OPTIMAL NOISE SHAPING IN ∆Σ MODULATORS VIA GENERALIZED KYP LEMMA
Masaaki Nagahara

Yutaka Yamamoto

Kyoto University
Graduate School of Informatics
Kyoto 606-8501, JAPANnagahara@ieee.org

Kyoto University
Graduate School of Informatics
Kyoto 606-8501, JAPANyy@i.kyoto-u.ac.jp

ABSTRACTIn this paper, we propose a new design method of ∆Σ modulators. First, we analyze the stability of ∆Σ modulators. We
show a parametric representation of all stabilizing loop filters
for a linearized model, and then an analysis of the nonlinear
stability is discussed. Next, by using the parameterization ofloop filters, we propose an optimal design to shape the frequency response of the noise transfer function (NTF). Generalized KYP (Kalman-Yakubovic-Popov) lemma is used toreduce our optimization to a linear matrix inequality.
Index Terms— ∆Σ modulation, quantization, H ∞ optimization, generalized KYP lemma
1. INTRODUCTION∆Σ modulators [1] are widely used in AD (Analog-toDigital) and DA (Digital-to-Analog) converters, in whichhigh performance can be obtained with coarse quantizers.
A fundamental issue in designing ∆Σ modulators is noiseshaping in the frequency domain [1]. A usual solution to thisis to insert accumulator(s) in the feedback loop to attenuate
the gain of the noise transfer function (NTF) in low frequencies. This methodology looks like PID (Proportional-IntegralDerivative) control [2], in which the performance of the designed system depends on the amount of experiences of thedesigner. That is, the conventional design is of an ad hoc nature.
Let us consider a general ∆Σ modulator shown in Fig.
1. In this modulator, Q is a quantizer and H = [H1 , H2 ]uHψQyis a linear filter with 2 inputs and 1 output. The filter H1shapes the signal transfer function (STF) from the input u tothe output y to be 1 in the frequency band of interest. On theother hand, the filter H2 eliminates the in-band quantizationnoise by shaping the NTF.
To shape optimally the NTF in the frequency band of interest, say [0, Ω], we minimize the maximum of the gain of
the NTF in [0, Ω]. This is related to a minimax optimization(or an H ∞ one). We have proposed an H ∞ optimization in[3], in which we have to choose a suitable weighting function
to obtain a good performance. On the other hand, we propose
in this article more useful method with no weighting function,
by generalized Kalman-Yakubovic-Popov (KYP) lemma [4].
Then the optimization can be reduced to one with a linear
matrix inequality (LMI). The idea to apply generalized KYP
lemma to ∆Σ modulator design is also proposed in [5], inwhich they assume one-bit quantizer for Q and optimize theaverage power of the reconstruction error in low frequencies.
In contrast to this approach, our optimization is for quantization noise shaping, which is more familiar to engineers andresearchers in this area. Moreover we assume more generalquantizers and under these quantizers we also analyze the stability of ∆Σ modulators.

2. FREQUENCY SHAPING IN ∆Σ MODULATORSIn this section, we discuss a role of the linear system H =[H1 , H2 ] (the loop filter) in the ∆Σ modulator shown in Fig.
1. To make the analysis easy, we use the linearized model
shown in Fig. 2. By using this model, the input-output equation of the modulator in Fig. 2 is given by y = TSTF u +TNTF n, where
TSTF (z) :=Fig. 1. ∆Σ modulator

H1 (z),
1 − H2 (z)TNTF (z) :=1.
1 − H2 (z)We call TSTF (z) and TNTF (z) the signal transfer function(STF) and the noise transfer function (NTF), respectively.
For a conventional first order ∆Σ modulator, the loop filters

unuH

ψ+yR1++

ψQy− +R2Fig. 2. Linearized model for ∆Σ modulator
Fig. 3. Structure of ∆Σ modulator with design parameters
R1 ∈ S and R2 ∈ S ′are given by
H1 (z) =z,
z−1, H2 (z) = −1.
z−1(1)

Then we have y = u + (1 − z −1)n, and TNTF (z) = 1 − z −1 isa highpass filter. This is a reason for setting the accumulator
1/(z − 1) in the loop. By this, the quantization noise is modulated to high frequencies, and if the input signal u contains
few high frequency components, we can separate the noise nfrom the output signal y by an appropriate lowpass filter. To
sum up, the loop filter H(z) plays a noise-shaping role in ∆Σmodulators.
Before discussing an optimal design of the loop filter, wediscuss the stability of ∆Σ modulators in the next section.

