Dithered Chaotic Sigma Delta Modulator

by allenlu2007

 

Dithered SDM

主要是為了消除或減輕 limiting cycle (frequency spur) 存在於 SDM.

Side effect: 

(1) SNR degradation

(2) Reduced dynamic range

(3) Stability issue

 

Conventional dither SDM

1. Add at signal input:  (1) and (2) side effect exists.  Particularly, to reduce the spur, the dither is large and results in big SNR degradation.

2. Add before the quantizer:  take advantage of the noise shaping function.  (1) is not very big; (2) may or may not be a problem depending on the nature of the dither signal (if overlapping with input signal, it may reduce the dynamic range).   (3) to be checked

In summary, it seems better to add dither signal before the quantizer to take advantage of the noise shaping feature.

 

Dither Noise

1. White noise totally uncorrelated to the input signal.  E.g.  additional PRBS

1a. High frequency sawtooth

2. Shape noise uncorrelated to the input signal.  Since the noise add before the quantizer sees an HPF; and the input signal is a low frequency signal.  It seems to be better to have a high frequency dither noise. 

High frequency dither noise has the advantage of minimum SNR degradation to the input signal.  In addition, since it’s frequency is not overlapping with input signal.  

 

 

How to Generate Dither Signal

Method 1: PRBS generator –> white noise

Method 2: 利用 MASH 架構產生 1-bit HPF random signal as below:

 

/Users/alu/work/matlab/simulink/sdm/sdm_mash111_v6.slx

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Catch 是:  K=1 and aa = -1e-6.  (假設 leaky integrator; aa=0 是一個 unstable fixed point, 只要一點 perturbation 就可以避開)

下圖是 amp=0.8 sine-wave 的 y1 (1st order, blue), t1 (2nd order, cyan), and yy (3rd order, magenta)

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Zoom in

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兩點觀察: 

1. 2nd (cyan) and 3rd (magenta) order noise shaping 太接近,表示第一個 quantizer 接近 saturation (?).  Overall SNR degrade at high input signal.

2. 在中頻 (0.1-0.2Hz) 附近仍有一些 spikes, 雖然 amplitude 已經不大。

 

如果 turn down dither to K=0.2 減輕第一個 quantizer 的 overloading 如下。

解決第一個問題,但讓第二個問題變差。

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採用 cocktail solution: chaotic dithering SDM, amp=0.8, K=0.2 and aa=+0.05

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amp=0.8, K=0.2 and aa=-0.05

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如果只有 chaotic SDM, K=0 and aa=+0.05 如下:

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加到 K=0, aa=+0.1 才比較好。但 in-band noise 明顯變差。

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另外試了 aa=0.02, 0.01 效果不好。但 a < 0 時也有類似的效果 (但稍差一些),why?

看來不全是 chaotic 造成的效果,而是 dithering 搭配 some noise shaping noise?  

The only works for sine wave input.  For DC it’s dead, see below!

 

真正挑戰 : DC

真正的挑戰是 DC, 特別是 DC=0, 0.125, 0.25, 0.5, 0.75, 0.875 cases

先用 K=1, aa=-1e-6 依上順序 Dither Only

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Fixed dither has BIG problem when input signal is large ==> overload the first quantizer!!!

NOT GOOD!

 

Now turn down K=1-DC, aa=-1e-6 repeat Dither Only

DC=0 is the same

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DC=0.125

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DC=0.25

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DC=0.5

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DC=0.75

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DC=0.875

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A lot better especially at high DC value!  No change at low DC value.  Not enough improvement!!!

 

Now only chaotic SDM, K=0 and aa=+0.05; Chaotic Only

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The result is NOT GOOD!  

 

Change aa to +0.1 instead

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DC=0.875 explode!!

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The result of a=+0.1 is slightly better than +0.05.  However, the dynamic range is reduced significantly (DC=0.875 explode!).

 

 

 

Now we check the cocktail approach

K=1-DC and aa=+0.05

DC=0

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DC=0.125

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DC=0.25

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DC=0.5

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DC=0.75

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DC=0.875

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The result seems a lot better.  Is it due to chaotic + dither effect?  

Let’s try to use aa=-0.05!!!  死穴。Leaky integrator + dither is totally useless!!

But it seems to be useful in sine wave input!!   Again DC is the most difficult case!

 

DC

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In summary:

Best setting:  Chaotic + Dithering SDM

Chaotic: aa=+0.05

Dithering (SDM):  K=1-dc or K=1-amp  (avoid overloading the quantizer ==> dynamic changing?) 

Dithering alters the initial condition of the chaotic SDM.

 

Next try uniform dithering noise

 

First Dithering Only Approach

K=1-dc, aa=-1e-6

DC=0

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DC=0.125

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DC=0.25

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DC=0.5

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DC=0.75

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DC=0.875

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Now we check the cocktail approach

K=1-DC and aa=+0.05

DC=0

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DC=0.125

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DC=0.25

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DC=0.5

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DC=0.75

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DC=0.875

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In summary, uniform noise seems to be a better choice compared with shaped noise. why?

Presumably shape noise for dithering is better than uniform noise.

 

 

 

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