Analysis and compensation of log_domain filter deviations due to transistor nonidealities

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Analysis and Compensation of Log-Domain Filter

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Abstract

Log-domain filters have recently received considerable research attention as an

intriguing alternative to the existing continuous-time filter implementations. Log-domain filtering explicitly employs the diode nature of bipolar transistors, resulting in a class of frequency-shaping translinear circuits. It demonstrates potential in high-speed and low- power applications. Most interesting of d l , it opens the door to elegantly realizing a linear system with inherently non-linear (logarithmic- exponential) circuit building blocks, and may benefit from the advantages offered by companding signal processing.

However, log-domain filters suffer directly from transistor-level nonidealities. This Lhesis will study the filter response deviations due to major transistor imperfections, which include parasitic emitter and base resistances, finite beta, Early effect, and area

mismatches. SPICE simulations, both large and small-signal andysis, are perforrned to

verify the results. By understanding the underlying deviation mechanisms, very natural and simple electronic compensation methods are proposed. The analysis will cover both biquadratic and high-order ladder log-domain filters.

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Résumé

Les filtres logarithmiques ont rộcemment reỗu un intộrờt considộrable e n tant qu'alternative pour la conception de filtres analogiques. Ils exploitent la nature diode des transistors bipolaires résultant en une classe de circuits de filtres translinéaires. Ils démontrent un potentiel dans les applications à haute vitesse et à faible puissance. Plus intéressant encore, ils offrent une alternative élégante pour la réalisation de systèmes linéaires tout en utilisant des circuits de base non linéaires (logarithmiques-exponentiels). Ils peuvent aussi bénéficier des propriétés avantageuses offertes par la méthode de traitement de signal connue sous le nom "companding".

Cependant, les filtres logarithmiques sont affectés directement par les effets non- idéaux au niveau des transistors. Cette thèse présente une étude sur la déviation des caractéristiques des filtres logarithmiques due aux imperfections des transistors. Ces imperfections incluent les résistances parasites de la base et de l'émetteur, le gain limité de courant, l'effet de Early, et la différence physique des transistors.

Les

résultats sont vérifiés avec des simulations de SPICE en utilisant de grands et petits signaux. La compréhension du mécanisme de déviation des différentes caractéristiques des filtres permet des méthodes simples de compensation. L'analyse inclue des filtres logarithmiques bi-quadratiques et des filtres d'ordres plus élevés.

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Acknowledgments

1 extend my sincere appreciation to my supervisor, Professor Gordon W. Roberts, whose expertise and insight has guided me throughout the course of this research. He

instilled in me the fun of analog circuit design as an i n t r i m g science and a charming art.

1 am inspired by his perseverance in pursuing a research goal, enthusiasm in teaching, and

tireless effort in ensuring our work is presented with precision and clarity.

1 am deeply grateful to the many memben of the MACS lab for providing a stimulating and supportive research environment. Thanks to al1 my friends and colleagues; Mourad, Arman, Choon, John, Loai, Benoit who have always been a source of leaming and encouragement.

Most irnportantly, 1 would like to thank rny family for their love and support. And my wife Venus, who has always been my true companioa through al1 the joyful as weil as the difficult times. To whom 1 lovingly dedicate this dissertation. Lastly, 1 must express my uunost gratitude to God for His steadfast love and wonderful guidance. 1 see it a special blessing having the chance to present my first research result in Hong Kong, the place where 1 was bom ruid felt so dear, in our honeymoon.

iii

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Table of Contents

Abstract

...

i .

.

Résumé

...

11

Table of Contents

...

iv .

. ...

1.2.3 LOG and EXP Operators

...

8

...

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2 Synthesis of Log-Domain Filters

...

17

2.1 Exponential State-Space S ynthesis

...

1 7 2.2 LC Ladder(SFG) Synthesis

...

2.2.1 Log-Domain Lowpass Biquadratic Filter

...

2.2.2 Log-Domain Bandpass Biquadratic Filter

...

23

4.2.1 Classical Theories of Nonideal LC Ladder

....

4.2.2 Applications to Log-Domain Filters

... ...

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4.3 Compensation of Hig h-Order Log-Domain Filters

...

66

4.3.1 Parasitic Emitter Resistances (RE)

...

67

...

