# What is formula for quantization step size?

I have read from different web links,different formulas of quantization step size as shown highlighted in attached snapshot,but what is actual/correct formula?

The actual quantization step size depends on the implementation of the quantizer, and would be by definition the input voltage change that would cause the digital output to increase by 1 bit.

You can find the specifics for the actual device you are using on the manufacturer datasheet, such as this example from ADI's AD9459 ADC, where the input range is from $$\pm V_{ref}$$ where $$V_{ref}$$ is an externally supplied reference voltage, and OR in the table is an out of range indicator:

The actual threshold down to 0.5 lsb levels is of no practical use since the self-noise within most ADCs typically exceed 1 lsb (otherwise the core could be used as a higher bit converter). Datasheets will also specific ENOB (Equivalent Number of Bits) which is dependent on sampling rate and input frequency. This captures all noise sources and provides a metric of the same number of bits the device provides compared to a "perfect" ADC. Note that this precision is maximized with an input signal that is typically 1 to 2 dB below actual full scale where clipping would occur (as we approach the actual full scale, spurious signals increase due to non-linearities which degrades overall dynamic range and therefore precision).

That said it is sufficient in most cases to simply use the total input range (up to the backed off full scale) and divide by $$2^{b}$$ where $$b$$ is the stated precision in ENOB.

• you mean the formula,where there is "L"(number of quantization levels) is sufficent in most cases?
– engr
Feb 25, 2021 at 6:00
• @engr In most cases the analog input signal and the ADC itself would have noise and offsets exceeding 0.5 lsbs so yes it wouldn't make sense to determine an exact voltage threshold- just simply take the entire range and divide by $2^b$ to determine the step size in those same input units (volts for example). When you are concerned with signal fidelity and SNR, you want to be sure you are quantizing your entire signal including the noise in the signal itself, otherwise you are degrading your signal with quantization noise. Feb 25, 2021 at 6:06
• but for small quantizer for other purposes, any specific mapping can be defined (it just must be defined), and ultimately is driven by the implementation (where are the actual thresholds causing the decoder circuit to toggle?). Feb 25, 2021 at 6:08