Relationships between Noise Shaping and Nested ...

ENGINEERING REPORTS

Relationships between Noise Shaping and Nested Differentiating Feedback Loops* J. VANDERKOOY,

AES Fellow

Department of Physics, University of Waterloo, Waterloo, Ont. N2L 3Gl, Canada AND M._O. J. HAWKSFORD, Department of Electronic Systems Engineering,

AES Fellow

University of Essex, Colchester C04 3SQ, UK

The application of heavy feedback is studied in two different topologies, namely, multiple-order noise shaping and nested differentiating feedback loops. Both have similar loop gain and stability considerations, although the two approaches have different implied circuit environments and areas of application. In noise shaping, emphasis is placed on the integrator characteristics of each gain stage, whereas fiat-gain stages with highfrequency poles form the usual basis of the nested differentiating loop concept. This engineering report helps in understanding the application of large amounts of feedback to control noise or distortion at baseband frequencies.

0 INTRODUCTION The concept of nesting differentiating feedback loops (NDFLs) has been introduced and promoted by Cherry [1], [2]. The basic idea with NDFLs is that for the stage that creates the most distortion (usually a power output stage), the encompassing differentiating feedback stabilizes the loop at high frequencies while allowing increased feedback at lower frequencies to reduce system distortion significantly. This engineering report makes comparisons between NDFLs and certain high-order noise-shaping loops, where it is shown that with minor _ topological transformations, similar loop behavior · and means of stabilization are observed. I NOISE SHAPING Our approach to the application of large amounts of feedback has come from a digital perspective, namely, that of_oise shaping, as typically applied to requantizers in both multibit and one-bit analog-to-digital and digitalto-analog converters; for reference, see Hawksford [3]. To commence the comparative discussion, Fig. 1 illus* Presented at the 93rd Convention of the Audio Engineering Society, San Francisco, CA, 1992 October 1-4; revised 1999 November 9. 1054

trates a progression from a generalized error-feedback noise-shaping feedback topology to an equivalent canonic form of feedback amplifier. In the noise shaper shown in Fig. l(a) the quantization error is filtered by a z-domain transfer function H(z) and then subtracted retrospectively from the input sequence to enable partial correction for the quantizer error. This classic topology has a signal transfer of unity and a noise-shaping transfer function of [1 - H(z)]. The negative feedback topology of Fig. l(d) is derived through the progression shown in Fig. l(b) and (c), where precise equivalence in terms of both signal and noise-shaping transfer functions is achieved when a(z) -

H(z) 1 - H(z) '

(1)

Fig. l(d) is based on a conventional digital feedback loop, although it is unusual in that a feedforward path x yields the required unity-gain signal transfer function. Since path x is outside the feedback loop, it does not modify the loop transfer function and often can be omitted at the expense of relatively benign high-frequency gain errors. It is common practice in noise shapers that are designed to maximize low-frequency performance, to implement the forward-path transfer function A(z) by cascading digital integrators of the form (1 - z-l) -I. d.AudioEng.Soc.,VoL47,No.12,1999December

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NOISESHAPING ANDNESTED DIFFERENTIATING FEEDBACK LOOPS

However, for a loop order N > 1 (where N is the number

r-.

are required in the closed-loop transfer function to enable the Bode stability criterion [4] to be satisfied. Fig. 2 shows two equivalent methods of loop synthesis based on cascaded integrators. Fig. 2(a) follows the structure presented in [3], configured tozeros supof integrators in whereas the loop),Fig.N 2(b) - 1istransmission

X(z) c

=r'_i._,_Y(z) (a)

port the discussion in Section 2. Again path x provides appropriate signals injected into the forward path to maintain a unity-gain signal transfer function. when The topoiogies of Fig. 2 have identical characteristics ot_ = [3rforr = 1..... N1. By way of example, both a first-order and a secondorder noise shaper are considered, as shown in Figs. 3(a) and4(a), respectively.In the first-ordersystemthe error resulting from quantizer Q is modified by just a sample delay, where Hi(z) = z- 1(there must always be at least a one-sample delay in the loop), whereas for the

X(z) c

[---_

_'_

I '_-_

Y(z)

(b)

From Eq. (1), second-order

Z-I case H2(z) = z-t(2 -- 1 - z-l

-

z -1)

is selected.

