What does “limit cycle” mean in delta-sigma modulator?

Question

Could anyone explain limit cycle related spurious tones? It is stated that a higher order modulator is better than low level one because of limit cycle related spurious tones, but I don't have picture of the limit cycle. Does it mean the tones in DC input and this tones in DC with period appear to be an increasing high frequency tones?

Answer 1

That's easy to answer with a general answer and immensely hard to answer in full detail that people have been writing books and papers about it for decades.

I've had some personal experience of this about 10 years ago when I implemented what was meant to be a simple and cheap SD converter (other side of same coin) with a mix of hardware and software and discovered a whole new world of opportunities to get unexpected results in the midst of normal behaviour.
Conclusions:

1. SD ADC have some major attractions and can be very cost effective, BUT avoid self implemented SD ADC if possible. Here there be Dragons - as material below amply indicates.

2. Use extra long barge poles when dealing with Z8 processors.

Here's the easy general answer to your question:

VERY roughly limit cycle oscillations (or spurious outputs or whatever) are spurious products that appear in an SD system in the presence of a constant input signal that should ideally be converted 'perefctly'. The constant input may be a DC level of a sinusoid or a mix of sinusoids which have a periodic function which interacts 'in some way' [tm] with the SD system due (probably) to the sampling period of the SD system not matching the period of the input system OR the SD having several stable states which it can flip between at certain points in the cycle. (Thing eg model railway circuit with many subloops and points which are switched randomly at certain locations).

That answer can be summarised as"makes funny outputs unexpectedly as a result of its nin-mlinera nature" and may sound excessively naive and simplistic.
As a demonstration of how well it's naive simplicity is reflected in the deeper ponderings of experts see this excellent discussion on pages 97 - 127. This is in a 2006 book "Analog Circuit Design". This chapter is intended to be about utilising limit cycle behaviour to impove SD modulator performance but along the way they give a good look at the deeper mysteries.

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Glimpse into deep mysteries:

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This very useful slideshow fromBerkley EE247 lecture 24 does a nice job of providing a "demystified" explanation and purports to provide a solution. It says:

Limit Cycle Oscillation

• Issue particular to SD modulator type data converters:

  • In response to low level DC inputs quantization noise becomes periodic and some of the components could fall with in the passband of interest and thus limit the dynamic range.

  • More pronounced in 1st order SD modulators compared to higher order (e.g. 2nd order)

BUT !!! - the following suggests that causes or solutions may not be quite that straight forward.

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Here's a sample of perhaps the best related content from page 100:

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This paper
Borkowski, Maciej, Digital Δ-Σ Modulation. Variable modulus and tonal behaviour in a fixed-point digital environment

says

• This work addresses a well known problem of unwanted spurious tones in the modulator’s output spectrum. When a delta-sigma modulator works with a constant input, the output signal can be periodic, where short periods lead to strong deterministic tones. In this work we propose means for guaranteeing that the output period will never be shorter than a prescribed minimum value for all constant inputs. This allows a relationship to be formulated between the modulator’s bus width and the spurious-free range, thereby making it possible to trade output spectrum quality for hardware consumption.

• Solution:

  • Use dithering (inject noise-like signal at the input ): to randomize quantization noise

  • If circuit thermal noise is large enough acts as dither

  • Typically, in the design of SD modulator integrating C values chosen carefully so that inband thermal noise level exceeds quantization noise