Optimal Design of ∆Σ Modulators via Generalized KYP Lemma
M. Nagahara†1 and Y. Yamamoto†2†

Graduate School of Informatics, Kyoto University, Kyoto, Japan1nagahara@ieee.org, 2 yy@i.kyoto-u.ac.jp

Abstract: In this article, we propose a new design method of ∆Σ modulators. We propose an optimal design to shape thefrequency response of the noise transfer function (NTF). Generalized KYP (Kalman-Yakubovic-Popov) lemma is used toreduce our optimization to a linear matrix inequality. Design examples are illustrated to show effectiveness of our method.
Keywords: Delta-sigma modulator, generalized KYP lemma, quantization.

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
noise shaping in the frequency domain [1]. A usual solution to this is 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-Integral-Derivative) control [2], in which
the performance of the designed system depends on theamount of experiences of the designer. 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 ]is a linear filter with 2 inputs and 1 output. The filter H1shapes the signal transfer function (STF) from the input u
to the output y to have a unity gain in the frequency bandof interest. On the other hand, the filter H2 eliminates thein-band quantization noise by shaping the NTF.
To shape optimally the NTF in the frequency band ofinterest, say [0, Ω], the NTF zero optimization [1] can beused. This method is to minimize the normalized noise
power, given by the integral of the squared magnitude of
the NTF over [0, Ω]. On the other hand, we minimize
the maximum of the gain of the NTF in [0, Ω]. This isrelated to a minimax optimization (or an H ∞ one), andmore effective than the NTF zero optimization in termsof uniform attenuation of the frequency response overthe band. 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 noweighting function, by generalized Kalman-YakubovicPopov (KYP) lemma [4]. Then the optimization can be
reduced to one with a linear matrix inequality (LMI). The

u
H

ψ

Q

y

H

ψ

+

y

Fig. 2 Linearized model for ∆Σ modulator
idea to apply generalized KYP lemma to ∆Σ modulator design is also proposed in [5], in which they assume
one-bit quantizer for Q and optimize the average powerof the reconstruction error in low frequencies. In contrast to this approach, our optimization is for quantizationnoise shaping, which is more familiar to engineers and researchers in this area. Moreover, the zeros of the NTF canbe assigned arbitrarily on the unit circle in the complex
plane with linear matrix equality (LME), and the stability condition can be described by an H ∞ constraint normof the NTF, which can be an LMI. That is, the proposed
method can be described by LMI’s and LME’s, which canbe solved effectively by numerical computation softwares
such as MATLAB. Design examples show effectivenessof our method.

2. CHARACTERIZATION OF LOOPFILTERSIn this section, we first characterize all H(z)’s which
stabilize the linearized model shown in Fig. 2. A necessary condition that a ∆Σ modulator is stable is that its
linearized model is internally stable. Note that the converse is generally not true, that is, even if the linearized
model is stable, the nonlinear system in Fig. 1 can be unstable. A stability condition for the nonlinear system is
discussed in 3.3.
We first characterize the filter H(z) which internallystabilizes the linearized feedback system. All stabilizing
filters are characterized as follows [3].
Lemma 1: The linearized feedback system in Fig.
2 is well-posed and internally stable if and only if
R2 (z)
R1 (z), H2 (z) =,1 + R2 (z)1 + R2 (z) (1)R1 (z) ∈ S, R2 (z) ∈ S ′ ,H1 (z) =Fig. 1 ∆Σ modulator

n

u

u

R1

ψ

+
+

Q



C(α)(zI − A)−1 B, where α = α0 α1 . . . αN ,


 
0 1
0
0


.. ..
 .. 
.
.
, B = 
A=
. ,


.