4.3.2 Finite Beta 68 4.3.3 Early Voltages

...

72

...

4.3.4 Extension to High-Order Bandpass Log-Domain Ladder Filters 73 4.4 Irnplementation Considerations Under Finite Beta

...

7 4 4.5 Summary

...

80

1 Synthesis of High-Oder Log-Domain Filter by Cascade o f Biquads

...

84

2 Deviation of Equation (3.15). Compensation of RE

...

87

3 Goodness of Fit Test

...

... ... 88

...

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1-4 Signal companding by LOG and

EXP

operators ... I O

1-5 Signal-flow-graph of a typical log-domain sub-circuit

...

10 1-6 (a) SFG of an arbitrary system to be implemented in 1og.domain . (b) An

implementation with individually-linearized log-domain sub.circuits

.

economical log-domain system implementation

...

I l . .

...

1-8 Log-domain positive and negative integrator pair 13

...

. .

...

2- 1 Simplified demonstration of exponential state-space synthesis method

...

2-2 Synthesis of lowpass log-domain biquad: (a) passive prototype. (b) the linear SFG.

...

vii

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Synthesis of bandpass logdomairi biquad: (a) passive prototype. (b) the linear SFG.

6th-order C hebyshev bandpass LC ladder prototype

...

27

continued next page

... .... ...

28

6th-order Chebyshev bandpass filter: (a) the log-domain SFG, (b) the actual circuit implementation

...

Sensitivity of high-order log-domain filters on capacitor variations

...

Sensitivity of high-order log-domain fdters on transistor area mismatches ... 31

Simulations of log-domain filters with ideal and realistic transistor models: (a) lowpass biquad filter, (b) bandpass biquad filter, (c) 7th-order Chebyshev lowpass ladder filter, and (d) 6th-order Chebyshev bandpass filter

...

Log-domain cells showing the parasitic emitter resistances

...

35

Effecu of RE on: (a) log-domain integrator, and (b) the log-domain biquad ... 37

Effects of RE on filter cutoff frequency

...

38

...

Simulated results of nonzero RE compensation 39 Effects of finite beta: (a) feedback mechanism of log-domain cells, (b) log-domain

...

integrator, and (c) log-domain biquad 41

...

Effects of fmite beta on filter Q factor 42 (a) Compensation of the nonideal effects due to finite beta, and (b) the simulated results

...

43

Effects of RB on filter cutoff frequency

...

44

...

Nonideal log-domain biquad SFG due to Early effects 46 3- 10 Effecu of finite Early voltage on: (a) cutoff frequency. (b) filter Q; and (c) fdter gain

...

viii

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3- 1 1 Nonideal log-domain biquad SFG due to area mismatches

...

48 3- 12 Filter response deviations of the log-domain biquad when the variance of area

mismatch equals 0.001

...

,.

...

3 1

...

4- 1 Models for (a) a dissipative inductor. and (b) a dissipative capacitor 61

4-2 Nonideal models for (a) an inductor and (b) a capacitor . It shows both the component . . .

tolerance and the parasitic dissipation

...

.62

4-7 Passive ladder equivalence of log-domain fdter under fmite beta

...

71

4-8 Compensation of the nonideal effects due to finite beta

...

71 4-9 Compensations of fmite beta on high-order log-domain ladder fdter

...

7 2

4- 1 O Compensations of Early effects on high-order log-domain ladder filter

...

73 4- 1 1 Passive ladder equivalence of nonideal log-domain bandpass ladder filter ... 7 3

4- 12 Four topologies of the log-domain bandpass biquad filter

... ...

76 4- 13 Simulated frequency responses showing the effects of beta under different

topologies (a) bandpass biquad. (b) lowpriss biquad ... 77

4- 14 Four topologies of the log-domain lowpass biquad filter

...

A- 1 7th-order log-domain fïlter by biquad-cascade

...

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List of Tables

Component values for the 7th-order lowpass LC ladder ... 23

Component values for the 6th-order bandpass LC ladder

...

27

Filter performances under different mismatch conditions

...

Monte Car10 simulations showing effects of area mismatches ... 52

Combined effects of device nonidealities

...

Summary of log-domain bandpass biquad deviations

...

Magnitude and phase errors of tog-domain integrators

... ...

58

Effects of RE on the high-order lowpass log-domain filter

...

