An empirical study of the stability of 4th-order Incremental-I:�-ADCs

Johannes Uhlig and Rene Schliffny

Department of Electrical Engineering

Technische Universitat Dresden, Germany

Emails:{uhlig.schueffn}@iee.ettu-dresden.de

Abstract-In order to find relationships between Noise­Transfer-Function (NTF) characteristics and stability of a Incremental-Delta-Sigma ADC (I-DS-ADC) 40.000 different NTF has been investigated. A fast and easy to use criterion to determine, if an NTF of an I-DS-ADC of 4th-order is likely to be stable was found. The novel criterion is fast and easy to use and covers a much bigger variety of possible NTFs compared to recent criteria.

Index Terms--delta sigma, stability

I. INTRODUCTION

Incremental Delta-Sigma ADCs (I-DS-ADCs) are Delta­

Sigma data converter whose memory elements are initially

reset before each new conversion. The input signal is assumed

to be stable during each conversion. I-DS-ADC show many

advantages for DC-Signal conversion such as lower offset

and gain errors, multiplex capabilities and much easier digital

filtering [l]. The analysis of the stability of a DS-ADC with an order

higher than 2 in general and of I-DS-ADCs in particular is

still an open issue when designing such ADCs.

Higher order DS-ADC consists of cascaded integrators with

various feedback or feed forward paths. Stability of a DS-ADC

means, that all output values of these integrators are limited

during each conversion for the whole dedicated input signal

range. Up to now, there is no successful analytical approach

to determine the stability of DS-ADCs with order higher than

2 [2]. The behavior of an DS-ADCs is uniquely defined by

its Noise-Transfer-Function (NTF) and its Signal-Transfer­

Function (STF) (see Fig. 1). The NTF is the transfer function

from the quantization noise input of the linearized quantizer

to the modulators output and the STF the one from the input

to the output of the modulator. Typically the STF is chosen to

be one. In this case the NTF determines the behavior of the

DS-ADCs and thereby its stability.

Available approaches to check an NTF for stability are either

to complex, require significant computing power or give very

conservative results.

Steiner and Young [3] presented a framework for analyzing

the stability by using a state-space transformation which

Manuscript received October 7, 20 II. This research is founded by the European Union (European Regional

Development Fund) and the State of Saxony

978-1-4577-0516-8/11/$26.00 ©2011 IEEE

Loop- V,",o"o

Filter

Fig. I. General block diagram of a DS-ADC. The behavior is completely defined by its Noise-Transfer-Function (NTF) and Signal-Transfer-Function (STF)

converts the modulator loop in a way that all state-space

variables are decoupled. Wong [4] combined this approach

with continuous-time embedding and Poincare map analysis.

However his simulation results seem not to confirm the theory

and the mathematics beyond the continuous-time embedding

rely more or less on graphical analysis or are nearly in­

tractable.

An comprehensive summary of recent approaches can be

found in [2]. It turns out that even if a pure analytical solution

to verify the NTFs stability will be found the mathematics will

be rather complex because of the strong non-linearity of the

internal quantizer.

Other approaches combine analytical analysis with simula­

tion or computation. Schreier [5] presented an algorithm for

computing an convex positively invariant set which contains

all possible states of the modulators integrators. However

the computational effort is pretty high especially for higher

order modulators. Ritoniemi [6] uses root-locus analysis to

investigate the stability but needs simulations to determine

the actual quantizer gain for each cycle. Yang [7] presents

a method for designing stable NTFs also based on root-locus

analysis but his results are very conservative and therefor not

applicable for the design of high resolution NTFs.

When designing an NTF with optimized characteristics like

resolution, signal swing or easy feasible coefficients an fast

and easy method to check if an NTF is very likely to be stable

is essential because a large number of different NTFs has to

be investigated in this design step. All previously described

approaches are either to complex and therefor not automatable

or require high computational resources. Currently, the most

used criterion is the modified Lee criterion [8] which is based

Fig. 2. Root-Locus for an NTF of 4th-order for varying quantizer gains A

on the criterion published in [9]. This criterion says that an NTF is likely to be stable if the maximum magnitude

found with

NTFquantIn =

Vq'i�tIn = NTF - 1 (6)

and its Infinity-Norm IINTFquantInlloo and Root-Mean­Square Gain rms NT F _qi can be derived.

III. POOL OF CHECKED NTFs

In order to analyze relationships between certain NTF characteristics and the stability 40.000 NTFs of 4th-order has been investigated. All NTFs are in the form

NTF (z) =

(z - 1)4 (7)

(z - pd (z - P2) (Z - P3) (Z - P4)

IINTFlloo = max (INTF (ejw)l) < 1.5 wE [O,IT]

(1) and their poles Pn where randomly chosen with the restriction that

of the NTF, also known as the infinity norm , is smaller than 1.5. Beside the fact that this criterion is not sufficient a lot of freedom of design will get lost when this criterion is used as we will show in the next paragraph.

Based on an empirical study using 40.000 random NTFs we will formulate a new criterion which is fast and easy to use but cover a much bigger variety of NTFs.

