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Idle Tones and Dithering in Sigma-Delta Modulators


Idle Tones and Dithering in Sigma-Delta Modulators – an Actual Overview –
Andreas Buslehner and Peter S¨ser? o

Abstract— This paper deals with idle tones and how to overcome them. Idle tones are unwanted discrete spectral peaks in the signal band which occur due to periodic patterns at the modulator output. First methods are discussed how idle tones can be detected. Further the origin of idle tones in sigma-delta (Σ?) modulators is alluded. In the end this paper presents common methods how to reduce or even eliminate tones in Σ? modulators named dithering. The purpose of dithering is to decorrelate and whiten the power spectral density spectrum of the quantization error. Further various implementations of dithering in Σ? analog-to-digital converters (ADCs) are evaluated and the pros and cons are discussed. Keywords— Sigma-delta modulators, noise-shaping modulators, dithering, idle tones.

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Fig. 1: Multiplication of 32 harmonically related sinusoids (K=32), each having the same amplitude, with a pseudorandom noise.

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Introduction

LTHOUGH white noise approximation is at best suspicious and at worst simply wrong in Σ? modulator analysis, estimations can be done and principles of Σ? modulators can be explained. Surprisingly simulations and actual circuit implementations show good agreements with additive white noise analysis. Unfortunately Σ? modulators also often exhibit unstable behavior not predictable by using the white noise analysis technique. Additionally this technique fails to predict non-harmonic tones. This fact is especially true for single-bit modulators. Non-harmonic tones, which are also sometimes referred as idle channel tones or pattern noise are not caused by circuit imperfections; they exist even in ideal sigma-delta modulators. These tones or repetitive patterns, respectively, of the modulator output are not harmonically related and occur as odd ”chirp” or ”pops” [1]. Idle tones are especially unwanted in audio applications, since human hearing can detect tones buried beneath white noise1 . Before investigating idle tones, it is important to ?nd methods how to detect and observe idle tones. It is instructive to investigate the following example [2]: Multiplying a random white noise sequence, which is
? Both authors are with the Department of Electronics, Graz University of Technology (TUG), Austria, A-8010 Graz, In?eldgasse 12. (e-mail: ?rstname.surname@electronic.tugraz.ac.at) 1 The ear can perceive tones that are more than 20 dB below the integrated noise ?oor (reported in [2]).

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independent and identically distributed (i.i.d.), with the sum of harmonic sine wave sequences. The sum of the harmonically related sine waves is given by K xharm [n] = k=1 cos(knω0 ), where ω0 is the fundamental frequency. Further it is assumed that the random white noise sequence has a rectangular probability density function bounded between ±0.5. The result of this multiplication in the time domain leads to the sequence plotted in Fig. 1 where periodic pattern noise sequences repeated approximately every 512 samples can be detected. Therefore someone would assume that peaks occur at multiples of ω0 in the power spectrum. However, looking at the computed power spectrum of this signal, shown in Fig. 2, it is revealing that the FFT shows no outstanding periodicity at multiples of ω0 . This can be explained by the following consideration: A multiplication of two signals in the time domain results in a convolution of these signals in the frequency domain. Since the harmonically related sinusoids are multiplied with a pseudo-random noise, the overall output signal spectrum also looks like another white-noise spectrum. Therefore there is a need for another method which allows to detect such periodic tones. The simplest method to overcome this problem is using the autocorrelation function. The discrete autocorrelation is given for a limited discrete signal sequence x(n) by [3] 1 ? Φyy [m] = Q
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Fig. 2: Power spectrum of Fig. 1 based simply on one windowed 128k Hann windowed FFT. Note, there are no outstanding periodicity at multiples of ω0 , which are marked with circles.