Effects of beta on the high-order lowpass log-domain filter

...

Corn bined effects of device nonidealities

...

Qualitative description of logdomain filter deviations

...

8 0 Capacitors of the 7th-order biquad-cascade log-domain füter

...

87

A-2 Goodness-of-fit test on the simulated data shown in Section 3.6

...

89

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Introduction

Log-domain filters have recently received tremendous research attention as an inuiguing alternative to the existing conrinuous-time filter implementation methods, such as MOS-C and Transconductance-C

111.

They were originally invented by Adams [2] in

1979 for easy electronic tunability, and recently rekindled by Frey who proposed a general exponential state-space design strategy [3]. Logdomain filtering explicitly employs the

diode nature of bipolar transistors, resulting in a class of frequency-shaping translinear cir- cuit [4]. It demonstrates promising results in high-speed [5]-[6] and low-power applica- tions [ 7 ] . Most interesting of d l , it opens the door to elegantly realïzing a linear system

with inherently non-linear (logarithmic- exponential) circuit building blocks, and may

achieve the advantageous potential of companding (cornpress- expand) signal processing [SI.

To make filter synthesis possible without getting into the complicated exponential state-space equations, Perry and Roberts presented the log-domain signal-flow-graph (SFG) approach [9]. This scheme is essentially a direct extension of, but not limited to [IO]-[Il], the classical operational simulation of LC ladders. High-order lowpass log- domain filters were designed and verified experirnentally. This systematic approach not

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only benefits from retaining the low passband sensitivity of LC ladders, but it also demon- strates how one can constmct a linear system from nonhnear elements with minimum amount of linearization circuit- More recently, it has been shown how to construct arbi- trary filter structures that are both DC and AC stable from log-domain integrators, such as bandpass filter realizations of [123.

We believe that before a new filtering technique can be industridy appiied, a thor- oug h and systematic understanding of its nonideal behavior is indispensable [ 13l. It cnables a designer to predict the deviations in advance, s o that he/ she can over-design the filter to allow margins for performance variations. Moreover, nonideality anaiysis is the pre-requisite of filter compensation and on-chip automatic tuning. This is where this rcsearc h begins.

This dissertation will focus on analyzing, quantifying and categorizing the log- domain filter deviations due to major transistor nonidealities. Based on the findings, elec-

tronic compensation techniques are also proposed.

For any practical system, an input to output iinearity is always desired. However, it would be intriguing to enquire: to achieve a linear system, must one always starts from

linear building blocks? [t is well known that transistors are nonlinear in nature: bipolar transistors are exponential, whereas MOS transistors are govemed by a square-law. S o far, trcmendous efforts have been invested in linearizing these inherently nonlinear devices by claborate circuit tricks, increasing power consurnption, reducing operating speed and kceping minute signal swing. Among them the technique of negative feedback is a good example. It is then natural to investigate if it is possible to utilire a transistor the way they intrinsicdly behave, while the resulting system is stili iinear. Maybe by doing so we can min in terms of speed, distortion, power and circuit sirnplicity.

C

A natural starting point would be to review the translinear theory, which carries with it the connotation of "lying somewhere between the familiar home temtories of the Iinear circuit and the formidable terrains of the nonlinear" [14].

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Chapter 1 : Inrruduction

1.2.1 Translinear Principle

Translinear circuits achieve a wide range of algebraic functions by exploiting the proportionality of transconductance to collector currents in bipolar transistors. This class of circuits share the properties that the inputs and outputs are entirely in the form of cur-

rents. And in fact, the small voltage variations due to the signais, which typically equal to tens of millivolts, are only of incidentai interest. Besides, the circuit function is essentialiy independent of the overall magnitude of the signals, but rather on the current ratios within the circuit. Desirably, the function is insensitive to temperature variations in the full range of operation on silicon. To illustrate the principle, we can begin with the fundamental expression relating the coilector current, IC, and the base emitter voltage, VBE, as described by

where VT is the thermal voltage, kT/q, and Is(T) denotes the saturation current. It should be noted that Is is a strong function of temperature: it c m Vary by 9.5% per O C [4]. When the device is driven by a certain VsE, this level of temperature dependency will make the rcsulting Ic virtually unpredictable. Although this relationship is at the very heart of every

bipolar device, it is not vastly popula. in everyday design practice.

