Yamamoto‡∗ † ‡∗Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japannagahara@i.kyoto-u.ac.jp, † twada@acs.i.kyoto-u.ac.jp, ‡ yy@i.kyoto-u.ac.jp

Abstract

assuming that the quantization noise is white, andindependent of the input signal. Although the assumption is not strictly valid, the method leads to alinear 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 2 or H ∞ control, and hence sucha linear model will be useful to design ∆Σ modulators. However, stability analysis cannot be easily
applied because coarse quantizers in feedback loops
are highly nonlinear. To solve this problem, we directly treat the nonlinearity by applying a Lyapunov
method. By using this, we will show a sufficient condition for stability.

In this paper, we propose a new method for designing ∆Σ converters via sampled-data H ∞ optimal control. 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 toattenuate quantization noise. In contrast, we give all
stabilizing controllers for the modulator, and proposean H ∞ design to shape the frequency response of thesystem from the noise to the output. The other is a
design for multirate filters in oversampling AD/DA
converters. While conventional designs are executedin the discrete-time domain, we take account of thecharacteristic of the original analog signal by using
On the other hand, a novel approach to designing
sampled-data H ∞ optimization. Design examples arepresented to show that our design is superior to con- ∆Σ converters is proposed by Quevedo and Goodwin[8]. They design a generalized ∆Σ converter, called
ventional ones.
Multi-Step-Optimal-Converter (MSOC), by formulating the design problem as a weighted H 2 optimization, which is reduced to a constrained multi-step
1 Introduction
optimization. Motivated by this work, we propose a∆Σ modulators are widely used in high-resolution AD new design method of ∆Σ converters. Since AD or(Analog-to-Digital) or DA (Digital-to-Analog) con- DA converters involve both continuous- and discreteverters, which are applied to measurement, digital time signals, it is necessary for analysis and design toaudio processing and wireless communications (see take the characteristics of both of them into account.
[6, 12, 7]). In combination with oversampling tech- For this purpose, the sampled-data control [1, 11] isnique, ∆Σ AD or DA converters can have high resolu- an optimal tool. In the last few years, several studtion despite quantizing signals by a coarse (by usual ies have been made on digital signal processing via
one-bit) quantizer. They reduce quantization noise sampled-data control theory [4, 5]. Based on theseby linear filters in feedback loops, which shape the studies, we propose sampled-data H ∞ optimization
frequency distribution of noise.
for designing ∆Σ converters. By sampled-data H ∞The analysis and design are commonly done by optimization, we can optimize intersample response

r
−

Σ

ψ

Q

y

n
r
−

Σ

ψ

+

y

Figure 1: ∆Σ Modulator
Figure 2: Linear Model for ∆Σ Modulatorof the signals in ∆Σ converters, while only sampledvalues are optimized by conventional methods. We
Since 1/(1 + Σ) = 1 − z −1 is high-pass, the quantizapresent design examples to show that our design istion noise is reduced at low frequencies and increased
superior to conventional ones.
at high frequencies. If the input signal r contains
few high frequency components, we can separate thenoise n from the output signal y by an appropriate
2 ∆Σ Modulators
lowpass filter.
Therefore, the accumulator Σ plays a noise-shaping
In this section, we study ∆Σ modulators. First, werole
in ∆Σ modulators. In the next section, we genintroduce a linear model for quantization. By useralize
this property.
ing this linear model, we characterize all stabilizing
controllers (including the conventional controller).
Then, regarding quantizers as nonlinear systems, we 2.2 Quantization noise shaping
investigate stability of ∆Σ modulators.
We here consider the linear model in Fig. 2. Let S
denote the family of all stable, proper, real-rational
2.1 Conventional modulatorstransfer functions andFig. 1 shows the block diagram of the conventional
S 0 := {G ∈ S : G is strictly proper}.
∆Σ modulator. In this figure, the difference between
the input r and the output y is fed back to Σ, which Then we characterize Σ for the linear system in Fig.
is conventionally an accumulator2.
Σ(z) =z −1,
1 − z −1