.
0

. 1
1
0
0

y

− +

R2
Fig. 3 Error-feedback structure of ∆Σ modulator withdesign parameters R1 ∈ S and R2 ∈ S ′ .
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 ′ , the STF
and NTF are given by
TSTF (z) = R1 (z),where

TNTF (z) = 1 + R2 (z),M1 (P, Q) = A⊤ P A + QA + A⊤ Q − P − 2Q cos Ω,and the input/output equation of the system in Fig. 2 isgiven byy = R1 u + (1 + R2 )n.

and C(α) = [αN , αN −1 , . . . , α1 ]. Then the inequality(3) can be described as a linear matrix inequality (LMI)
by using the generalized KYP lemma [4].
Theorem 1: The inequality (3) holds if and only ifthere exist symmetric matrices Q > 0 and P such that
M1 (P, Q) M2 (P, Q) C(α)⊤
 M2 (P, Q) M3 (P, γ 2 ) < 0,
1
C(α)
1
−1

(2)

By (2), the structure of the ∆Σ modulator with thedesign parameters R1 ∈ S and R2 ∈ S ′ is shown in Fig.
3. This structure, called error-feedback structure, is often
applied in the digital loops required in ∆Σ DA converters[1]. By this block diagram, we can interpret the filter R1
as a pre-filter to shape the frequency response of the input
signal, and R2 as a feedback gain for the quantizationnoise Qψ − ψ.

3. OPTIMAL LOOP FILTER DESIGN VIA
LINEAR MATRIX INEQUALITIESAND EQUALITIESIn this section, we propose an optimal design of theloop filter H(z) by using the parameterization in Lemma
1.

M2 (P, Q) = A⊤ P B + QB,
M3 (P, γ 2 ) = B ⊤ P B − γ 2 .
By Theorem 1, the optimal coefficients α1 , . . . αN can beobtained efficiently by standard optimization softwares,
e.g., MATLAB (See [6]).
3.2 NTF zeros
To ensure perfect reconstruction of the DC input level,
and to reduce low-frequency tones, TNTF (z) should havezeros at z = 1, or the frequency ω = 0 [1]. A similar requirement is in bandpass ∆Σ modulator; we set
NTF zeros at a given frequency ω0 , or z = e±jω0 .
The zeros of TNTF (z) can be assigned by linear equations (linear constraints) of α1 , . . . , αN . Define n(z) :=PN
z N + k=1 αk z N −k . Then, TNTF (z) has M zeros atz = z0 if and only if
dk n(z)= 0,
dz k z=z0

k = 0, 1, . . . , M − 1,0

3.1 Optimal noise shaping via generalized KYP
lemma
For simplicity, we assume R1 (z) = 1. This meansthat the STF is assumed to be allpass. Then our problem
is formulated as follows.
Problem 1: Given Ω (0 < Ω < π) and γ > 0, findR2 (z) ∈ S ′ which satisfies
sup |TNTF (ejω )| < γ.

(3)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=0Note that R2 (z) is always in S ′ . We then introduce
state-space matrices {A, B, C(α)}, such that R2 (z) =n(z)where d dz:= n(z). The LMI with these linear con0
straints can be also effectively solved.

3.3 Stability condition
The linearized model Fig. 2 is useful for analyzing thenoise shaping properties of ∆Σ modulators. The stabilityof ∆Σ modulators, however, should be analyzed by considering their nonlinear behaviors. To analyze the stability, the H ∞ norm of TNTF (z) is available. For the stability of binary ∆Σ modulators, the following criterion (Leecriterion) is widely used [7], [1]:kTNTF k∞ < 1.5.

(4)Note that this is not a sufficient nor necessary condition
for the stability. For multi-bit modulators with M -step
quantizer, the following is a sufficient condition for thestability [8], [1]:khk1 ≤ M + 2 − kuk∞ ,

(5)

n
+

+

R

R

R

Q

y

0

−10
Magnitude (dB)

u
+

10

−
−
−
+
+
+
Fig. 4 Cascade of Error Feedback

−20

−23.8

−30

where h is the impulse response of TNTF (z) and u is the
input signal. Let N denote the order of TNTF (z). Then,
we have the following relation [9]:

−40
Ω/π

−50
10

−2

−1

10
Normalized Frequency (rad/s)

100

khk1 ≤ (2N + 1)kTNTF k∞ .
By combining this with (5), we have another stability
condition.
kTNTF k∞ ≤1
(M + 2 − kuk∞ ).
2N + 1(6)

From the conditions (4) and (6), attenuation of kTNTF k∞helps the stability. Therefore, we add the following stability constraints to the design of modulators:
kTNTF k∞ < C,
where C > 0 is a constant (e.g., by Lee criterion, C =1.5). This inequality is also reducible to LMI [6] andeasily combined with the LMI optimization mentioned
above.

