# Linear-to-logarithmic FOM mapping

There are two mathematical forms commonly used for ADC figures-of-merit (FOM) – the linear form, and the equivalent base-10 logarithmic expression. As an example:

$G_{A1} = SNDR+ 20\times\log{\dfrac{f_s}{P}}$

is the logarithmic equivalent of

$F_{A1} = \dfrac{P}{2^{ENOB} f_{s}},$

and both are used in the literature. The mapping between the two is derived as follows: Assume that B is a performance measured in “bits”, and that B is determined by a performance $X_{dB}$ (e.g., SNDR) as

$B = \dfrac{X_{dB} - 1.76}{6.02}$

Then

$2^{\alpha_B B} = 2^{\alpha_B \dfrac{X_{dB} - 1.76}{6.02}} = {(10^{\log2})}^{\alpha_B\dfrac{X_{dB} - 1.76}{20\times\log2}} = 10^{\alpha_B\dfrac{X_{dB} - 1.76}{20}}$

If a generic FOM [1] has the linear form

$F = K \times{P}^{\alpha_P}\times{f}^{\alpha_f}\times{V}^{\alpha_V}\times{A}^{\alpha_A}\times{L}^{\alpha_L}\times{D}^{\alpha_D}\times{2}^{\alpha_B B},$

then we get the equivalent logarithmic form through the transformation

$G = \dfrac{1}{\alpha_B}\times 20\times\log{F},$

which yields

$G = X_{dB}+M_0+M+ \dfrac{20}{\alpha_B}\times[\alpha_P\times\log{P}+\alpha_f\times\log{f}+...\\ +\alpha_V\times\log{V}+\alpha_A\times\log{A}+\alpha_L\times\log{L}+\alpha_D\times\log{D}].$

The scaling with $\alpha_B$ gives the most natural expression for G, starting with an unscaled “dB-performance” term rather than a scaled one in case $\alpha_B \neq 1$. For FOMs that does not include a “bits” or similar resolution-related performance, the scaling with $\alpha_B$ is omitted.

A strict mapping between F and G requires  $M =0, M_0 = 1.76$, but $M_0$ is usually omitted and M is used to handle various scaling permutations.

Read more in [1] and [2].

# References

[1] B. E. Jonsson, “A generic ADC FOM”, Converter Passion blog, On-line, https://converterpassion.wordpress.com/a-generic-adc-fom/, Jan., 2011.

[2] B. E. Jonsson, “Generic ADC FOM classes”, Converter Passion blog, On-line, https://converterpassion.wordpress.com/generic-adc-fom-classes/, Jan., 2011.