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ing output pattern (idle tone) can occur in the baseband spectrum. In Σ? modulators with more active input signals, the output sequence is typically more complex. Nonetheless, the concept underlying tonal behavior is that repeating patterns in the quantizer output cause discrete spectral peaks. Fig. 4 shows an example of tones occurring in the baseband of a second order modulator having a DC input of about -55.6 dBFS. Finally, it should be mentioned that these tones might not lie at a single frequency but instead be short-term periodic patterns. That is one of the reasons why it is so important to choose only a moderate number of samples if analyzing Σ? modulators. Tone magnitudes can be reduced by several techniques for instance choosing a higher-order architecture due to the fact that a chain of integrators decorrelates the input signal and the quantization error. For many years, it was generally thought that high-order single-bit Σ? modulators show no tonal behavior. It was assumed that thermal and 1/f noise scramble the tones and whitens the noise spectrum [4]. However, even in high-order Σ? modulators tonal behavior was found [5]. Although cascaded as well as multi-bit architectures have improved tone properties the only technique to prevent idle tones is a technique called dithering.

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Fig. 3: Autocorrelation of Fig. 1, where (a) is computed from 128k samples and (b) is computed from 32k samples.

Φyy (0) is well known as the mean-square value of the signal x[n]. The autocorrelation sequence of a stochastic noise sequence should asymptotically tend toward zero, while the autocorrelation of a periodic sequence results in periodic patterns and peaks for Φyy (= 0) . However, Fig. 3(a) shows the autocorrelation of the noise modulated sinusoidal sequence from Fig. 1. It reveals a periodicity every 512 samples shifts, which is the period of the fundamental frequency ω0 . Comparing Fig. 3(a) and Fig. 3(b) we see that with increasing number of samples the detected peaks of the periodic sequences tends toward zero. So following important conclusion can be drawn: Idle tones in Σ? modulators can only be reliably detected with the aid of the autocorrelation function having a moderate number of input samples.

A possible solution to the idle tone problem is to include, usually at the quantizer input, some nonperiodic signals, such as a pseudo-random noise. With this technique, which is called dithering [5], it is possible to partially decorrelate the quantization error at the price of additional noise in the in-band [6]. The design parameters under consideration for the dither signal are as following [5]: The relative dither level δ/? which is needed to decorrelate the idle channel tones and pattern noise. Here ? denotes the binwidth and δ the peak-to-peak value of the dither signal. A relative dither level between 0.5 and 2 seems to be a good choice as reported in [5]. The probability density function (PDF) of the dither signal,
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Considering a ?rst-order Σ? modulator with a small DC input. The digital output of the modulator will be a periodic sequence of 1’s and -1’s. This repeat-

Fig. 4: Tones in the baseband of a second-order modulator with a DC input of -55.6 dBFS.

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Fig. 5: Adding dithering (d) to a Σ? modulator at the quantizer input.

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where usually two di?erent dither signals are used. The one is a dither signal having a rectangular PDF (RPDF) and the other dither signal having a triangle PDF (TPDF). The dither signal having a TPDF needs a higher magnitude to let the idle tones disappear, but on the other hand this dither signal was found to be better at whitening the quantization as well as at smoothing idle tones. The number of quantization levels required for the dither if the dither signal is a quantized signal. The trade-o? between improving tonal behavior versus degrading the modulator’s dynamic range (DR). Large dither magnitudes let the tones disappear, but let the in-band noise increase as well as decrease the dynamic range of the integrators. However, the choice of the design parameters should be supported using computer simulation. Several implementations are proposed and implemented in Σ? modulators and are treated in the next sections. Additional dithering helps improving the digitization of slowly varying signals as well as DC signals. The added dither signal or noise, respectively, causes the digital output to randomly toggle between adjacent levels. Therefore noise provides more ”information” of the input signal.

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Fig. 6: A Σ? modulator with dynamic dithering (dm ) taking the input signal directly.

3.3

Dynamic Noise-Shaped Dithering

3.1

Noise-Shaped Dithering

Assuming the random signal to be introduced has a white noise spectrum, then the most suitable place to add the dithering signal2 is just before the quantizer, as shown in Fig. 5. The penalties for adding the dither signal at the modulator input are on the one hand an increased hardware complexity as well as the need for greater dynamic range inside the loop.