Conversely, when the transistor is driven by Ic to produce VBE, the temperature dcpendcncy is now greatly reduced. Rewriting (1.1) as

a very exact and linear relationship between the logarithm of Ic and VBE is shown. When a couple of these devices are connected in a speciai manner to be demonstrated shortly, the resuiting circuit can be made completely temperature independent. Furthermore, an

impressive list of mathematical functions can be readily achieved. This leads us to the dis- cussion of the translinear principle.

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This junction voltage will typically represent the base-emitter voltage VBE of a bipolar

device. By the same token, the junction current will correspond to the bipolar transistor collector current Ic. Therefore, based on (1.2), (1 -3) can be rewritten as

In a monolithic process where transistors are implemented in close proximity, it is generally valid to assume equal thermal voltage of

all

junctions. Therefore, we can wnte

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To eliminate the dependency of (1 -6) on temperature, the saturation current terms

should cmcel out. This would require that N I =

Nỵ,

and

N

(= N I + N2) be an even nurnber.

In other words, there must be equal number of CW and CCW elements c o ~ e c t e d

together? and the loop must comprise an even number of elements. Therefore, w e c m write

where h is a dimensionless number denoting their ratio. Most often. when

A

= 1 , the uea of the bipolar transistors are identical, or they are well matched for pairs of oppositely connected elements. Equation (1.6) can then be rewritten as

The last equation is called the translinear pnnciple. To summarize, it is re-stated as

As an extension to the principle, when a voltage source

V,

is inuoduced into the loop, (1 -8) would become

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Chapier 1 : Introduction

after straight-forward derivation which is left to the reader.

1.2.2 Translinear Circuit Examples

One of the eariiest use of the translinear principle was in realizing a wideband amplifier and an anaiog multiplier [15]-[16]. Here, we wiU describe the elegant example of

Type

"B'.

two-quadrant translinear multiplier, as shown in Figure 1-2 [17]. It's operation will bc described here to demonstrate the simplicity and the practicality of the principle. The multiplier consists of four transistor arranged in a loop, two in each direction. Assurn-

in2 the transistors are appropriately biased, collector currents of ( 1 f X ) I , and

( 1

+

Y ) I , / 2 are generated in Ql-Q4, where X and Y are modulation indices lying between

-

1 to + l t . According to the translinear principle (1.8), we c m write by inspection the fol-

Iowing trmslinear relationship,

Now. if we substitute the appropriate transistor collector current as given in Figure 1-2, we

which implies

If the output is taken as the differential c u m n t (1,) between collectors of Q3 and

Q,,

the two-quadrant multiplication is achieved,

j-. The use of dimensionless modulation indices is often helpful in the analysis in translinear circuit, where

the actual magnitudes of the currents are of secondary concem than theu ratios.

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Figure 1-2: Type

"B"

two-quadrant translinear multiplier

where X represents the AC input signal, and the bias current Iy controis the multiplier gain. Four quadrant multiplier is achieved by superimposing two of these loops and sharing transistors Q I and

Q,

[16]. It should be noted that the above analysis is an exact large-sig-

nal analysis, and is completely temperature insensitive. However. it does assume ideai

translinear elements with perfect diode exponential property, zero ohmic resistances, infi-

nite beta, and that they are perfectly matched.

Shown in Figure 1-3 is another example of the translinear principle. It is appropn-

ately narned the voltage-programmable current rnirror [18]. Similarly, this circuit is com-

Figure 1-3: Voltage-programmable current mirror

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posed of a single translinear loop with four complementary transistors. However, a voltage

source,

VG,

is now inserted into the loop. Taking this into account and according to the modified translinear principle in (1.9), w e c m directly wnte

which is giving a current rnirror relationship, in which the output current Io,, is now mod- ulated by the difference of two voltages, V G = V I

- V Z .

Obviously, a wide range of cur-

rcnt gain is now reahzable by altering VG.

The second circuit just demonstrated happens to be the corner-stone of the log- domain filtering technique. It is the key eiement for converting linear signals into their compressed form for signal processing, and expanding them to restore overall linearity. It will be re-introduced from a different perspective in the next section, and we will see how the translinear circuit can be applied in the frequency domain, producing compact and intnguing filter circuits.