Lemma 1. The linearized feedback system Fig. 2 iswell-posed and internally stable if and only if
½
¾
R
0
Σ∈
:R∈S
1−R

and outputs a signal ψ.
Then the signal ψ is quantized by a quantizer Q,
which is a piecewise constant function R → Q where
Q is a finite subset of R.
Proof. See chapter 5 in [2].
The quantizer Q is a nonlinear system. To makeThis lemma gives all stabilizing feedback filters. By
the analysis easy, we introduce a linear model for0
Q. Define the quantization error n, that is, n := using the parameter R ∈ S , the input/output relaQ(ψ) − ψ. Assuming that the error n is independent tion of the system Fig. 2 is given byof the input ψ, we take the additive noise model for
y = Rr + (1 − R)n =: Rr + Hn,(1)
the quantizer as shown in Fig. 2. By using this model,
we see that the input-output equation is obtained bywhere H is the noise transfer function (NTF) to be
designed. Note that the conventional first order ∆Σ1Σr+n = z −1 r + (1 − z −1 )n.
y=modulator has R(z) = z −1 ∈ S 0 .
1+Σ
1+Σ

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

ak z −k ,H(z) = 1 −NXak z −kthe ∆Σ modulator shown in Fig. 1 is given by
σx = (AΣ − BC)x + B(r − n),y = Cx + n,n = Q(Cx) − Cx.

(2)

Since R ∈ S 0 , all eigenvalues of A := AΣ − BC lie
inside the unit circle. By the Lyapunov stability theNote that R(z) is always in S 0 . If a desired NTF
ory, there exist X = X T > 0 and Y = Y T > 0 such
Hdes (z) is given by an FIR filter, R(z) is obtainedthat
by R(z) = 1 − Hdes (z). Since R(z) must be strictly
AT XA − X = −Y.
(3)
causal, we have to restrict Hdes (∞) = 1.
On the other hand, if Hdes (z) is given by an IIR fil- We assume the following:
ter, our problem is to approximate Hdes (z) by an FIR
filter H(z). Since desired NTFs are given by their fre- Assumption 1. For the quantizer Q, there exist
quency characteristics, approximation of H(z) should M > 0 and δ > 0 such that if |ψ| ≤ M thenbe done in the frequency domain. Therefore, we for- |Q(ψ) − ψ| ≤ δ.
mulate our problem as an H ∞ optimization:
Under this assumption, we have the following
proposition.
Problem 1. Given a stable transfer function Hdes (z)(desired NTF) and a stable weighting function W (z), Theorem 1. Assume thatfind H ∈ S with H(∞) = 1 which minimize k(H −M
Hdes )W k∞ .
(4)
Ωγ ≤
kCk
Then the optimization is reducible to a linear mawheretrix inequality (LMI) with respect to a matrix variable and the coefficients a1 , . . . , aN [10], and can beΩ := kBk(krk∞ + δ),
effectively solved by standard optimization software
γ := max{α, β},s(e.g., MATLAB).
|λ(X) − λ(Y )|β 2 + 2kXAkβ + λ(X)Moreover, the zeros of H(z) can be assigned by linα :=,ear equations (linear constraints) of a1 , . . . , aN . Deλ(X)PNqfine nH (z) := z N − k=1 ak z N −k . Then, H(z) has
kXAk2 + λ(X)λ(Y )kXAk+
M zeros at z = z0 if and only if
β :=
.
λ(Y )
¯
dk nH (z) ¯¯= 0, k = 0, 1, . . . , M − 1,
Then the region
dz k ¯z=z0
¾
½
M2
n
0
d0 nH (z)R := x ∈ R : x Xx ≤ λ(X)(5)
where dz0 := nH (z). The LMI with these linear
kCk2
constraints can be also effectively solved.
is an invariant set for the system (2).
k=1

2.3

k=1

Stability of quantized feedback Proof. Let x be in R. Then, x0 Xx ≤ λ(X)M 2 /kCk2
and |x| ≤ M/kCk. It follows that |Cx| ≤ M and bysystems

We here consider stability of ∆Σ modulators as nonlinear systems. Let a state space realization of Σ be{AΣ , B, C}. Then the state space representation of

the assumption 1, we have |n| = |Q(Cx) − Cx| ≤ δ.
Put w := B(r − n). Then we have|w| ≤ kBk(krk∞ + δ) = Ω.

