Could anyone explain this sampled noise simulation result

It's aliasing. See the rather similar question and my explanation here: Noise Figure of track and hold circuit

Is this some university project that you're both doing? Or is it some wild coincidence?

Andrew

Thank you Andrew. They are indeed the same question and it's a coincidence..

Now I konw aliasing happen in this case, but I can't understand clearly the principle behind this. it's wide-band noise when switch is open, but it's band limited noise when closing, and it seems that dutycycle has some effects. Could you explain this more:

"the output noise is higher because of the sampling of the wide-band thermal noise into the band up to the Nyquist frequency"

Thank you!

Dear  SpiceMonkey,

SpiceMonkey said:

. it's wide-band noise when switch is open, but it's band limited noise when closing, and it seems that dutycycle has some effects.

If you had a chance to review the prior post that Andrew provided a link to whose question was essentially the same as yours, I included a file in that response where I tried to show the impact of sampling and why the aliasing was occurring. Did you happen to view it and did it help to provide any insight into your question? The file is at URL:

noise_figure_nose_figure_sampled_sml_081521.pdf

In direct to your question, the band limiting you mention is after the switch and hence will not limit the aliasing. In order to eliminate the aliasing, the bandwidth of the signal before the switching action must have a bandwidth  of much less than the switch activation frequency. Hence, in your case, it appears the switching frequency is 400 kHz. Hence the bandwidth of the signal applied to the switch input must be much less than 200 kHz to prevent any significant aliasing. With no capacitor prior to the switch, there is nothing band limiting the signal and hence significant aliasing is occurring. You note the following:

SpiceMonkey said:

(r=920, c=10p)

However, you then indicate the cutoff frequency of the RC product is 27 kHz. Am I understanding this correctly? It seems to me that this only provides a first order lowpass filter with a corner frequency of 1/(2*pi*920*10e-12) = 17.3 MHz - far greater than 27 kHz. With a 920 ohm resistor, to get a 27 kHz first order lowpass filter cutoff frequency requires a capacitor of 6.4 nF.

I would suggest you try analyzing the noise in a modified circuit where you move the capacitor to before the switch, change its value to 6.4 nF so as to provide a 27 kHz first order lowpass filter cutoff frequency prior to the switch. Since 27 kHz is just under a factor of 10 less than 200 kHz (half the 400 kHz switching frequency), the aliasing will be far less. You can compare the two spectra and hopefully, this will provide you some added insight.

Shawn

Thank you ShawnLogan!

Resistor is 920K rather than 920 posted before. I have revised the post. You are correct! If I move the capacitor prior to the switch, then get the 4KTR density.

Now I know that 0.5/400k (the time of switch being closing) is too short to establish a signal of 27k. I have have read your PDF, it explains the aliasing well, thank you again!

Im still trying to understand the principle behind this, quantitatively, maybe something about both "correlated" and "aliasing".

Dear SpiceMonkey,

SpiceMonkey said:

Resistor is 920K rather than 920 posted before. I have revised the post.

Ahah! That makes more sense now to me - thank you for updating your post! However, my computation of 920K and 10 pF provides a first order low-pass filter cutoff frequency (-3 dB) of 17.3 kHz. I think your post suggests the -3 dB corner is 27 kHz.

SpiceMonkey said:

f(RC) 27KHz, no alising should happen. (r=920K, c=10p)

Am I misinterpreting your data? 

SpiceMonkey said:

ou are correct! If I move the capacitor prior to the switch, then get the 4KTR density.

Excellent and great! Thank you for modifying your schematic and re-running the simulation!

SpiceMonkey said:

have have read your PDF, it explains the aliasing well, thank you again!

Im still trying to understand the principle behind this, quantitatively, maybe somtting about both "correlated" and "aliasing"

I am glad the document provided you some insight SpiceMonkey - great!

With respect to your last comment concerning an understanding of the quantitative principle and the terms "correlated" and "aliasing", I must admit I am not fully understanding what you are looking for. I'll make an attempt to see if this helps at all....

You may consider the sampling switch as multiplying the analog signal by a series of time-delayed impulses. As this is a multiplication process in the time-domain, it represents a convolution operation of the two Fourier transforms (i.e. the Fourier transform of the signal and the Fourier transform for the series of sample pulses) in the frequency domain. A property of the sample function delayed by some value of tau is to place a copy of the signal's spectrum centered about the frequency corresponding to the frequency of 1/tau. Hence, if there are a series of impulses that periodically sample the analog signal every t= Ts, the analog spectrum of your signal is translated to appear at multiples of the frequency corresponding to Ts. Mathematically, this means the resulting spectrum of the sampled signal is from reference [1]:

I hope this provides a bit more insight consistent with what you were looking for!

Shawn

Reference [1]: ece-research.unm.edu/.../sample.pdf

Thank you! 

Effectice Bandwidth of the rectangle, with hight = 4KTR, area = KT/C 

EBW=(KT/C)/(4KTR)=1/4RC=1/(4*920k*10p)=27.17k, sorry for ambiguousness.

Now I'm trying to understand this, Noise Analysis in Switched-Capacitor Circuits, correlation is mentioned, page 32.

From Tutorial of Noise Analysis in Switched-Capacitor Circuits, page 37, I reprodece the same results in Matlab.

clc;clear;
fs=400e3;R=920e3;C=10e-12;T=300;K=1.38e-23;duty=0.5;
N=duty/fs/(R*C);
syms f;
X(f)=2/fs * K*T/C * (1-exp(-2*N)) / (1-2*exp(-N)*cos(2*pi*f/fs)+exp(-2*N));
fplot(X(f),[1,10*fs/2])
set(gca,'xscale','log')

If possible, is anyone could explain this "intuition", page 32

- RC<0.5/fs, little correlation, whilte spectrum;

- RC>0.5/fs, significant correlation, colored spectrum.

Dear SpiceMonkey,,

It appears the document you are referring to is not a Cadence publication. Nevertheless, I did examine the document and specifically page 32. The author shows exactly what is meant by "correlation" graphically on the author's following two pages (33 and 34). For small RC time constants with respect to the sample period, each sample of the noise is not well correlated to the prior sample or following sample since the RC time constant settling occurs very quickly relative to the switching period. However, if the RC time constant is large with respect to the sample period (the author chose a ratio of 1 to shown on page 34), then the relatively long RC settling will prevent the prior sample, present sample, and following sample from varying significantly from one another. Hence, there is a high degree of correlation between the three samples. Contrast this to the example on page 33 where the ratio of sampling period to the RC time constant is 10. The settling time in this case does allow the prior, present and following samples to differ significantly from each other and hence they are not as well correlated.

Does this help SpiceMonkey?

Shawn

Thank you ShanwLogan and I'm sorry for late reply.

it takes me long time to re-leran the basic theories... and try to understand them intuitively rather than mathmaticaly..