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aaron_do

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Nyquist sampling SNR issue
Jul 19^th, 2013, 12:30am

Hi all,

according to theory, I need to sample more than twice the signal bandwidth. Suppose that I am sampling marginally higher than twice my signal frequency. If I have a limited number of samples, they may be close to the zero crossing of the signal. In this case, will there be any issue with the SNR? I have illustrated this below. In the fist picture I have a sine(2pif0t) signal (blue) that is being sampled slightly higher than 2f0 (red). In the second picture I have added 0.25cos(2pif0t) to the original signal (blue) and I am still sampling as before (red). Notice that in the beginning, the output samples are all noise (poor SNR) and after some time the SNR improves.

My question is am I analyzing this correctly or am I missing something?

thank,
Aaron

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Ideal_and_noisy_SNR.png

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aaron_do Senior Fellow Offline Posts: 1398	Re: Nyquist sampling SNR issue Reply #1 - Jul 19^th, 2013, 12:46am After some more checking, this seems to be the image problem...So in order to get the correct output, I need to filter out the high frequency image. My question remains however, will the SNR be affected? Note that if I oversample by 2x, I think the problem is avoided. thanks.
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boe

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Re: Nyquist sampling SNR issue
Reply #2 - Jul 22^nd, 2013, 4:26am

Aaron,
Quote:

according to theory, I need to sample more than twice the signal bandwidth. [...] If I have a limited number of samples, they may be close to the zero crossing of the signal. In this case, will there be any issue with the SNR?

Strictly, if you have a limited number of samples the bandwidth is infinite. So yes, the sampling rate should affect the SNR.

This is particularly true if you have wide-band noise (due to aliasing of the noise).
- B O E

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aaron_do

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Re: Nyquist sampling SNR issue
Reply #3 - Jul 22^nd, 2013, 6:08am

Hi boe,

thanks for the reply.

I'm not really sure what you mean. I understand that by windowing my signal I am convolving it with a sinc function in the frequency domain, which would make the bandwidth infintite. Then when I sample the signal, there will be noise folding which will degrade the SNR.

But I don't think this is the problem I'm referring to since you would see that problem to some degree even if you sample at 4fmax. Also, referring to the problem I mentioned, you can avoid it by sampling at the right instance which doesn't follow from your frequency domain explanation.

regards,
Aaron

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boe

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Re: Nyquist sampling SNR issue
Reply #4 - Jul 23^rd, 2013, 3:45am

Aaron,
the problem occurs only if the frequency of the signal is close to fs/2. Otherwise the folded sinc (due to aliasing) is small enough.

Why this depends on the phase of the sampling is less obvious, but the phase of the aliased signal depends on the sampling phase (note that a time delay is equivalent to a linear phase shift over frequency).

- B O E

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aaron_do

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Re: Nyquist sampling SNR issue
Reply #5 - Jul 23^rd, 2013, 4:24am

Hi boe,

Quote:

the problem occurs only if the frequency of the signal is close to fs/2. Otherwise the folded sinc (due to aliasing) is small enough.

Well that only depends on how good my anti-aliasing filter is and the nature of my out-of-band noise. But I think this is clearly not the same problem I'm describing.

Quote:

Why this depends on the phase of the sampling is less obvious, but the phase of the aliased signal depends on the sampling phase (note that a time delay is equivalent to a linear phase shift over frequency).

I'm not really sure that phase is the correct term since we are dealing with two incommensurate frequencies.

A look at the time-domain picture I included suggests that there really is an SNR problem. Is this a real problem?

thanks for the reply,
Aaron

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aaron_do

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Re: Nyquist sampling SNR issue
Reply #6 - Jul 23^rd, 2013, 5:22pm

Hi boe,

I have slept on it, and I think that maybe your explanation may be correct. My apologies. Im still not sure whether it tells the full story though. My concern is what if you have a real signal which is changing. It is not the same as having a limited number of samples, so if you convert to the frequency domain, with more samples the spreading will be less. But for some parts of the signal you may encounter the problem...

Thanks,
Aaron

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boe

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Re: Nyquist sampling SNR issue
Reply #7 - Jul 26^th, 2013, 8:33am

Hi Aaron,
aaron_do wrote on Jul 23^rd, 2013, 5:22pm:

I have slept on it, and I think that maybe your explanation may be correct. My apologies. Im still not sure whether it tells the full story though. My concern is what if you have a real signal which is changing. It is not the same as having a limited number of samples, so if you convert to the frequency domain, with more samples the spreading will be less. But for some parts of the signal you may encounter the problem...

I don't think so. All samples of the signal contribute to every frequency component. The influence of the sampling error of every single sample does not depend on the level of the signal at that instant because the Fourier transform is linear.
Assuming white noise (with its flat power spectral density), the SNR of a certain frequency component depends directly on the power of that component (to which all samples contribute).
This is applies at least to the case of a band-limited signal of which you have all (infinitely many) samples.
I'm not sure whether that covers the case you think of, though.
- B O E

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aaron_do

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Re: Nyquist sampling SNR issue
Reply #8 - Jul 26^th, 2013, 7:08pm

Hi boe,

Quote:

The influence of the sampling error of every single sample does not depend on the level of the signal at that instant

this is exactly the point i've been trying to make. You're trying to get maybe a BER of 10e-4, and at some instances you're probability of en error is 10e-5, but for a short duration it might be 10e-2. This might not show up in the frequency domain because its averaged over all samples.

Actually in truth you can't really sample at just 2 fmax without seriously degrading your signal with the anti-aliasing filter. So if this is a problem, it probably isn't encountered very often.

Aaron

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boe

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Re: Nyquist sampling SNR issue
Reply #9 - Jul 29^th, 2013, 5:43am

Hi Aaron,
if you used an infinite delay you could use all samples to reconstruct those regions you mentioned without 'local' loss of SNR (that is what the Fourier Transform does).
In practical cases, however, you have only a limited number of samples for each bit (or symbol) of data. So we are back to your initial question with a limited number of samples...
And I agree, it is impractical to go too close to the Nyquist limit (just think about the quality of the aliasing filter you would need).
Does that help?
- B O E

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aaron_do Senior Fellow Offline Posts: 1398	Re: Nyquist sampling SNR issue Reply #10 - Jul 29^th, 2013, 8:16am Hi boe, you've helped clear some stuff up for me so thanks. We are more or less on the same page now, but that still leaves me with my initial question. I'm just wondering if the scenario I described has any effect at all, or is there something I am missing... Aaron
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