1.2.3 LOG

a d

EXP Operators

The circuit shown in Figure 1-3 whose behavior is described by Eqn. ( 1-14) relates signals in the linear form (such as I,,, IWf) as well as those buried in the exponential func-

rion

(VG=

V I

- V2).

It is possible to employ this propeny to implement logarithmic signal compression, and likewise, exponential signal expansion. Therefore, we will explicitly distinguish the compressed and uncompressed signals as log-domain and iinear signals, respectively. These compression/ expansion functions cm be defined by the following pair

of cornp1ernentax-y mathematical operatorst:

+.

Notice that these operators are slightiy different from those presented by Perry and Roberts in (91

to more appropriately describe the log-domain integrator circuit in the next chapter.

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2. forcing I,,, to be the bias current

l,;

3. connecting V, to ground, and finally,

4. the logarithmically-compressed (log-domain) signal will appear as voltage

V7.

By the same token, the exponential signal expansion operator (EXP) is reaiized by:

1. applying the log-domain input voltage to V I ;

2. setting IM to be the bias current 1,;

3. connecting V , to ground, and finally,

4. the current Io,, will be the exponentially-expanded linear output current, which equals ( 1

+

.

1, , where Y is the modulation index from O to 1.

The companding scheme discussed above is illustrated in Figure 1-4, in which a LOG and MP circuits are connected together. Voltage is the log-domain (compressed) signal, while the linear signals are represented by XI, and YI,. Due to the inverse nature of LOG m d EXP operators, Le.,

EXP(L0

G(x)) = x , X is identical to

Y:

1.2.4 Linearization of Log-Domain System

A typical log-domain sub-circuit c m be characterized by the SFG shown in Figure 1-5. The linear function H(s), which can b e summation, scaling, integration o r any combi- nation of h e m , is embedded between t h e

E X .

and LOG operators. Signals

Gi

and

fo

t. It should be noted that physically, Io is the bias current that carries the ac input current 1, on it.

As common to al1 Class A circuits, the condition II,) < 1, must be satisfied.

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Chapter 1: Introduction

LOG operator

EXP

opemtor

Figure 1-4: Signal companding by LOG and

E X .

operators

represent the log-domain inputs and output, respectively. Due to the LOG-linear-EXP for- mat of the cell, an isolated log-domain transfer function,

fo/ci,

would be non-linear. In

order to irnplement a practical linear systern from this building block, linearization is undoubtedly necessary.

Suppose an arbitrary system as shown in Figure 1-6(a) is to be built using the log- domain circuit of Figure 1-5. Without loss of generality, H f i ) can be any Linex mathemat- ical function. One straight-forward way to tackle this problem (but rather redundant as

will become obvious shortly) would be to abut external LOG and

EXP

blocks to the

UO

euch log-domain sub-circuit. This will result in an overall linear input-output relationship as displayed in Figure 1-6(b).

u

Figure 1-5: Signal-flow-graph of a typical log-àomain sub-circuit

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Chapter 1: Inrroditction

A more economical way is to simply connect the log-domain sub-circuits together, and let the LOG and EXP operators cancel themselves naturally [93. Note that tfüs linear- ization takes place in both feedforward and feedback signal path. The only extra compo-

nents to add would be the input LOG and the output

EXP

blocks, as demonstrated in Figure 1 -6(c).

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In summary, suppose we have a list of log-domain sub-circuits of the form shown in Figure 1-5 that implement a variety of functions. Then we would simply need to join these blocks together according to the specific topology, add the inverse operators at the input and output, m d the desired linear system will result- As will be demonstrated in sub- sequent chapters. this linearization technique is the foundation of log-domain filter synthe- sis, while the sub-circuits to be employed are known as log-domain integrators.

Therefore, in the next section, we will analyze the log-domain integrator in detail. I t will be dernonstrated that the integrator indeed c o n f o m s to the paradigm EXP-H(s)-

LOG as illustrated in Figure 1-5. Besides, some of its interesthg features, such as the darnping and scaling functions, will be highlighted. It tums out that these characteristics,

as will be appreciated in the coming chapters, are the keys to electronic compensation.