X(z) i_

I

'

Y(z)

(c)

Al(Z)[fu'st°rder

Path x A2(z)lsec°nd°raer= 1 -- 2Z-l + z-2

X(z)

(' )

- 1 z-l(2 -z- z1-1 - '1z-l)-z -l + 1 .

Y(z) .

A(_

(d) Fig. 1. Progression from classic noise-shaping configuration to exact equivalent negative-feedback loop with feedforward path x designed to maintain unity-gain signal transfer function.

Al(z) is a single integrator whereas A2(z) consists of two cascaded integrators together with a unity-gain feedforPath x

X(z)

Y(z)

i

(a) Path x

IntegratorI

Integrator2

IntegratorN

(b) Fig. 2. Nth-order digital noise shaper showing two equivalent forms of stabilization. (a) Uses feedforward paths across integrators. (b) Uses equivalent multiple feedback paths; preferred for demonstrating equivalence with NDFL. J. AudioEng.Soc.,Vol.47,No.12,1999December

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ward path, justifying the choice of H2(z). Figs. 3(b), 4(h), and 4(c) show the first- and second-order topolog-

lored at high frequency, for example, by using one of the techniques illustrated in Fig. 2. It follows from the

les redrawn in feedback form, which can be compared · directly with a sigma-delta converter when Q is a twolevel comparator. The order of the loop may be increased further by cascading additional integrators that are tai-

earlier discussion that both these noise shapers have signal transfer functions of unity and that the quantization error (normally considered as noise) is shaped by a response R(z), where

X(z) o

Y(z) Such noise-shaping structures in the literature as well [5].

have been much discussed

2 DISTORTION SHAPING ._

R(z) = (1 - z-t) N . (2) Fig. 5 illustrates a further progression of ideas. Fig. 5(a) shows a third-order digital shaper for which the quantizer output-related error q[z] is treated as an addi-

(a)

X(z) +o____¢_.

Q

_

Y(z)

(b)

live where path is omitted for simplicity, since error, with abut third-order loopx the signal transfer function within the audio band is virtually unity. Fig. 5(b) shows a similar system, but implemented in the analog domain that uses ideal continuous integrators, in which the quantizer is replaced by a continuous but nonlinear output

Fig. 3. Progression of equivalent circuits for first-order digital noise shaper,

stage. The analog circuit can be interpreted as a limiting digital case in which the sampling frequency has become

,_Y(z)

X(z)_+

'_7_ (a) Path x

x

(b) Path x

+ (c) Fig. 4. (a) Progression of equivalent circuits for second-order digital noise shaper. (b) Feedforward compensation. (c) Multiplefeedback-path configuration as presented in Fig. 2. 1056

J. Audio Eng. Soc., Vol. 47, No. 12, 1999 December

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NOISESHAPINGANDNESTEDDIFFERENTIATING FEEDBACKLOOPS

infinite, hence removing all delays, and where the discontinuous output quantizer is replaced with a continuous but nonlinear analog output stage, In the subsequent analysis the following notation is

time constants xlf, x2f, and x3ffor the feedback paths and xls, X2s,and x3sfor the signal paths, and where the output stage has gain G, then the transfer function is

Consider integrator which receives an inputintegrators. Xr(S ) from used to describe the r,function of the analog the signal path and Yr(s)from the feedback path. Then the integrator output Ir(s) is given by

[S3Tl_2sT3s

+

S2TlsT2sT3 'r3f s

-_'

Ir(S

)

=

Xr(s)

$Trs

4.