(6)

Define V (x) := x0 Xx. Thenrc

V (σx) − V (x)= (Ax + w)0 X(Ax + w) − x0 Xx= −x0 Y x + 2w0 XAx + w0 Xw
2

Sh/N

r

∆Σ

y

K1 (z)↓Nv

Figure 3: Oversampling ∆Σ AD converter2≤ −λ(Y )|x| + 2ΩkXAk|x| + λ(X)Ω .
If |x| ≥ Ωβ, we have V (σx) − V (x) ≤ 0, and hencex ∈ R. If |x| ≤ Ωβ,(σx)0 X(σx)ª©
≤ λ(X) − λ(Y )}|x|2 + 2ΩkXAk|x| + λ(X)Ω2©
ª≤ |λ(X) − λ(Y )|β 2 + 2kXAkβ + λ(X) Ω2≤ λ(X)γ 2 Ω2
M2
.
kCk2

input/output relation (1), the system from n to y islinear in R, and the design problem is formulated as a
one-block sampled-data H ∞ optimization. Then, via
fast discretization method [3], we reduce the problem
to a discrete-time H ∞ optimization.

3.1

Design problem of ∆Σ AD converters

Fig. 3 shows an oversampling ∆Σ AD (analog-todigital) converter. The input analog signal rc is sampled by the sampler Sh/N with sampling time h/N ,It follows that σx ∈ R.
and becomes a discrete-time signal r. Then, a ∆Σ
modulator
modulate the signal r to become another
By this theorem, taking x(0) ∈ R results the state
digitalsignaly, whose word length is usually 1 [bit].
x(k) is bounded for all k > 0. In practice, we can set
Thesignalyis downsampled by the decimator [9]the initial state x(0) := 0, and hence the feedback
(↓
N
)H(z)
to
becomea discrete-time v with samplingsystem is stable in the sense of Theorem 1.
time h. The filter K1 (z) is called a decimation filter,
Example 1. Consider the conventional ∆Σ modu- which eliminates aliases that the downsampler ↓ N
lator shown in Fig. 1 with Σ(z) = 1/(1 − z) and the will cause. By the downsampler ↓ N , the sampling
quantizer
time is reduced to h by taking every N -th output(
sample of the filter K1 (z). Finally, we have a digital1,
ψ > 0,(7) signal v with sampling time h and word length b [bit].
Q(ψ) = sgn(ψ) =Our objective here is to design the decimation fil−1,
ψ < 0.
ter K1 (z) to reconstruct the original analog signal
Then we have AΣ = 1, B = 1, C = 1, and M = 2, rc . Conventionally, the filter is designed assumingδ = 1. Since A = AΣ − BC = 0, the Lyapunov that the analog signal rc is fully band-limited byequation (3) is satisfied for arbitrary X > 0 and Y > the Nyquist frequency π/h. It follows that the fil0. By Theorem 1, if krk∞ ≤ 1 and |x(0)| ≤ 2, then ter K1 (z) is designed to approximate the ideal low|x(k)| ≤ 2 for all k ≥ 0.
pass filter whose cutoff frequency is π/h [9]. The real
analog signals, however, have the spectra beyond the
Nyquist frequency, and hence this design will not be
3 Design of ∆Σ AD/DA Con- necessarily optimal.
Therefore, we propose an alternative method thatverters
takes account of the analog performance, in particBy using the controller characterized in Lemma 1, ular, the frequency component over the Nyquist frewe here design AD or DA converters with ∆Σ mod- quency. For this purpose, we consider the error sysulators. To take account of intersample response, we tem shown in Fig. 4. In the figure, we take the linear
introduce sampled-data H ∞ optimization. From the model for the ∆Σ modulator. The delay e−ds in the≤ λ(X)

wc

F

e−ds

rc
Sh/N

r y
v
R + K1 ↓N Hh

ec
−

wc

P vc

F

rc

e−ds

u
Sh ↑N K2

r y
R + Hh/N

1−R
n

↑N

K(z)

r

∆Σ

P yc

1−R
n

Figure 4: Error system T1 for AD converter
u

ec
−

y

Hh/N

P (s)