1.2.5 Log-domain Integrators

Log-domain integntors are at the heart of the log-domain filtering technique. To

start, the voltage-programmable current mirror (Figure 1-3), together with its opposite-

polarity counterpart, are re-drawn in Figure 1-7. In the log-domain literature, they are also

called "log-domain cells". By writing

KVL

equations around the QI-Q4 translinear loop,

the basic log-domain equation is given by

Combining the two log-domain cells of Figure 1-7, and adding a capacitor, the log-domain integrator [12] is formed as shown in Figure 1-8. Applying KCL a t node P, we can write

where

Y,, Yin

and

Y ,

denote the log-domain positive input negative input, and log-

km v,)

domain output, respectively. Multiplying through by e and applying the chain

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Chapter 1: Introduction

Figure 1-7: Log-domain cells with opposite polarities

rule will result in

If we define a pair of inverse LOG and EXP mappings as in (1.15) which is recaptured below.

we c m rewrite (1.18) as

Figure 1-8: Log-domain positive and negative integrator pair

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Chapter I : Introduction

This can also be symbolically represented by the SFG shown in Figure 1-9.

There are two points that worth paying attention to: (i) As revealed from (1.20),

the bias current I o c m be Viewed as to "scale" the capacitor. It is this factor that accounts for the electronic tunability of this integrator and the log-domain filters'. (ii) According to

( 1-17}, damping (when

Vin

Po)

c m be realized by replacing the righr log-domain ce11 of Figure 1-8 by a dc current source. The damped log-domain integrator is s h o w in

Figure 1 - 10. These two simple insights are the keys to analyze and compensate for the log- domain filter nonidealities.

This thesis will descnbe the synthesis of log-domain filters based on the linearization technique and the log-domain integrator discussed previously. Then, the cffects of major transistor imperfections on filter responses will be studied. Insiphts about

C...-...-...-...ri...--..-..-....~....---...--.-..-.-.,.-...~...~.~-.~.---...-.--.-..-....-...---.---.

Fi y r e 1-9: Log-domain integrator Signal-Flow-Graph

t. The factor 1 1 VT also makes the integrator and the resulting filter temperature-dependent.

According to the design method outlined in [9], where the log-domain filter is designed to oper-

ate at 2S°C, any fluctuations in operating temperature (T, in OC) will inuoduce a scaiar error k to the in tegrator, which equals

X: = 273.15 + 25

273.15 + T

As will becorne evident later, k will also represent the final filter cutoffl center tiequency deviâ- tion. For a temperature insensitive design, PTAT current bias is therefore necessary.

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Cliapter 1: Introduction

Figure 1-10: Damped log-domain integrator

the various deviation mechanisms are offered, which wili also lead to straight-forward

compensation schemes.

Chapter 2 demonstrates the log-domain filter synthesis based on simulation of LC ladder. Four log-domain filter circuit examples will be illustrated. They are namely the

lowpass biquad, the bandpass biquad, the 7th-order lowpass Chebyshev, and the 6th-order

bandpass Chebyshev filters. Circuits presented here will serve as the examples for the nonideaiity studies in subsequent chapters.

Chapter 3 discusses the effects of transistor nonidealities on log-domain biquadratic filters. Toward that goal, the log-domain integrator is thoroughly analyzed under parasitic emitter and base resistances, finite beta, Early effect, and area mismatches.

The results are used to predict the nonideal lowpass and bandpass biquadratic filter responses. Through Our understanding of the deviation mechanisms, very natural

clectronic compensation methods are presented. SPICE simulations, both large and smali- signal analysis, are provided to support the findings.

Chapter 4 reflects on the previous nonideality study, and extends the biquadratic filter analysis to the high-order filter regime. Based on classicd LC ladder theories, high- order log-domain filter deviations due to transistor nonidealities are quantitatively achieved. To promote better understanding of the deviation mechanism, quivalent nonideal passive ladders are presented as a visual aid. Effective electronic compensation

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schemes, similar to that of the biquadratic case, are proposed.

The final chapter summarizes the resufts of the logdomain nonideality analysis, and addresses areas of future research.

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CHAPTER 2 Synthesis of Log-Domain Filters

Synthesis of analog filters can be achieved in many different ways. Among h e m , state-space synthesis

and

operational simulation of LC ladders are getting wide-spread use. Synthesis of log-domain filters can be understood as a direct extension of what exist for their linear ancestors. Here, we would briefly review two approaches: the exponential state-space and the log-domain signal-flow-graph synthesis. For illustration, several log- domain filter circuits will be presented. They will also serve as the objects of our nonideality studies to be presented in later chapters.