+

STisT2s 'r3f

X(S)

'_-

+ Tls] 'rlf_lY(s)

S3TIsT2sT3s

G

D(s)

(3)

Yr(s)

where D(s) is the Laplace transform of d(t), the additive error of the analog output stage. It should be observed that in the context of a thirdorder loop, the x parameters define both the stability

S'rrf

Applying this convention to the circuit in Fig. 5(c), where the integrators are shown labeled with respective

q(z)

X(z)

--_

Y(z)

(a) d(t)

x(t)

/

_if

.

J

._

/

x_s -

y(t)

shift

shift

/

/

I

I

I

_

.--or(s)

(c) r ....................

:

n

D(s)i

'tis

' / ', _-.-__ r. r_' 'C_2s _.. "1; i_3s_.,r-_ _!

x.,

g

j

(d) Fig. 5. Derivation of NDFL concept. (a) Digital third-order noise shaper. (b) Analog form. By moving feedback paths labeled x2e and xsf in (b) to the left, and adding compensating differentiators to maintain the same circuit transfer function, NDFL representations of (c) and (d) are achieved. (d) Dotted lines encompass what might be considered a realistic output stage of a

normal amplifier. J.AudioEng.Soc.,VoL47, No.12, 1999December

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criteria and the signal transfer function of the circuit, and that the cube of the signal frequency weights the output-stage error D(s). Consequently this analog feedback circuit can be viewed as a distortion shaper, where the multiple integrators, by virtue of their large lowfrequency gain, reduce the effects of nonlinearity at lower frequencies. The parallels between noise shapers and feedback amplifiers are evident. The similarity with NDFLs is now demonstrated. In the circuit of Fig. 5(c) the two feedback paths applied to the inputs of the last two integrators are shifted to the · left by one integrator stage, and compensating differentition unaltered. Observe that the output signal fed back at°rssx2fands?3fareinsertedt°keepthetransferfunc', directly to the input is related to (1 + S'flf'r2s/T2f ). This has a similar function as the parallel resistor-capacitor phase-advance network typically used in the feedback path of an amplifier. Finally in Fig. 5(d) an extra amplitier with gain g has also been inserted to give the circuit a more usual configuration. It is this modified circuit that we wish to compare with the NDFL circuits of Cherry [1]. In this reconfiguration the output stage is modeled with frequency-independent gain G, but where the dashed box shown in Fig. 5(d) associates the third integrator with the output stage so that it can display real poles. It is also possible to think of each integrator as having a finite low-frequency gain of form a/(1 + sT), more like the actual stages of an amplifier. The first integrator might be associated with an intermediate amplifier stage, the second integrator with the voltage-gain stage, and the third integrator and output block with a more realistic output stage,

·

ity of the loop with respect to the output stage. It is the output stage that generally is considered the dominant source of the distortion, and a high loop gain will act to reduce it. The output-stage loop gainA(s) in the circuit of Fig. 5(b) [and hence also of Fig. 5(c) and (d)] determines the distortion-shaping transfer function, where Y(s) D(s)

1 1 + A(s) '

Setting the input X(s) = 0, then A(s) can be shown to be

A(s) = D(s) Y(s) - 1 = G [s-_3el+

-1 + S2'r2f'T2s

-1 -

s3'r3f'lr2s'r3s

] (4)

and this is easily generalized to higher or lower order. At the highest frequencies, only the first term survives, and for good stability the phase shift must be considerably less than - 180 °, ideally being close to -90 °, the lag of a single integrator. It is the _r3ffeedback path that ensures this stability at high frequencies, but the other loops-allow increasing feedback at lower frequencies, reducing distortion in the process. As an example, let us consider a fifth-order distortionshaper feedback circuit, for which G _ 1. The signal transfer function Y(s)/X(s) is by proper choice of 'r parameters a fifth-order Butterworth unity-gain low-pass filter with cut-off frequency of, say, 50 kHz. The 10op gain can be written as A -sa