Figure 6: Error system T2 for DA converterycThen 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]. Thenthe discrete-time signal y is converted to a continuous
Figure 5: Oversampling ∆Σ DA converter
signal by the zero order hold Hh/N with hold timeh/N , smoothed by a continuous-time filter P (s), andupper portion of the diagram corresponds to the fact finally becomes an analog signal yc .
that we allow a certain amount of time delay for sigOur objective here is to design the interpolation filnal reconstruction. To make the design easy, we take ter K2 (z) to interpolate samples taking account of thethe linear model 1 for the ∆Σ modulator.
analog performance. If we a priori have the knowlLet T1 = [T1w , T1n ] denote the input/output oper- edge about the characteristic of the original analog
ators from the input [wc , n]T to the error ec . Our signal (e.g., u is a sampled data of an orchestral mudesign problem is then as follows:sic), we can use it for reconstruction.
Therefore, in the same way as our design of AD
Problem 2. Given a stable, strictly proper F (s), sta- converters, we consider the error system Fig. 6 forble, proper P (s), stable, strictly proper R(z), upsam- designing the filter K (z). Obviously, K (z) cannot
22pling factor N , delay d, sampling period h, find K1 (z) control the quantization noise, and hence we here dewhich minimizes
sign not only K2 (z) but also R(z). Our design problem is then as follows:
kec kL2q
kT1 k∞ :=sup.
wc ∈L2 ,n∈`2
kwc k2L2 + knk2`2
Problem 3. Given a stable, strictly proper F (s), stable, proper P (s), upsampling factor N , delay L, sam3.2 Design problem of ∆Σ DA con- pling period h, find K2 (z) and R(z) which minimizes

verters
Fig. 5 shows an oversampling ∆Σ DA (digital-toanalog) converter. Assume that the input signal u
has sampling time h and word length b [bits]. The
digital signal u is first upsampled by a interpolator[9] (↑ N )K2 (z). By ↑ N , N − 1 zeros are introduced
between two consecutive input values [9]. The following digital filter K2 (z), called interpolation filter,
operates on the N − 1 zero-valued samples insertedby ↑ N to yield nonzero values between the original
samples.

kec kL2
wc ∈L2 kwc kL2

kT2w k∞ := supandkec kL2.
n∈`2 knk`2

kT2n k∞ := supNote that T2n depends only on R(z) while T2w on
K2 (z) and R(z). To obtain K2 (z) and R(z), we first
optimize kT2n k∞ for R(z), then we fix R(z) and optimize kT2w k∞ for K2 (z).

3.3

£

¤

Fast discretization of sampled- and S is a matrix S := 1 0 . . . 0 .
By using these formulae, our problems are reduced
data systems

Problem 2 and 3 involve a continuous-time delay component e−ds , and hence they are infinite-dimensional
sampled-data problems. To avoid this difficulty, we
employ the fast discretizaition method [3]. By thismethod, our design problems are approximated byfinite-dimensional discrete-time problems assumingthat the delay time d is mh, m ∈ N.
First, we define the discrete-time lifting (or blocking) operator [1].
Definition 1. Define the discrete-time lifting LN
and its inverse L−1
N by
¤T
£
LN := (↓ N ) 1 z . . . z N −1 ,£¤−1
. . . z −N +1 (↑ N ).
L−1
N := 1 z
Then, the fast-discretized system T1 of thesampled-data system T1 is given as follows.
£¤
T1 (z) = T1w (z) T1n (z) ,fd (z)H K
f1 (z)R(z)}efd (z),
T1w (z) = {z −m − PFfd (z)H K
f1 (z){I − R(z)},eT1n (z) = −P
fd , P
fd , Re and Kf1 are discrete-time LTI syswhere Ftems given by
fd := LN Fd L−1 ,FN

Fd := ST /N F HT /N

fd := LN Pd L−1 ,PNe := LN RL−1 ,RN£f
K1 := 1 0 . . .