4. CASCADE OF ERROR FEEDBACK
FOR HIGH-ORDER MODULATORS
Assume that R1 = 1 and R2 = R. To design a highorder modulator, we can use a cascade of the error feedback modulator in Fig. 3. The proposed cascade structureis shown in Fig. 4. By using this structure, we have
TSTF (z) = 1,

TNTF (z) = (1 + R)M ,where M denotes the number of filters R. If R ∈ S ′ , thenthe linearized feedback system is stable. An advantage ofthis structure is that the number of taps of R can be reduced, and hence the implementation is much easier thana filter with a large number of taps. This structure can beapplied to ∆Σ DA converters.
To satisfy the stability condition kTNTF k∞ < C, thefilter R is designed to limit
k1 + Rk∞ < C 1/M .
If this is satisfied, we havekTNTF k∞ ≤ k1 + RkM∞ < C,
by the sub-multiplicative property of the H ∞ norm. In
this section, we show examples of designing ∆Σ modulators by the proposed method.

Fig. 5 Frequency response of TNTF : proposed (solidline) and conventional (dash)

5. DESIGN EXAMPLES
5.1 ∆Σ Modulator with DC zero
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
is 3π/32. The NTF 1 + R2 (z) is designed to have a zeroat z = 1 to attenuate DC noise most, and also to satisfy
the stability condition kTNTF k∞ < 1.5 (these constraints
can be described as 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. 5 shows
TNTF ’s by the proposed method and the first order ∆Σmodulator. The TNTF of our design shows a lower gain in
the low frequency and a higher gain in the high frequency.
The frequency response in Fig. 5 is that of 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. 6 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. 7 and Fig. 8 shows outputs respectively of proposed and conventional ∆Σ converters against a sinusoidal wave.
5.2 Higher order modulator
We here show a design example of a higer order modulator with cascade structure in Fig. 4. We set R1 = 1, andR2 (z) = R(z) be an FIR filter with 32 taps. The cutoff
frequency Ω is set to be π/32. The FIR filter R(z) is designed by using the LMI in Theorem 1, with the stability
condition kTNTF k∞ < 1.5. The number of cascades M
is 4, that is, the order of the modulator is 32 × 4 = 128.
We also design a modulator by the NTF zero optimization [1] which minimize the normalized noise power of
the NTF. This modulator is designed by the MATLAB
function synthesizeNTF in the Delta-Sigma Toolbox[1], with the order of TNTF is 4, the over sampling ratio(OSR) is 32, and kTNTF k∞ < 1.5.

1.2

0.5
1

0.4

0.8

0.2
Amplitude

NSR

0.3

0.6

0.1
0
−0.1
−0.2

0.4

−0.3
−0.4

0.2

−0.5
0
0

0.04

0.08

0.12

0.16

0

0.20

0.5

1

Frequency(normalized)

Fig. 6 NSR against sinusoidal waves: proposed (solid)
and conventional (dash)

1.5
Time(sec)

2

2.5

3

Fig. 8 Output against sinusoidal wave (conventional)
Frequency Response of NTF
10
0

0.5

−10

0.4

−20
Magnitude (dB)

0.3

Amplitude

0.2
0.1
0
−0.1
−0.2

−30
−40
−50

−49.4
−56.7

−60 −59.6
−70

−0.3

−80

−0.4

−90

−0.5
0

0.5

1

1.5
Time(sec)

2

2.5

3

Fig. 7 Output against sinusoidal wave (proposed)
Fig. 9 show the frequency responses of the proposed
modulator and the NTF zero optimized one. By this figure, the gain of the proposed NTF is uniformly attenuatedover [0, π/32] while the conventional one shows a peak inthe band. The maximal difference between two gains isabout 10 [dB].

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. Based on this, we haveformulated our problem of noise shaping in the frequency
domain. By using generalized KYP lemma, our design isreducible to an LMI optimization. Assignment of NTF
zeros and stability condition are described by LMI’s andLME’s, respectively. Design examples have shown effectiveness of our method.

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