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Subtractive Dithering

The idea of subtractive dithering is to add the dither signal to the quantizer input, and to subtract it at the quantizer output. Therefore the dither signal is totally compensated at the output of the Σ? modulator and does not increase the noise in Σ? modulators. Although this technique is widely applied in PCM quantizers it is not relevant to Σ? modulators [5] because of the increased hardware complexity as well as on its poor e?ciency.
2 Since the dither signal is here noise-shaped by the noise transfer function; therefore this kind of dithering is called noiseshaped dithering.

Dynamic noise-shaped dithering is a powerful method to decorrelate and whiten the power spectral density spectrum of the quantization error without loss in dynamic range as seen in previous sections. The concept of dynamic noise-shaped dithering [7] is the following: It is well known that quantization noise becomes most tonal and correlated with the input, when the magnitude of the input signal is very small. On the other side tones are in?nitely small as the magnitude of the input signal is large. Hence, for small magnitudes the dither generator should produce a maximum of dither amplitude, while large magnitudes demand almost no dither. Simulation results [7] show that almost no dither is needed for the top 20 dB of the dynamic range of the Σ? modulator, whereas a large magnitude of dither is needed especially for input signals in the range of the quantization error. The designer has to choose the variable dither amplitude in such a way to guarantee on the one hand that dithering smooth all tones for low-level inputs while on the other hand dithering should not decrease the DR performance for high-level inputs. There are mainly two basic principles to implement dynamic dithering. The ?rst implementation is processing the analog input signal directly as illustrated in Fig. 6. The input is quantized by a nonlinear (nonuniform) coarse quantizer. A coarse nonlinear quantizer with output levels to the powers of 2 is preferable, since this simpli?es the digital multiplication. It should be mentioned that only a few quantization levels are needed (for example 4) to achieve good tonal behavior. The second kind of implementation generates the dither signal by taking the digital word after the ?rst stage decimation ?lter dk , which is a delayed estimation of the input signal, and multiplying it with a digital random dither signal. The bene?t of this implementation can be found in no need of complex analog hardware. However a penalty of this implementation can be seen in the phase shift of dk related to the input signal. To sum it up, dynamic dithering is an e?ective method to smooth tones with a little increase on ad-

ditional hardware and the bene?t of nearly no reduction in dynamic range compared to the other dithering techniques described above. The dynamic dithering technique has been manifold successfully implemented on silicon for instance by G. Gomez [8].

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Conclusion

3.4

Adaptive Bit Flipping Dithering

Magrath et al. [9] proposed another technique for dithering in a single-bit Σ? modulator based on the detection of limit cycles. A Σ? modulator behaves like a nonlinear system without having a stable equilibrium point. However, a stable Σ? modulator has so called limit cycles, which represents a special case of a positively invariant set. More precise a limit cycle represents a periodic oscillatory motion of the state. Limit cycles can be divided into two groups: Idle pattern (idle tones) when the limit cycle results in a small amplitude signal at the output of the Σ? modulator, and large signal limit cycles when the limit cycles occur with a very large amplitude at the output, which further results in overload and instability. The hardware detection of limit cycles can be implemented simple using a shift register and additional Boolean expressions looking for long strings of consecutive 1’s and 0’s at the modulator output3 . If a limit cycle is detected a pseudo random sequence ?ips the sign of the quantizer output sequence. To sum it up, this strategy terminates the limit cycle by manipulating randomly the sign of the quantizer output until the limit cycle disappears. This dithering technique pro?ts from a very simple hardware. Another bene?t can be found in the utilization of the limit cycle detection hardware for identifying large limit cycles, which can be used to manage instability4 . That is half the battle. A penalty of this method can be found in an increased noise ?oor and in the fact that the hardware must be designed in such a way, that all limit cycles can be detected in order to guarantee a good tonal behavior.