s + as 4 + (a + 2)s-3 + (a + 2)s_ + aSln+ 1 5 Sn

- 1

3 DISCUSSION that is, The circuits of Fig. 5(c) and (d) can be identified directly with NDFL circuits, even though there may be differences of small detail, such as introducing zeros in the integrators at high frequency. The nested loops may not all take their feedback signal directly from the output, for example, although there is then a difference in implied circuit environment. In Fig. 5(b) each of the integrators (which in practice will have finite dc gain) is regarded as similar. However, in Fig. 5(c) and (d) thc last integrator is considered to be part of a low-gain output stage describing, for example, a typical emitterfollower output stage of an audio amplifier. Although it is customary to include a Miller compensation capacitor across the voltage-gain stage, if this capacitor also encompasses the output stage, and feeds back to the virtual ground input of the voltage-gain stage, it becomes the s'r2fdifferentiating feedback shown in the diagram. The effect on amplifier distortion is very beneficial, and this point has been emphasized by Cherry [2]. There are other strong parallels between distortionshaping NDFLs and digital noise-shaping topologies, In principle by _adding more stages? the ord er. of each structure is increased, giving even more feedback, at · lower frequencies. Another aspect is the internal stabil1058

a a + 2 a + 2 a 1 A = _i +. _Sn + Sn 23 Sn + _ Sn _ + _ _ Sn

(5)

where xr ' a = 1 + 2 cos

+ 2 cos

and the normalized Laplace variable s, = s/_oo = s/(2'tr fo), and fo = 50 kHz. Fig. 6 is a plot of Eq. (5), showing the desired 6-dB per octave roll-off of the loop gain at high frequencies, but with a gain at lower frequencies proportional tof -5. If the integrators in the circuit have finite dc gain, the graph is similar, but will limit near dc at somehigh value of gain, giving a loop gain for the output stage very similar to that indicated by Cherry [1]. In this example, unity gain occurs at 157 kHz, and the attendant phase shift is - 120 °, representing good stable behavior. Note that at 20 kHz there is already 40 dB of distortion reduction, rising in an ever-increasing way at lower frequencies. For high-order loops Such as discussed her e, this system displays conditional stability, as pointed out by J. Audio Eng. Soc., VoL 47, No. 12,1999 December

ENGINEERING REPORTS +7[ -

0

NOISESHAPING ANDNESTED DIFFERENTIATING FEEDBACK LOOPS I

I

I

I

I

I

I

I

I

I

_ Loop gain'

m

0dB

_

[ Amplitude response

]

l fo = 50 kHz -60 dB I

I

I

I

200000

I

Z0000 0

I (Hz)

200000

Fig. 6. Loop gain as seen by output stage of fifth-order distortion shaper having a transfer function of a fifth-order Butterworth low-pass filter with 50-kHz cut-off frequency. Rising loop gain at low frequencies vastly reduces distortion at these frequencies and effectively introduces distortion shaping.

Cherry [ 1] and in a later discussion of NDFLs [6]. Hence it is important to consider large-signal clipping behavior experimentally or by simulation to see whether the system can be provoked into self-oscillation or other bizarre or chaotic behavior. It is evident that there is no simple limit to the amount of distortion reduction at high orders, but increasing care must be taken in defining stable circuit parameters and behavior for large signals,

a differentiating loop all along, but the general concept of the NDFL puts a firm footing on new aspects of feedback that relate particularly to analog amplifiers. However, the equivalence is important at a conceptual level, especially as the means of stabilizing high-order digital loops was known [8] at the time of invention of NDFL [1]. What this engineering report shows is that there is another way of looking at the application of large amounts of negative feedback, and that the two

4 OVERVIEW

are closelyrelated.