Pd := ST /N P HT /N ,¤0 LN K1 L−1
N ,

£
¤T
and H is a matrix H := 1 . . . 1 .
In the same way, the fast-discretized systems of T2w
and T2n are given as follows.
fd (z)R(z)e K
f2 (z)S}F
fd (z),
T2w (z) = {z −m − P
T2n (z) = −Pd (z){1 − R(z)},f2 (z) is a discrete-time LTI systems given by
where K£f2 := LN K2 L−1 1 0
K
N

...

¤T

0

,

to discrete-time H ∞ optimization ones. By discretetime H ∞ optimization, we can obtain the optimal
f1 (z), K
f2 (z) and R(z). We reconstruct K1 (z) andK
f1 (z) and K
f2 (z) respectively, by the folK2 (z) from K
lowing equations.
£¤
f1 (z N ) 1 z . . . z N −1 T ,K1 (z) = z −N K£¤
f2 (z N ).
K2 (z) = 1 z −1 . . . z −N +1 K

4

Design Example

4.1

Design of AD converters

We here present a design example of ∆Σ AD converters. The design parameters are as follows: thesampling period h = 1, the upsampling ratio N = 8,
the reconstruction delay d = h/N = 1/8 (m = 1), thefilter in ∆Σ modulator R(z) = z −1 (i.e., conventional∆Σ), and the analog filters are1, T = 22.05/π,(T s + 1)(0.1T s + 1)
P (s) = 1.

F (s) =

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 orchestral music. For comparison, we
compare our filter with the equiripple one obtained
by Parks-McClellan method [9] of order 41. ParksMcClellan method is widely used for designing FIR
filters.
Fig. 7 shows the frequency responses of the decimation filters. The equiripple filter shows the sharper
decay beyond the cut-off frequency ω = π/2, while
the filter by sampled-data design shows a rather slow
decay.
To see the difference, we simulate the oversampling ∆Σ AD converter shown in Fig. 3 with the
one-bit quantizer given by (7) for Q. Fig. 8 shows
the time responses against a rectangular wave. The
sampled-data designed ∆Σ AD converter well reconstructs the rectangular wave, while the decimatorFrequency Resposne
10

Time Response

0

1

−10

0.8

−20

0.6
0.4

−40
0.2
Magnitude

Gain (dB)

−30

−50
−60

0
−0.2

−70
−0.4

−80
−0.6

−90
−0.8

−100 −1
10

0

1

10

10

−1

Frequency (rad/sec)

0

10

20

30

40

50

60

70

80

60

70

80

Time (sec)Figure 7: Decimation filters for AD converter:
sampled-data H ∞ design (solid) and equiripple design (dotted)(a) Sampled-data H ∞ design

Time Response1with the equiripple filter exhibits an amount of ringing around the edges. This is because the equiripple
design does not take account of higher frequency thanthe Nyquist frequency.

0.8
0.6
0.4

Design of DA converters

We here present a design example of ∆Σ DA converters. The design parameters are the same as thoseused above except for
P (s) =

1,
(Tc s + 1)2

Tc :=

h
.
π

Magnitude

0.2

4.2

0
−0.2
−0.4
−0.6
−0.8
−1
0

10

20

30

40

50

We design the filter in ∆Σ modulator R(z) and theTime (sec)interpolation filter K2 (z) by the sampled-data H ∞optimization. By assuming that R(z) is an FIR filter(b) Equiripple designof order 7, we design R(z) ∈ S 0 via LMI optimization[10]. For comparison, we take R(z) = z −1 and theequiripple filter for K2 (z) of order 21 as a convenFigure 8: Time response
tional design.
The obtained interpolation filters are shown in Fig.
9. The gain around ω = 1/Tc = π of our filter isThen, we simulate the oversampling ∆Σ DA conrelatively large because the filter is designed by converter shown in Fig. 5 with the one-bit quantizer (7)sidering the lowpass characteristic of P (s).

