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What is the PSD of noise generated using rdist_normal function? (Read 736 times)
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What is the PSD of noise generated using rdist_normal function?
May 29th, 2018, 6:51pm
 
In the designer's guide document on jitter model of VCO, I understand that the Sigma from integrated noise is already assumed to be extracted. This is used to generate phase noise. Oscillator does the accumulation every cycle. But, Ideally I could have multiple PSDs giving me the same integrated noise power. So does generating a sequence of numbers using just the defined variance, guarantee any frequency distribution? Is it white?
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #1 - May 31st, 2018, 3:23am
 
So to add more details, the an undefined PSD for the noise generated means that I could have any shape of PSD and just hope that my PLL filters out the noise from the VCO noise model. The below given examples are different possibilities of having the same sigma but different PSDs. In case anyone would like to comment on anything I'm missing, please do as it is not of any use for a designer to spend time using an unknown or unrealistic AMS or verilogA model in his PLL  design.
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #2 - Jul 31st, 2019, 2:01am
 
Once you have converted phase noise into an integrated jitter then you can not obtain the phase noise back from it. I think such a model should only be used to analyse system jitter in time domain.
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #3 - Jul 31st, 2019, 2:17am
 
Yes. I agree it is used in time domain model. But if i analyze the noise profile of the model in frequency domain by taking FFT of the samples, what kind of spectrum do you think I'll get? In fact the sigma is the input of the noise model while we don't know what kind of PSD it has.

Since I received no reply from anyone regarding this, I did my own little research to find topics like maximum length sequence. I believe there are algorithms which are used to generate the sequences. The autocorrelation of the sequence is defined. In other words the power spectral density is closer to white based on the length. I think it's a topic of study by itself. There is also this book called shift register sequences which may be of interest to anyone who stumbles across this post and same doubt. Smiley
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #4 - Jul 31st, 2019, 11:29am
 
The rdist functions generate (hopefully) uncorrelated random numbers. There is no concept of PSD for the generator as there is no concept of time. You could generate a time-sequence that approximates a discrete time white PSD by generating the numbers at fixed intervals with no filtering (a new value is completely unrelated to any previous values). If you want to approximate a continuous time white process the fixed time intervals must be small.

If you would like a non-white PSD, you would need to add filtering. Passing white noise through an integrator produces red noise (1/f2).  Passing it through a fractional half pole produces pink noise (1/f).

The random number generators used by the simulators are not perfect and may have unexpected correlations. It is a worthwhile exercise to generate a long sequence of random numbers and plot the spectrum and confirm that it is indeed white.

-Ken
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #5 - Aug 1st, 2019, 2:45am
 
Hi Ken,
I completely agree to the point that without time there is no point talking about the PSD of a sequence. My first reference to some understanding as to why the random numbers sequence when used as a waveform should work is from the below links :

https://en.wikipedia.org/wiki/Maximum_length_sequence
https://www.amazon.com/Shift-Register-Sequences-Solomon-Golomb/dp/0894120484
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #6 - Aug 12th, 2019, 2:44am
 
Since the model uses normal distribution, its FFT would produce a normal distribution again but with a 1/std.dev. I am not sure if this is what you are lookign for.
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #7 - Aug 12th, 2019, 5:25am
 
Nope. I could literally have any shape of PSD which gives me the same area under the curve (variance) for the gaussian curve you're talking about. My question was as to how do we know if the PSD or the power of the resultant signal generated using those sequences have white PSD. Rather than PSD, such sequences are characterized by Autocorrelation function.
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #8 - Aug 12th, 2019, 11:47am
 
Conceptually if you generate a new random number every TS seconds and the numbers are uncorrelated then the resulting stochastic process will appear roughly white for f ≪ 1/TS. I assume you know that and the question you are asking is how do you know that the samples produced by the random number generators are truly uncorrelated. You don't. You have to just try it and look at the resulting PSD and see what you get. I tried this many years ago and was disappointed with the result. The common random number generators of the day were not very good when it came to producing white sequences. I believe the random number generators have gotten much better, but the simulators may still be using the old generators.

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Re: What is the PSD of noise generated using rdist_normal function?
Reply #9 - Aug 13th, 2019, 1:38am
 
Hi Ken,
Actually I only have a rough mathematical intuition behind how random number generated maybe roughly white till f<<1/Ts. To get to there, I tried picturing how I could get finite valued samples from white noise theoretically. Sampling white noise using a dirac delta will have infinite PSD. But then an real oscillator couldn't do a delta sampling. I then felt maybe the Impulse Sensitivity Funciton(ISF) itself can be assumed to be the mixing quantity than dirac delta. This does low pass filtering preventing infinite-magnitude spectrum even after aliasing, given the sum of square of fourier series component of periodic ISF is finite. The cut off of the filter is inversely proportional to how thin the ISF pulse is. The pulse width is anyway < Ts.But still we have continuous waveforms rather than discrete kronecker delta which maybe roughly equivalent when continuous time simulations are being performed. This averaging over a period decides the whiteness for f<<1/Ts. I would be delighted to know if this is a correct direction of thinking.
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #10 - Aug 13th, 2019, 10:36am
 
I could not follow your thinking. For one thing, the ISF is a characteristic of the circuit, not the signals, so it really does not apply here.

I think of it very simply. Imagine you start with a white continuous time stochastic process and you sample it at TS. Then by Nyquist the frequency characteristics of the sampled sequence should match those of the continuous time process for fTS. If you sample a white continuous time stochastic process at TS you will end up with a sequence of uncorrelated random value, same as if you generated the values with a random number generator. Hence, for fTS both of the sampled sequences will be white, just like the continuous time process.

-Ken
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Re: What is the PSD of noise generated using rdist_normal function?
Reply #11 - Aug 13th, 2019, 11:38am
 
DIRAC DELTA
I currently find it hard to picture sampling of white noise by dirac delta train as :
1. It results in infinite PSD or Infinite Variance (All hell breaks lose)
2. Dirac delta trains rather than finite valued-finite width samples.

PULSE SAMPLING
ISF for my perspective looked like a gate which periodically opens and closes allowing noise to get sampled for an oscillator like a pulse. Pulse sampling was chosen to avoid infinite aliasing because if we had a periodc ISF train, then its fourier coefficients die down at high freq. Sum of Squares of the coeffiecients will be the multiplication factor on the white noise in the band. But then any pulse couldn't be it, but ISF.

BAND LIMITING
Definitely pulse sampling followed by the inverter's RC LPF-hold(say for a ring oscillator) results in a band determined by the RC (< Ts). Ignoring the above two para, we have a finite valued sample valid for one time period Ts and band limited to f<1/Ts.

In case you think, this thought could be erronous, let's not pursue it. Smiley I value your precious time. I'm also trying to construct a theory.
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