# Signal to noise in relationship with limit of detection and/or signal strength

I wonder about the following: The limit of detection is given as being 3 sigma of the blank peak away from the blank signal (which is noise finally, I think?). What struggles me is: In my head this is also very closely connected to SNR but I may confound them.. however, do the peak heights not play a role in this context? For example, the height of the signal can be arbitrary high or low, compared to the blank signal. Hence, one of them might overlap the other, then what? My thoughts: When the signal peak is so low that the blank signal will overlap it, it can't be 3 sigma apart from it. This is obvious. But the signal might be much lower in general. So maybe my question isn't about LOD or SNR but about: When is my signal strong enough that I can say "There is a signal"?

The minimum detection limit is the concentration corresponding to a peak that can just be distinguished statistically from background fluctuations. This is generally taken to be a peak height equal to three times the standard deviation of the background count.

I think this explains it, doesn't it? So to conclude: The peak height has to be as high as 3 sigma of the blank measurement (noise)? When the signal is lower, it isn't the LOD.

I'm not sure whether the topic fits here as it also touches chemistry or physics but, finally, it is about to detect/separate signals so I think the electronics thread makes the most sense.

• You can read about limits of detection in many references, a few of which are listed in my answer here. My book on the topic is in the list.
– Ed V
Commented May 6, 2020 at 20:05

It's a good question and there is a strong correlation between SNR and the ratio of errors /bit or BER and the steepness of the BER curve in dB per decade of BER. This curve can also be used with different data patterns and be used to find an asymmetric error in the detector or ISI or harmonic, random and periodic sources of noise.

Normally (no pun intended) each communication standard may have a defined discriminator model and standard detector error value in dB and assume the resolution or quantization error is minimal.

So its not a simple 3 or 10 sigma value, but for a given type, the concept is similar but often expressed in dB for a BER of 1e-9 soft error rate and a few decades higher for hard errors that are uncorrectable. The variation in standard deviation can be expressed for SNR in dB for a given BER then the slope of dB/ decade is a quality indicator of the discriminator and filters.

Electronic detectors are a little different than chemical detectors with a wide range for error probability. They may be linear, non-linear, and multidimensional with phases, amplitudes frequencies, antenna diversion or coloured-space and time-multiplexed and even error corrected.

Each detector must be characterized for Resolution, Jitter, SNR, ISI, Intersymbol Interference, and any source of error than might be non-Gaussian like PS noise. The ISI might be minimized using a Raised Cosine Filter that is matched to the channel BW for a "Matched receiver response. Or it might not and just have a simple-might level comparator sampled once per bit.

There may be other sources of error like adjacent channel interference or environmental stress.

It's usually best to use logarithmic values for thresholds and tie this into an overall Channel error budget combining Friis losses, detector errors from jitter and sub-optimal methods not integrating over the entire time interval, A/D errors. etc.

So each source must be allocated an error budget in the design then verified.

For a very steep slope of BER vs SNR, we expect few surprises from unwanted error sources. When this slope flattens out or changes with data patterns then there is some non-Gaussian effects occurring like Riciian Fading from echoes off walls or trees.

The difference between no signal and no errors with FEC or ECC might be 10 to 20dB depending on the optimization of each block in the design.

With a lot more thought, I might be able to make a more concise answer for you. ;)

Yes this is SNR and LOD are related. As SNR=20log(Srms/Nrms). Rms value (root mean square) it's equal to the standard deviation (sigma). So say "you need 3 sigma to mesure your signal" it's equivalent to say "You need a SNR of 20log(3)=10dB to measure your signal"

SNR is usually measured by taking the power of the signal and noise. Power is a well behaved robust estimator.

The peak is very fragile metric. Look at a noisy signal over different periods of time and you will record different peak levels. A better metric is the CDF or Cumulative Distribution Function, which tells you what fraction of time the signal spends above some level. Then you can talk about a level which is exceeded 1% of the time, or 10-6 of the time.

The standard deviation of a signal is closely related to power. People often use a working definition of 3 sigma as a more robust way of defining a 'peak level'. This is a convention, but it works well in many circumstances.