ENOB describes an analog-to-digital converter’s performance with respect to total noise and distortion.
In the earlier parts of this series on analog-to-digital converters (ADCs), we looked at the basics (part 1); gain error, offset error, and differential nonlinearity (part 2); and integral nonlinearity (part 3); and then we looked at some ADC topologies and introduced AC errors (part 4).
Q: We concluded with a mention of the effective number of bits (ENOB). What is that?
A: To understand an ADC’s effective number of bits (ENOB), we first need to consider how quantization error contributes to an ADC’s signal-to-noise ratio (SNR). Figure 1 repeats plots from earlier parts of this series showing the expected output code on the y axis vs analog input voltage on the x-axis for an N-bit ADC, where N=3. To find SNR, we need a plot of quantization error vs. analog input voltage. Suppose we define quantization error Q as the input analog voltage minus the voltage associated with the corresponding output code. In that case, we get the plot in red at the bottom of Figure 1, a sawtooth wave with a peak-to-peak value of Q and with peaks of -Q/2 and Q/2.
To calculate SNR, we will need the root-mean-square (RMS) value of the sawtooth wave, which we can calculate using any one complete segment of the sawtooth wave. We’ll choose the one highlighted in blue in the figure, which extends from x equals one-sixteenth of the full-scale (FS) analog input voltage (VFS), where the quantization error is -Q/2, to x equals three-sixteenths of VFS, where the quantization error is Q/2. We can calculate the mean-square value as follows:
Keeping in mind that the definite integral of x2 dx is x3/3 evaluated over the area of interest, we can derive the mean square:
To find the RMS value, we take the square root of the mean square:
From Figure 1, we can see that Q relates to VFS as follows:
Substituting Equation 2 into Equation 1 gives us QRMS as a function of VFS:
The largest sinusoidal signal we can apply to our ADC’s input is one whose peak-to-peak value equals VFS, equivalent to one whose peak value is VFS/2. To get the RMS value, we then need to divide the peak value by the square root of 2. So we can express our maximum input signal in RMS as follows:
We can now calculate the SNR by dividing Equation 4 by Equation 5:
SNR is typically expressed in decibels, so we can rewrite Equation 6 as follows:
Keeping in mind that with logarithms, powers become multiplications and multiplications become divisions, we can rewrite Equation 7 as follows:
Equation 8 is described as “infamous” by one reference[1] because it’s often presented without explanation.
We still haven’t calculated ENOB, but we can solve Equation 7 for N:
Q: Wait, so what’s the point of that? We just used N to calculate SNRdB.
A: Keep in mind that SNR only accounts for quantization error. A real-world ADC will also gain, offset, and nonlinearity errors, as discussed in the earlier parts of this series. These errors will combine with quantization error to form a total signal-to-noise plus distortion (SINAD) ratio that will always be lower than the SNR for real-world devices. If we substitute SINAD in dB for SNRdB in Equation 8, we get ENOB:
ENOB will always be less than N for a real device, and it can take on fractional values.
Q: What else should I know about ENOB?
A: In addition to an ADC, ENOB can apply to an entire signal chain, and even a complete digital oscilloscope. You can review an application note to see details on measuring an oscilloscope’s ENOB[2].
References
[1] Taking the Mystery out of the Infamous Formula, “SNR=6.02N + 1.76dB,” and Why You Should Care, Delft University of Technology[2] Understanding Oscilloscope ENOB Values, Rohde & Schwarz
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