II. CHARACTERISTICS OF THE N OISE TRA NSFER FUNCTION

As stated before the behavior of an DS-ADC and therefor its stability is solely defined by its NTF. Goal of our research is to find relationships between certain NTF characteristics and the stability of the NTF.

Important characteristics of an NTF are the Infinity-Norm

which is the maximum magnitude of the NTF according to Eq. 1, and the Root-Mean-Square Gain which is

IT

rmS-f}ainNTF = � J INTF (ejw)12 dw (2)

o

The NTF describes the transfer function of the quantization noise under the assumption that the gain of the linearized quantizer is 1. Actually the gain of the quantizer is

A =

sign (VquantIn) . Vref VquantIn

which leads to a modified NTF depending quantizer gain with [8]

NTF NTF,\ =

A + (1- A)NTF

(3)

on the actual

(4)

If A decreases the poles move towards the unit circle (see Fig. 2). If A < Amin there are poles which lies out of the unit cycle and the modulator tends to get unstable [10]. From Amin a maximum input voltage of the quantizer

1 QuantInmax_stable = � (5)

/\mzn can be calculated for which all poles remain in the unit circle if this voltage is not exceeded.

Like the NTF, a transfer function from the quantization noise input to the input of the quantizer (see Fig. 1) can be

• there are two conjugate complex pairs or one conjugate complex pair and two real poles

• the real part of each pole is Re (Pn) > 0 • the absolute value of each pole is I Pn I < 1. Each NTF has been checked for stability under the as­

sumption that the normalized input voltage range is ±0.5Vref and the I-DS-ADC operates for 1000 cycles for each input value. Stability of an I-DS-ADC can more easily be checked by simulation than for conventional DS-ADC because the input is assumed to be stable and the modulator will operate for a certain number of cycles only. Each NTF has been simulated with 220 linear distributed input values in the range [0 -+ 0.5Vrefl and in addition with 230 linear distributed input values in the range [0.4Vref -+ 0.5Vrefl because instability is more likely if the input voltage increases. An NTF is considered to be stable if the maximum observed quantizer input voltage is max (!VquantInl) < QuantInmax_stable. However, also a much larger bound for max (IVquantInl) e.g. 1000 could be chosen because our investigations show that there is a certain input value for which the quantizers input value starts to oscillate and rises towards infinity provided

�4 "

:0 3 .8 .,

I H OJ

� 2 ,:: ....... ....., ,:: tj ;:l 1

0' 1

Lee crit {jon

-'" " ' .

. •

•

2 3 4

infinity norm NT F

stable

unstable

•

5 6

Fig. 3. Maximum quantizer input voltage {lQuantInmax_stable {Ill Amin for which all Poles of the NTF are within the unit circle ploted

vs. the infinity norm IINTFlloo' Green NTFs are stable while red NTFs are unstable. The blue line shows the border for potential stable NTF specified by the modified Lee criterion which would classify 11% of the stable NTFs as stable

� 4 • •

stable •

• • • •

• • • unstable '"

::c 3 .8 '"

I H d � 2

� ....... ....., � (j ;:l I

0' 0.5 I 1.5 2 2.5 3 3.5 M

rmsGain NT Fquantln

Fig. 4. Maximum quantizer input voltage vQuantInmax_stable Vl/ Amin for which all Poles of the NTF are within the unit circle ploted

vs. rms NT F _qi for all random NTFs which are flat and therefor does not have poles in their magnitude vs. frequency. The boundary can be described

with VI/Amin = 1.06· rmSNTF_qi + 0.08 and Rpos = 80%

� 4 • stable

• • unstable • <u 3 :8 H d � 2 �

....... ....., � (j ;:l I

0' 0.5 I 1.5 2 2.5 3 3.5 4

rmsGain NT Fq'uantI n

Fig. 5. Like Fig. 4 but both flat NTFs and NTFs with poles are used. The Ratio Rpos drops to Rpos = 45%. The boundry can be described with

VI/Amin = 3.4· rmSNTF_qi - 2.26

� 4

'" 3 :8 H d �

� 2 ....... ....., � (j ;:l

0'

infinity norm NT FquantIn

Fig. 6. Maximum quantizer input voltage vQuantInmax_stable VI/Amin plotted vs. IINTFquantlnlloo' The boundary can be described

with VI/ Amin = 1.16· liNT Fquantln 1100 - 0.38

that max(!VquantInl) > QuantInmax_stable occur in a certain

conversion step.

IV. NTF CHARACTERISTICS VS. STABILITY

The quality of a criterion to select potential stable NTFs

can be measured as the ration

#positive classified NTFs Rpos = (8)

#all stable NTFs

between the number of NTFs which will be classified as stable

and the total number of stable NTFs.

If we apply the former described modified Lee criterion

on the pool of NTFs to select potential stable NTFs, only

Rpos = 11 % of the stable NTFs will be selected (see Fig.

3). However we did not find any NTF with II NT FII 00 < 1.5 which is unstable under the condition named in Sec. III. This

confirms the modified Lee criterion but a lot of freedom of

design will get lost because only a small part of the stable

NTFs will be classified as potential stable.