For example, the conventional modulator in (1) has R1 (z) =1 ∈ S and R2 (z) = −z −1 ∈ S ′ .
By (3), the structure of the ∆Σ modulator with the design
parameters R1 ∈ S and R2 ∈ S ′ is shown in Fig. 3. By thisblock diagram, we can interpret the filter R1 as a pre-filter toshape the frequency response of the input signal, and R2 as afeedback gain for the quantization noise Qψ − ψ.
Next, we discuss the stability of nonlinear ∆Σ modulatorsin Fig. 1. We here assume the following.
Assumption 1.
1. The linearized model is stable. That is,
the filter H(z) = [H1 (z), H2 (z)] satisfies (2).
2. There exist real numbers M > 0 and δ > 0 such that if|ψ| ≤ M then |Qψ − ψ| ≤ δ.

3. CHARACTERIZATION OF LOOP FILTERS
In this section, we first characterize all H(z)’s which stabilize
the linearized model shown in Fig. 2, and then we consider
the stability of the nonlinear system in Fig. 1. A necessary
condition that a ∆Σ modulator is stable is that its linearized
model is internally stable. The converse is generally not true,that is, even if the linearized model is stable, the nonlinear
system in Fig. 1 can be unstable.
We first characterize the filter H(z) which internally stabilizes the linearized feedback system. All stabilizing filters
are characterized as follows [3].

Lemma 2. Assume Assumption 1. If |ψ(0)| ≤ M andkr1 k1 kuk∞ + kr2 k1 δ ≤ M then for all k ≥ 0, we have|(Qψ)(k) − ψ(k)| ≤ δ and |ψ(k)| ≤ M , where r1 and r2 arerespectively the impulse responses of R1 and R2 , and k · k1
and k · k∞ are respectively the ℓ1 and ℓ∞ norm of signals.

Lemma 1. The linearized feedback system in Fig. 2 is wellposed and internally stable if and only if

Proof. The proof uses the technique discussed in section
4.2.2 of [1]. We omit the proof.

R2 (z)
R1 (z), H2 (z) =,1 + R2 (z)1 + R2 (z)R1 (z) ∈ S, R2 (z) ∈ S ′ ,This lemma gives a sufficient condition so that the amplitude of the input ψ of the quantizer Q is always less than thesaturation level M of Q. This property is said to be of nooverload, and if this is satisfied a ∆Σ modulator is said to bestable [1]. We here consider the stability more precisely. We
introduce a state space model of the ∆Σ modulator shown inFig. 1, and analyze the stability in the state space.
Let state space equations of the system in Fig. 1 be as
follows.

H1 (z) =(2)where S is the set of all stable, causal, real-rational transfer
functions, and S ′ := {R ∈ S : R is strictly causal}.
By using these parameters R1 ∈ S and R2 ∈ S ′ , we have
TSTF (z) = R1 (z),TNTF (z) = 1 + R2 (z),and the input/output equation of the system in Fig. 2 is givenbyy = R1 u + (1 + R2 )n.
(3)

For example, the uniform quantizer with step size 2δ andno-overload input range [−M, M ] (see section 2.1 of [1]) satisfies the second assumption.
Under these assumptions, we have the following lemma.

x(k + 1) = Ax(k) + Bu u(k) + Bn n(k),ψ(k) = Cx(k) + Du(k),n(k) = (Qψ − ψ)(k).

Consider the ideal state xI (k), which is the state when
there is no quantization, that is, when Q is identity. Define thestate-space error e := x − xI . Then, we have the following
theorem.
Theorem 1. Suppose that the ∆Σ modulator in Fig. 1 satisfies Assumption 1. If |ψ(0)| ≤ M andkr1 k1 kuk∞ + kr2 k1 δ ≤ M(4)Note that R2 (z) is always in S ′ . We then introduce statespace matrices {A, B,
 C(α)}, such that R 2 (z) = C(α)(zI −A)−1 B, where α = α0 α1 . . . αN ,
A=


001..
.

0..

.

..

.



,
10 0 ..  B = . ,0
1then for all k ≥ 0,|e(k)| ≤ βk := δk−1
Xand C(α) = [αN , αN −1 , . . . , α1 ]. Then the inequality (5)can be described as linear matrix inequalities (LMI) by using
the generalized KYP lemma [4].

kAi Bn k.

i=0

Proof. By using the inequality |Ax| ≤ kAk|x| for a vector x
and a matrix A, and by Lemma 2, the theorem can be easily
proved. We omit the proof.
By this theorem, when a ∆Σ modulator satisfies the condition in Theorem 1, the quantization error e(k) in the state
space is bounded by βk . This bound is finite for all k ≥ 0and β∞ := limk→∞ βk is also finite, since the matrix A is
Schur-stable 1 by Assumption 1.1. As a result, the state x(k)is also bounded, and we can conclude that the system is stable
in a weak sense (i.e., bounded but not guaranteed to converge
to zero).
Assuming that kr1 k1 = 1, the condition (4) can be described by the H ∞ norm of R2 , that is, kR2 k∞ ≤ C whereC > 0 is a constant (see [3]). This means that to guarantee
the stability one cannot arbitrarily increase the feedback gainkR2 k∞ . This property is due to the nonlinear nature of ∆Σ
modulators.
4. OPTIMAL LOOP FILTER DESIGN VIA
GENERALIZED KYP LEMMA
In this section, we propose an optimal design of the loop filter
H(z) by using the parameterization in Lemma 1. For simplicity, we assume R1 (z) = 1. This means that the STF is
assumed to be allpass. Then our problem is formulated asfollows.
Problem 1. Given Ω (0 < Ω < π) and γ > 0, find R2 (z) ∈S ′ which satisfies
sup |TNTF (ejω )| < γ.