The existence of tones can be manifold. The two most important factors for the unwanted idle tones are the location of the noise transfer function zeros and the DC input itself. Simulation shows that moving the noise transfer function zeros away from DC improves the tonal behavior dramatically. Another, often overlooked factor is the overall delay of the integrators, DACs and ADCs which has been reported by Gosslau et al. [12]. Also nonlinearity and coupling can modulate high frequency tones into the baseband. Since dynamic element matching scrambles tones by itself, most multi-bit Σ? modulator designs do not make use of any dithering technique. However, care must be taken if one assumes that thermal and 1/f noise whiten the tones in the baseband. Only careful simulation, analysis and dithering presented in this paper guarantees a good and tonal free Σ? modulator design.

References
[1] Gabor Temes. Electronics Laboratories Advanced Engineering Course on Delta Sigma Converters for Telecom. ? Ecole Polytechnique F?derale De Lausanne, July 2000. e [2] Steven R. Norsworthy. E?ective Dithering of Sigma-Delta Modulators. In Proc. International Symposium on Circuits and Systems, volume 3, pages 1304–1307. IEEE, May 1992. [3] A.V. Oppenheim, R.W. Schafer, and J.R. Buck. DiscreteTime Signal Processing. Prentice Hall International, Inc., New Jersy, 2nd edition, 1999. [4] Jeni?er Y. Dong and Ajoy Opal. Dithering E?ect Simulation for Sigma-Delta Modulators with the Presence of Thermal Noise. In IEEE Canadian Conference on Electrical and Computer Engineering, volume 1, pages 24–28, May 1998. [5] Steven R. Norsworthy, Richard Schreier, and Gabor C. Temes, editors. Delta-Sigma Data Converters: Theory, Design and Simulation. IEEE Press, New York, 1997. [6] Wu Chou and Robert M. Gray. Dithering and Its E?ects on Sigma-Delta and Multistage Sigma-Delta Modulation. In IEEE Transaction on Information Theory, volume 37, pages 500–513, May 1991. [7] S.R. Norsworthy. Dynamic Dithering of Delta-Sigma modulators. presented at the 99th Convention of the Audio Engineering Society, October 1995. reprinted 4103. [8] Gabriel J. Gomez. A 102-dB Spurious-Free DR Σ? ADC Using a Dynamic Dither Scheme. In Transaction on Circuit and Systems-II: Analog and Digital Signal Processing, volume 47, pages 531–535, June 2000. [9] A.J. Magrath and M.B. Sandler. E?cient Dithering of Sigma-Delta Modulators with Adaptive Bit Flipping. In Electronic Letters, volume 31, pages 846–847, May 1995. [10] Richard Schreier. Destabilizing Limit Cycles in DeltaSigma Modulators with Chaos. In IEEE International Symposium on Circuits and Systems, volume 2, pages 1369–1372, May 1993. [11] Mariam Motamed, Avideh Zakhor, and Seth Sanders. Tones, Saturation and SNR in Double Loop Σ? Modulators. In IEEE International Symposium on Circuits and Systems, volume 2, pages 1345–1348, May 1993. [12] A. Gosslau and A. Gottwald. Linearization of a SigmaDelta Modulator by a Proper Loop Delay. In IEEE International Symposium on Circuits and Systems, volume 1, pages 364–367, May 1990.

3.5

Chaotic Σ? Modulators

To prevent idle tones Schreier [10] and Motamed et al. [11] presented another technique by moving a zero of the noise transfer function outside the unit circle. Therefore any small perturbation of the unit causes the chaotic Σ? modulator to go in an arbitrarily large change cycle. Consequently limit cycles are destabilized and tones are eliminated. Comparing chaotic modulators with dithered ones show that chaotic modulators su?er from a higher baseband noise ?oor, especially if the op-amp gains are moderate. However, due to the fact that a normal Σ? modulator can be made chaotic by simply moving a zero of the noise transfer function outside the unit circle, this method o?ers an attractive alternative to dithered modulators.
a 3rd order Σ? modulator ?12-32 same valued bits have been found to be a good choice by the authors. 4 The large limit cycle detector signal can for instance reset all integrators to remove instability in the Σ? modulator.
3 For


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