When the research of logdomain filters was rekindled by Frey in 1993 [3], it was synthesized using the "exponentiai state-space" method. Its general idea is described

below. Consider a dynamic state-space representation of an arbitrary filter function:

where x = ( x l , x2,

. . .

, x,)' is the vector of state variables' u and y are the input and output scalars respectively, and A,

, ,

, B , , ,

Cl,,

are the matrices with their dimensions shown by the subscripts. Also, if we map the state variables xi and scalar u to

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the exponenual of voltages

V i

and

U

according toT

Eq. (2.1) can then be rewritten as

-v,/ v,

Multiply the first expression in (2.3) by ( C i V T ) e , we obtain

where the

Ci

are arbitrary constants. In order to interpret (2.4) as a set of KCL equations to be realized by actual circuit elements, we will rewrite it as

w here

For the circuit implementation, the following observations are utiiized:

I . T h e term

ci Y ,

on the left side of (2.5) represents the current flowing into the grounded capacitor

C i

tied to node i.

T. Notice that our convention of specifying logdomain signais by circumflex (A) is not followed here, in order to present the original look of the exponentiai state-space synthesis method in [3].

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Chapter 2: Synthesis of Log-Domain Filters

2. Each item on the right side of (2.5) denotes a cument flowing through a diode, with a

"composite" voltage of, Say, Vj

+

Va, - Vi or

U +

Vbi - Vi across it.

3. Vau represents a diode voltage due to a forward-biased current of magnitude lCiAijVd.

Similar argument also holds for

Vbi.

4. When bipolar transistors are used, Vj

+

Vau corresponds to the jfh node voltage

(5)

being Ievel-shifted by a diode-connected transistor (QI) wiih base-emitter voltage

equals V a , . This voltage is then applied to the base of another bipolar transistor (Q2)

whose emitter is in turn connected to the if" node voltage

4.

The coliector current of Q2 will then implement the desired items on the right side of (2.5).

This

is shown in Figure

2-1.

The above procedure is repeated for al1 state-space equations. Similar

manipulations ais0 apply to the second equation in (2.3), which is the input/output

relations. in summary, the exponential state-space synthesis method involves transforming the state-space variables into the node voltages by an appropriate exponentiai mapping, so that the resulting expressions can be realized by bipolar transistors.

So far, only biquadratic filters have been designed from this approach. Strictly

speaking, although high-order log-domain filter is possible by cascading biquads [3], high-order synthesis from a single state-space mode1 has never been successfully tried.

Figure 2-1 : Simplified demonstration of exponentid state-space synthesis method

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Chapter 2: Synthesis of log-Domain FiZrers

This fact exposes a major weakness of this approach: the complexity of the state-space equations can easily become un-manageable as the filter order increases, despite that the synthesis is theoretically sound regardess of filter order. Besides, due to its purely mathematical manner, no t muc h physical insight is offered. Practically, the synthesis requires a fair amount of ad-hoc circuit tricks, thus making the design procedure rather

unsystematic and limited to log-domain experts.

One of the most popular filter synthesis techniques is the method of operational simulation of LC ladders, or thc signai-flow graph (SFG) approach. This method finds the active circuit realization that mimics the internai i-v relationships of the individual L and C clements in the LC network. The benefits are many-fold. Widely accepted by the design community, lossless doubly-tenninated LC ladders that are designed to deliver maximum possible power to the load exhibit a low passband sensiûvity to the inevitable process and elcment variations. The resulting circuit, commonly known as a leapfrog filter, has a one- to-one corres pondence to i ts passive LC ladder ancestor. This promo tes physical understanding about the functionality of various part of the resulting circuit. Designers c m

tell from inspection which part of the circuit is implementing integrations, scaling o r summation etc. As will become obvious in the subsequent chapters, the nonideality analysis, as well as the resulting compensation design, would not be feasible without the physical, o r graphical, insights offered by this SFG approach.