Frequency Response
30

Time Response
1

20
0.8

10
0.6

0
0.2
Magnitude

Gain (dB)

0.4

−10
−20
−30

0
−0.2

−40

−0.4

−50

−0.6
−0.8

−60 −1
10

0

1

10

10

−1
0

Frequency (rad/sec)

10

Figure 9: Interpolation filters for DA converter:
sampled-data H ∞ design (solid) and equiripple design (dotted)

20

30

40
Time (sec)

50

60

70

80

60

70

80

(a) Sampled-data H ∞ design

Time Response
1

Table 1: Comparison of error
Conventional
2.67 × 10−1
7.21 × 10−1
8.06 × 10−2

for Q. We take for the digital input u a sinusoidal
wave

0.6
0.4
0.2
Magnitude

kec k∞
kec k2RMS(ec )Proposed
2.08 × 10−1
5.68 × 10−1
6.34 × 10−2

0.8

0
−0.2
−0.4
−0.6
−0.8

u[k] = sin(0.1πk),k = 0, 1, 2, . . . , 80.

The time responses are shown in Fig. 10. The conventional DA converter shows large errors aroundt = 10, 20, . . .. To see the difference, we show the absolute error in Fig. 11, and some norms of the error in
Table 1. In the table, RMS is the root-mean-squarevalues defined as follows (cf. the definition of pow):
For fixed Tf > 0, RMS is defined as(
RMS(φc ) :=

1
Tf

Z Tf
0

)21
|φc (t)|2 dt

.

−1
0

10

20

30

40
Time (sec)50(b) Equiripple design

Figure 10: Time response

These comparisons show that our design is superior
to the conventional one.

controllers, IEEE Trans. Autom. Control, AC37, pp. 214–223, 1992.

Error[4] M. Nagahara and Y. Yamamoto, A new design
for sample-rate converters, Proc. of 39th IEEE
Conf. on Decision and Control, pp. 4296–4301,
2000.

0.25

Magnitude

0.2

0.15

[5] M. Nagahara and Y. Yamamoto, Optimal design of fractional delay FIR filters without bandlimiting assumption, Proc. of IEEE ICASSP,
2005.

0.1

0.05

0
0

10

20

30

40
Time (sec)

50

60

70

80

Figure 11: Error: sampled-data H ∞ design (solid)
and equiripple design (dotted)

5Conclusions[6] S. R. Norsworthy, R. Schreier and G. C.
Temes, Delta-Sigma Data Converters, IEEE
Press, 1997.
[7] J. G. Proakis, Digital Communications, McGraw
Hill, 2000.
[8] D. E. Quevedo and G. C. Goodwin, Multistep optimal analog-to-digital conversion, IEEE
Trans. Circuits Syst. I, Vol. 51, No. 12, 2004.

[9] P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993.
We have proposed a new design method for ∆Σ converters via sampled-data H ∞ optimization. We havealso discussed the stability and the performance of [10] Y. Yamamoto, B. D. O. Anderson, M. Nagahara and Y. Koyanagi, Optimizing FIR Approx∆Σ converters as nonlinear systems. We have preimation for discrete-time IIR filters, IEEE Signal
sented design examples and shown the advantages ofProcessing Letters, Vol. 10, No. 9, 2003.
the present method.
A further direction of this study will be to apply [11] Y. Yamamoto and P. P. Khargonekar, Frequency
sampled-data control to the MSOC converters [8].
response of sampled-data systems, IEEE Trans.
Their converter show a better performance than conAutom. Control, AC-41, pp. 166–176, 1996.
ventional ones, and sampled-data optimization will[12] U. Zölzer, Digital Audio Signal Processing, John
improve the performance further.
Wiley & Sons, 1997.

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Control Systems, Springer, 1995.
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