V_{FS(pk)} 
Fullscale peak input voltage 
V_{FS(pkpk)} 
Fullscale peaktopeak input voltage 
V_{FS(rms)} 
Fullscale input rms voltage 
P_{FS(mW)} 
Fullscale input power in mW units at fullscale input voltage 
P_{FS(dBm)} 
Fullscale input power in dBm units at fullscale input voltage 
f_{samplerate} 
Sampled analog input signal frequency in Hertz (Hz) 
V_{LSB(mVpkpk)} 
Peaktopeak input voltage at one "q" (LSB) level; i.e., n=1 
V_{LSB(mVprms)} 
Rms input voltage at one "q" (LSB) level; i.e., n=1 
V_{n_bits(mVpkpk)} 
Peaktopeak input voltage at n "q" levels; i.e., 0≤n≤2^{N} 
V_{n_bits(mVrms)} 
Rms input voltage at n "q" levels; i.e., 0≤n≤2^{N} 
ΔP_{n1_ton2_bits(dB)} 
Difference in P_{n1(dBm)} and P_{n2(dBm)} expressed in units
of dB 
snr_{quant} 
Signaltonoise ratio due to quantization (sampling) 
SNR_{quant} 
snr_{quant} expressed in decibels 
SNR_{aperture_jitter} 
Signaltonoise ratio due to aperture jitter 
NSD_{ADC} 
Noise spectral density expressed in decibels 
NF_{ADC} 
Noise figure expressed in decibels 
These equations predict the RF electrical performance of an AnalogtoDigital
Converter (ADC, A2D, A/D converter, etc.). Since A/D converters are often the last
stage in a receiver chain, it is extremely useful to be able to predict the contribution
for noise figure, signaltonoise ratio, power levels, etc., since those values
are needed for a complete cascade analysis. Lots of variations on the equations
can be found across the Internet, so I have endeavored to reduce them to a few most
common quantities. Calculations for dynamic range vary considerably amongst sources,
so they are not presented here. It is best to consult device datasheets when possible
for specific values.
Note: The following equations
are valid for pure sinewave inputs with no DC
offset voltage. "R" is the input
resistance in ohms. Be sure to note units and subscripts for both the input parameters
and for the equations, or you will end up with really bad results.
Full Scale Voltage & Power
Power of a sinewave is calculated based on the rootmeansquare (rms) value of
the fullscale voltage. V_{FS(rms)} calculated from the peak (pk) input
voltage is:
Using the peaktopeak voltage (pkpk):
Fullscale input power in units of milliwatts (mW) based on fullscale peaktopeak
input voltage is:
Fullscale input power in units of dBm is:
Quantization Levels of an AnalogtoDigital Converter (ADC)
The value of a 1bit (LSB, aka "q" level) voltage step anywhere between 0 and
N bits for an Nbit ADC is:
The value of an nbit voltage step anywhere between 0 and N bits for an
Nbit ADC is:
Because decibel units represent a logarithmic and not linear relationship between
of number of ADC bits ("n") and power level, a simple multiplication of "n"
or "n_{2}  n_{1}" times some fixed power reference value does not
work. Instead, you must calculate the value in watts (or mW, nW, etc.) for each
number of bits using the voltage at each level, then conversion to dBm units can
be made for an absolute value at each bit count:
The difference in Pn1(dBm) and Pn2(dBm) is expressed in units of dB as follows:
SignaltoNoise Ratio (SNR) of an AnalogtoDigital Converter (ADC)
Most sources give the ideal quantizationbased signaltonoise ratio (SNR) equation
as 6.02*N + 1.76 dB (yellow highlight
below). A little more research turns up the
source of that equation (purple highlight below). Here, I show the
steps between purple and yellow, using common rules of logarithms and rules of exponents.
Another equation exists for
calculating SNR based on
aperture jitter that looks like the following.
Note in the graph to the right that the SNR goes negative  which is invalid  when
f_{input_signal}*t_{aperture_jitter} >
1/2π.
It might be best to use the worst case of either SNR_{clock_jitter} or
SNR_{quant} for system budget planning. Datasheets often provide SNR information,
which should be used instead of any generalized equations.
Noise Figure of an AnalogtoDigital Converter (ADC)
Probably the most difficult equation to find for an ADC is for noise figure (NF),
which is typically the last component in a cascade calculation of a receiver chain.
My source for the equation is a
Texas Instruments (TI) document authored by Mr. Tommy Neu
(it also appeared in
MWJ). You need the SNR value either from the
ADC datasheet or from the above equation is required. Noise spectral density (NSD)
is also needed, so its equation is provided as follows. NSD units are formally W/Hz
or, equivalently, V/√Hz; however, the equations are provided without units
because of the manner in which bandwidth is absorbed into them in these simple forms.
,
where:
Finally, the noise figure (NF) is calculated, where kTB is −174 dBm/Hz:
Example
An example for the
ADS4149
from the aforementioned TI paper (page 4) helps to clarify the application.
f_{sample_rate} = 250 Msps
N = 16 bits
V_{pkpk} = 2 V
SNR_{full_scale} = 71.9 dB
kTB_{T=290K,B=1_Hz} = 174 dBm
R = 200 Ω
Updated August 16, 2019