At first glance it almost seems trivial that the reassignmerit of the feedback loops around the integrators in Fig. 5(b) results in the NDFL structure of Fig. 5(c). However, there are differences in circuit realization and circuit tradition. Cherry [1] also works out a great deal of the mathematical aspects of NDFLs with sensitivities and the analysis of appropriate models. Presumably most of this work applies to the distortion-shaping topology as well, with appropriate measuring points or circuit associations in the two approaches. We do not in this engineering report work out such details or attempt a mapping between the two topologies. In some ways the distortion-shaping approach is easier to grasp initially. But NDFLs are perhaps better if one is faced with a traditional class AB audio power amplifier and wishes to improve its performance. In fact this is suggested in the title of one of Cherry's papers [7]. Also, when the NDFL was invented, it appeared as a new result since the traditional methods of stabilizing amplitiers had limitations on the amount of simple feedback that could be applied, as Bode [4] had shown. There have been engineers who have employed selected aspects of

5 REFERENCES

J. AudioEng.Soc.,Vol.47,No.12,1999December

[1] E. M. Cherry, "A New Result in NegativeFeedback Theory and Its Application to Audio Power Amplifiers," IEEE J. Circuit Theory Appl., vol. 6, pp. 265-288 (1978 July). [2] E. M. Cherry, "Nested Differentiating Feedback Loops in Simple Audio Power Amplifiers," J. Audio Eng. Soc., vol. 30, pp. 295-305 (1982 May). [3] M. O. J. Hawksford, "Chaos, Oversampling, and Noise Shaping in Digital-to-Analog Conversion," J. Audio Eng. Soc., vol. 37, pp. 980-1001 (1989 Dec.). [4] H. W. Bode, "Network Analysis and Feedback Amplifier Design (van Nostrand, Princeton, NJ, 1945). [5] M. W. Hauser, "Principles of Oversampling A/D Conversion," J. Audio Eng. Soc., vol. 39, pp. 3-26 (1991 Jan./Feb.). [6] J. Scott and G. Spears, "On the Advantages of Nested Feedback Loops," J. Audio Eng. Soc. (Engineering Reports), vol. 39, pp. 140-145 (1991 Mar.); E.M. Cherry, Discussion, ibid., pp. 145-147. [7] E. M. Cherry, "A Power Amplifier 'Improver,'" 1059

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J. Audio Eng. Soc. (Engineering Reports), vol. 29, pp. 140-147 (1981 Mar.). [8] S. K. Tewksbury and R. W. Hallock, "Over-

sampled Linear Predictive and Noise Shaping Coders of Order N > 1," IEEE Trans. Circuits Sys., vol. CAS25, pp. 437-447 (1978 July).

THE AUTHORS

John Vanderkooy received a B.Eng. degree in engineering physics in 1963 from McMaster University in Hamilton, Ontario, Canada. He continued studies there in low-temperature physics of metals, receiving a Ph.D. in 1967. He spent two years as a postdoctoral fellow at the University of Cambridge in England, and returned to Canada in 1969 to join the faculty at the University of Waterloo. For some years he followed his doctoral interests in magnetic properties of electrons in metals, but his research interests have slowly shifted since the late 1970s to audio and electroacoustics. He is currently a professor of physics at the University of Waterloo. Dr. Vanderkooy is a fellow of the Audio Engineering Society, a recipient of its Silver Medal, and several Publication Awards. He has contributed a wide variety

1060

of papers at AES conventions and to the Journal in such areas as loudspeaker crossover design, electroacoustic measurement techniques, dithered quantization, digital signal processing, loudspeaker impedance, acoustics, and diffraction. He is a founding member of the Audio Research Group at the University of Waterloo, working together with his colleague Stanley Lipshitz and a humbet of graduatestudents. Dr. Vanderkooy's current research interests are digital audio signal processing, dithered quantization, transducers, acoustic diffraction from edges, stochastic resonance, and loudspeaker ports.

The biography for M. O. J. Hawksford was published in the 1999 September issue.

J. Audio Eng. Soc., Vol. 47, No. 12, 1999 December