# What effect does bit depth of an ADC have on noise floor, THD, or any other specification?

I was told recently that there is a direct relationship between bit depth of a measurement adc and minimum measurable noise floor, which in turn affects accuracy of THD measurements of DUT. I'm afraid I'm just not seeing it. Could someone explain please, optimally with pictures?

## 3 Answers

A analog to digital converter (A/D) takes the continuous analog input signal and produces a number that quantizes it at best to the nearest of a bunch of pre-determined levels.

Let's say you can give me any real number from 0 to 100, and I quantize that spread to the nearest one of a few discrete levels we picked up front. To make the illustration more intuitive, let's consider a 2 bit A/D. That means it can only produce one of 4 possible output values. To minimize the worst case error, I divide up the 0-100 input range into 4 equal bands, 0-25, 25-50, 50-75, and 75-100. I report the input value as either 0, 1, 2, or 3 depending on which of these bands it is in. This is precisely what a A/D does, at best. The difference is that most have more than 2 bits, and therefore have more bands to break the input range into. In general, the number of bands is 2n, where N is the number of bits.

Now look at what error that adds. If I report a value is in band 0, all you know is that it is from 0 to 25. The best you can do is assume 12.5, and you know you'll be within 12.5 of the original. Put another way, 12.5 is the amount of uncertainty on each measurement. By assuming the input was in the middle of each band, you essentially have to assume there is up to 12.5 of noise on the signal. Related to the full input range, the noise is therefore 12.5/100 = 1/8. This is assuming a perfect A/D, limited only by the number of unique numbers it can produce. The inherent noise level of the digitally encoded signal is therefore 1/8, which is 18 dB. This particular unavoidable noise due to quantization only is called, surprisingly enough, the quantization noise. In general, it is 1/2n+1. Expressed in dB, is is pretty close to 6(n+1) dB.

There are other noise figures for a A/D, like how close the overall input to output curve is to the ideal straight line (linearity), and what the worst case error on the size of any band is. These get into quality of the specific A/D and are not inherent to the process like quantization noise is.

## Added:

As pointed out in a comment, the 6(n+1) dB noise figure is not the only story. If you are making individual uncorrellated measurements, this is the maximum possible noise, and generally what you have to work with since you don't know that it doesn't apply to any one measurement.

However, if you are sampling a continuous signal that is band-limited to less than half the sampling frequency, then you probably care more about average noise. In that case, you consider that 1/8 of full scale is the worst case error on any one reading, but statistically errors on individual readings will be linearly spread between 0 and 1/8. Taking the RMS of such a noise signal results in a significantly lower effective noise level than the worst case I computed above. It looks like Rawbrawb has gone into some derivation of this noise, so I won't go further.

Which noise figure you need to use to design your system depends on system level issues that we don't know here. The point is to beware there are different legitimate ways of looking at the quantization noise, and you have to think carefully what each individual measurement means to decide which view is applicable to the error or noise analisys of your system.

• Many thanks for your extremely informative info, but I was referring to the noise floor and THD of DUT. I have edited my question to reflect this. – Joe Stavitsky Mar 29 '13 at 23:49
• @JoeStavitsky, Olin's 3rd paragraph pretty directly shows the relationship between bit depth and (quantization) noise, which you could call a noise floor since it is present even if there is no other noise. – The Photon Mar 30 '13 at 4:10
• @ThePhoton that paragraph is wrong. The residual error of an ideal ADC is a sawtooth waveform and 1/2 is but one value on that waveform. To speak of noise one must speak of the E (Expectation) of that residual error. That result is $\frac{1}{\sqrt{12}}$ – placeholder Mar 30 '13 at 10:51
• @rawbrawb: This is the difference between worst case error and average error after assuming some characteristics of the input signal. Both are valid. Which is more useful depends on system-level issues. For example, if you are processing a audio signal and want to know "noise", then your view is more applicable. If you doing instrumentation and taking inividual uncorrellated measurement, you have to take the more pessimistic view I presented. – Olin Lathrop Mar 30 '13 at 13:22
• @rawbrawb: I updated the answer to discuss both views of the quantization noise. – Olin Lathrop Mar 30 '13 at 13:59

The noise performance of an ideal ADC follows:

1) $SNR = 6.02N +1.76$ [dB] N= # of bits. SNR = Signal to Noise Ratio.

• note: this assumes a AC waveform on the input.

A handy measure is ENOB (which is the effective # of bits) which comes from inverting eqn #1 above:

$ENOB = \frac{SINAD-1.76} {6.02}$ Where SINAD = Signal to Noise AND Distortion.

The 1.76 dB term arises from the quantization noise of $\frac{LSB}{\sqrt{12}}$ when corrected for rms vs. full scale.

You can see that quantization noise in not $\frac{LSB}{2}$ as commonly thought.

$SINAD = 20Log\frac{S}{N+D}$

In general Analog Devices have excellent applications notes on this.

Basically the Distortion terms reduce the ENOBs that the ADC can produce.

On Edit: Here is a link to a paper from Analog Devices giving you a broader background. Warning *.pdf!

On second edit: Here is a link to the derivation of the $\frac{1}{\sqrt{12}}$ term. And eqn #1 above. Warning *.pdf!

• So when you say noise performance you mean noise floor due exclusevely to ADC, correct? – Joe Stavitsky Mar 30 '13 at 18:19
• That's what the question was about, Noise and Distortion components limits on the ADC. All noise in the system add as RSS (root sum of squares) so if your system noise is larger it rapidly swamps out the ADC noise, by the time you hit 10X you can ignore the ADC (as a rule of thumb). – placeholder Mar 30 '13 at 18:29

Basically (and simply) an ADC deals with integers and a real analogue signal is like a continuous series of numbers having infinite decimal places. The ADC can't resolve those decimal places. That means a measurement error.

This error occurs every time the ADC samples the analogue so if it samples every milli-second there is an error (that might be plus or minus) every millisecond - this can be regarded as adding noise to the real analogue signal and it is this noise that determines the signal to noise ratio of the numbers produced by an ADC.