Suggested in [9], but using the revised log-domain integrator given in [12], the log-domain filter synthesis by operational simulation of LC ladder involves the following steps:

1. Find an LC ladder that meets the design specifications

2. Denvc the corresponding SFG from the LC prototype

3. Modify the SFG to its log-domain equivalence by:

a. Placing

EXP

and LOG blocks in front and behind each integrator respectively

b. Placing LOG and

EXP

blocks at the filter input and output respectively

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Chaprer 2: Synritesis of Log- Damain Fil rem

4. Map the log-domain integrator circuit ont0 the log-domain SFG.

Notice that by performing step 3, we are transforrning the filter SFG from the linear-domain into the log-domain, while ensuring overall input-to-output linearity by the

linearization techniques discussed in Section 1.2.4. Utilizing the integrators proposed in Section 1.2.5, four log-domain filter examples based on LC ladder synthesis will be

presented. They are respectively the lowpass/bandpass biquadratichigh-order log-domain

filters. The circuits presented here will serve as the objects for Our nonideality studies in the coming chapters.

2.2.1 Log-Domain Lowpass Biquadratic Filter

The synthesis of the log-domain lowpass biquad filter is outlined in Figure 2-2. It begins with finding the passive prototype (Figure 2-2(a)), followed by deriving the corre-

spondinp linear SFG (Figure 2-2(b)) and the log-domain SFG (Figure 2-2(c)), and finally,

the log-domain filter circuit (Figure 2-2(d)). This fiIter wiU ideaüy realize the transfer Func tion,

wherc

and K (=1) denotes the filter dc gain. Its behavior subject to physical frequencies, Le.

s = ja, c m then be written as

Referring to Figure 2-2, the first integrator (and the associated signal subuaction) is implemented by QI-Q,, Q9-Q12 and capacitor C l . The input LOG operation is also

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Chapter 2: Synthesis of hg-Domain Filters

naturally incorporated by

QI-&

[9]. The second damped integrator is then cornposed of

araB.

capacitor

C2

and the constant current source a t the node of

P2.

Finally, transistors

Q,,-Q16 perform the output EXP function. Notice that the resulting circuit is indeed

identical to the one first proposed by Frey using exponential state-space synthesis method

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Chapter 2: Syntlresis of @-Domain Filters

2.2.2 Log-Domain Bandpass Biquadratic Filter

By means of the synthesis method proposed in [9], the bandpass biquad filter is realized from a LC prototype as shown in Figure 2-3. It implements a transfer function of the form

where oo, Q are identical to that given in (2.7), and K (=I) is the center frequency gain.

Expressed in terms of physicai frequencies s = jo, Eq. (2.9) becomes

which is a conventional bandpass filter response. Notice that a lowpass output is also

available if the signal

Pz

is EXPed (exponentially expanded) [ 121.

2.2.3 High-Order Log-Domain Lowpass Filter

A 7th-order log-domain Chebyshev lowpass filter with 1dB passband ripple and

cutoff frequency of 1 MHz is chosen for our study. Figure 2-4 shows the LC ladder proto- type. By means of filter design program [19]. or filter design handbook [20], the compo-

nents vdues are found and summarized in Table 2- 1.

Table 2-1: Component values for the 7thlorder lowpass

LC

ladder

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Chaprer 2: Synthesis of log-Domain Filrers

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Chapter 2: Synthesis of Log-Domain Filters

Figure 2-4: 7th-order Chebyshev lowpass

LC

ladder prototype

By means of modified nodal anaiysis, each capacitor node is assigned a voltage variable ( V , ,

Vj

, V5, V7 = Vour ), and each inductor is given a current variable (i2, i4, I , ). Their inter-relationships c m be written directly as follows:

(2.1 Ob)

(2.10c)

I

i4

- ..j( v3 -

V 5 ) d t (2.1 Od) 14

Using the above equations, we can draw the corresponding linear SFG. Routinely

adding the LOG and

EXP

blocks according to the niles presented before, and after the following mappings are made,

the log-domain SFG shown in Figure 2-5(a) results.

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Chapter 2: Syntiiesis of Log-Domain Filters

</div>

Analysis and compensation of log_domain filter deviations due to transistor nonidealities

<b>the </b>

<i><b>the </b></i>

<b>Analysis and Compensation of Log-Domain Filter </b>

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<b>Abstract </b>

<b>Résumé </b>

<b>Les </b>

<b>Acknowledgments </b>

<b>Table of Contents </b>

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