Under the assumption of white quantization noise the

transfered noise to the quantizer input should depend on

rms NT F _qi . Because instability is very likely if the quan­

tizer input voltage exceeds QuantI nmax_stable stability of

an NTF should depend on the ratio between rms NT F _qi and

QuantI nmax_stable. Even though the quantization noise is not white for DC­

Inputs, we found a distinct relationship between rms NT F _qi and QuantInmax_stable. If only flat NTFs (see Fig. 8) are

considered a ratio of Rpos = 80% can be achieved by

using a linear relationship between {/ QuantI nmax_stable and

rmSNTLqi (see Fig. 4). If also NTFs with poles will be taken

into account this ratio drops to Rpos = 45% (see Fig. 5).

Apparently modulators tend to be unstable even for mod­

erate values of rmSNTF_qi if whose NTF is not flat. We

could identify the liNT FquantIn 1100 to be significant be­

side rms NT F _qi for the stability. Fig. 8 shows one sta­

ble and one unstable NTF with nearly identical character­

istics beside a significant difference for the infinity norm

IINTFquantInlloo' The significance of rmSNTF_qi together

with the IINTFquantlnlloo for the stability is also obvious in

Fig. 7.

The boundary between stable and unstable NTFs can be

described with

canst = (rmsNTF_qi - rmSNTF_qi,max( {/l/Amin))' . (1lNTFquantInlloo -IINTFquantInlloo,max ({/l/Amin))

(9)

with rmSNTF qi max and IINTFquantInl1 based on the - , oo,max boundaries shown in Fig. 5 and Fig. 6

rms NT F _qi, max

liNT FquantInlloo,max

�+2.26 3.4

�+O.38 l.16

(10)

(11)

; �> 1.1 < 1.1 � �> 1.2 < 1.2 � �> 1.4 < 1.5

• � � " 6 " 6 .. " 6 , " ,

"," 0-

0. "," f-< 4 f-< 4 f-< 4 <: .. <: <: � � � : i �

2 � 2 �

2 0 0 0 . .. , . � � � I I I ....,

0.5 1 1.5 2 2.5 ....,

0.5 1 1.5 2 2.5 ....,

0.5 1 1.5 2 2.5 .� � .� rms _gainNTFquan-tfn rms _gainNTFwuan-tIn rms _gain NTFquan-tIn

" " � o/i7;\ > 1.6 < 1.6 �> 1.8 < 1.9 � 2.1 < 2.2

• � " " 6 " 6 ........... , "

"," 0-

" f-< 4 f-< 4 <: <: � � � 2 ·� tI �S � � �··

� 2 0 0 -; ...... ;. '

� � ... .. .

I I ....,

0.5 1 1.5 2 2.5 ....,

0.5 .� .�

.... ,

1.5

.

.. . · ,1 .:

2 2.5

" 6 " "," f-< 4 <: � � 2 0

� I ...., .�

.. , .. . . . '

...... ·.:.I. s-

. :.;,;,-

';p.'

0.5 1 1.5 2 2.5

rms _gainNTFquan-tfn rms _gain NT F quan-tI n rms _gainNTFquan-tfn

Fig. 7. IINTFquantlnl1 vs. rmSNTF_qi for different intervals of {.Ii7\, blue line shows boundary between areas treated as stable or unstable according to Eq. 12 with canst = 0':04

rti' o

:"'.. �50 � h �100

Frequency .:��

Frequency :��

Fig. 8. Magnitude of two exemplary NTF to illustrate the difference between NTFs which are flat and NTFs which has poles in their magnitude. The green NTF is stable, the red NTF unstable. Both NTFs show nearly identical characteristics with rmS--9ainNTF = 1.6, rmSNTF_qi = 1.3, liNT FII= = 2.0 and QuantInmax_stable = 3.9 resp 4.0 but differ significantly with IINTFquantInll= = 1.6 resp. 2.1.

According to Eq. 9 - 11 an NTF is likely to be stable if

{h/Amin + 2.26 rmSNTF_qi <

3.4 or

canst IINTFquantInlloo <

�+2.26 + (12)

rmSNTF_qi - 3.4

{h/Amin + 0.38

+ 1.16

If canst = 0.04 is choosen there is no unstable NTF in the

investigated pool which will misleadingly classified as stable.

With this criterion (see also Fig. 7) a ratio Rpos = 85% can be

reached which is a considerable improvement compared to the

11 % which can be reached with the Lee criterion. Increasing

canst will improve Rpos further but at the cost that it will

be more likely to get unstable NTFs which are misleadingly

classified as stable.

V. CONCLUSION

A novel criterion for selecting stable NTFs for 4th-order

I-DS-ADCs has been presented. The criterion is is fast and

easy to use and increases the variety of NTFs which can be

considered significantly compared to the so far used Lee crite­

rion. Further research will be done to generalize the criterion

to cover also NTFs of different order and to consider arbitrary

input signal ranges. The criterion was found empirical. Work

will be done to investigate the analytical background of the

found criterion.

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