(5)ω∈[0,Ω]

In implementation, finite impulse response (FIR) filters
are often preferred, and hence we assume that R2 (z) is FIR,
that is, we set
R2 (z) =NX

αk z−k,α0 = 0.

k=0
1 A matrix A is Schur-stable iff ρ(A) < 1 where ρ(A) is the spectral
radius of A.

Theorem 2. The inequality (5) holds if and only if there exist symmetric matrices Q > 0 and P such that the LMI (6)(printed in the top of the next page) holds.
By Theorem 2, the optimal coefficients α1 , . . . αN can beobtained efficiently by standard optimization softwares, e.g.,
MATLAB (See [6]).
5. DESIGN EXAMPLE
In this section, we show examples of designing ∆Σ modulators by the proposed method.
We here design the filter R2 (z) which is an FIR filter with
12 taps, and set R1 (z) = 1. The cut-off frequency Ω is3π/32. The NTF 1+R2 (z) is designed to have a zero at z = 1to attenuate DC noise most, and also to satisfy the stability
condition kTNTF k∞ < 1.5 (these constraints can be describedas linear matrix equality and inequality, see [3]). By this optimization, we obtain the minimum value of γ = 6.48 × 10−2(−23.8 [dB]). Fig. 4 shows TNTF ’s by the proposed methodand the first order ∆Σ modulator. The TNTF of our designshows a lower gain in the low frequency and a higher gain in
the high frequency. The frequency response in Fig. 4 is thatof the linearized system shown in Fig. 2. To see the nonlinear effect in the quantizer, we simulate responses against sinusoidal waves with various frequencies. The reconstruction
filter after the ∆Σ modulator is chosen to be H ∞ optimal one
proposed in [3]. Fig. 5 shows NSR (Noise-to-Signal Ratio)
against sinusoidal waves. The NSR shows that our ∆Σ modulator shows a better response than the conventional one in
all frequencies. Fig. 6 and Fig. 7 shows outputs respectivelyof proposed and conventional ∆Σ converters against a sinusoidal wave.
6. CONCLUSION
In this paper, we have propose a new design method of ∆Σ
modulators. We have characterized the all stabilizing loop filters for linearized model, and analyzed the stability of nonlinear ∆Σ modulators. Then we have formulated our problem ofnoise shaping in the frequency domain. By using generalizedA⊤ P A + QA + A⊤ Q − P − 2Q cos Ω(A⊤ P B + QB)⊤C(α)
A⊤ P B + QBB⊤P B − γ21C(α)⊤
 < 0.
1−1(6)

100.5
0.4

00.3
0.2

−20

Amplitude

Magnitude (dB)

−10−23.8

0.1
0−0.1
−0.2

−30

−0.3
−0.4
−0.5

−40

0

0.5

1

Ω/π

−50
10−2−110
Normalized Frequency (rad/s)

10

1.5Time(sec)

22.5

30Fig. 6. Output against sinusoidal wave (proposed)
Fig. 4. Frequency response of TNTF : proposed (solid line) andconventional (dash)

0.5
0.4
0.3

1.2

Amplitude

0.2

10.8

0.1
0−0.1−0.2

NSR

−0.3
0.6

−0.4
−0.5

0.4

00.5

11.5
Time(sec)

22.5

30.2

000.04

0.08

0.12

0.16

0.20

Frequency(normalized)

Fig. 7. Output against sinusoidal wave (conventional)

Fig. 5. NSR against sinusoidal waves: proposed (solid) andconventional (dash)mization,” Proc. of IEEE ICASSP, vol. III, pp. 612–615,
2006.

KYP lemma, our design is reducible to an LMI optimization.
Design examples have shown efficiency of our method.

[4] T. Iwasaki and S. Hara, “Generalized KYP lemma: unified frequency domain inequalities with design applications,” IEEE Trans. Autom. Control, vol. AC-50, pp. 41–59, 2005.

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[3] M. Nagahara, T. Wada, and Y. Yamamoto, “Design ofoversampling delta-sigma DA converters via H ∞ opti-[5] M. Osqui, M. Roozbehani, and A. Megretski, “Semidefinite programming in analysis and optimization of performance of sigma-delta modulators for low frequencies,”
Proc. of the American Control Conf., pp. 3582–3587,
2007.
[6] 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.