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A/D Converter Calculations for RF Applications

A/D Converter Calculations for RF Applications - RF Cafe

VariableDefinition
VFS(pk)Full-scale peak input voltage
VFS(pk-pk)Full-scale peak-to-peak input voltage
VFS(rms)Full-scale input rms voltage
PFS(mW)Full-scale input power in mW units at full-scale input voltage
PFS(dBm)Full-scale input power in dBm units at full-scale input voltage
fsample-rateSampled analog input signal frequency in Hertz (Hz)
VLSB(mVpk-pk)Peak-to-peak input voltage at one "q" (LSB) level; i.e., n=1
VLSB(mVprms)Rms input voltage at one "q" (LSB) level; i.e., n=1
Vn_bits(mVpk-pk)Peak-to-peak input voltage at n "q" levels; i.e., 0≤n≤2N
Vn_bits(mVrms)Rms input voltage at n "q" levels; i.e., 0≤n≤2N
ΔPn1_to-n2-_bits(dB)Difference in Pn1(dBm) and Pn2(dBm) expressed in units of dB
snrquantSignal-to-noise ratio due to quantization (sampling)
SNRquantsnrquant expressed in decibels
SNRaperture_jitterSignal-to-noise ratio due to aperture jitter
NSDADCNoise spectral density expressed in decibels
NFADCNoise figure expressed in decibels

These equations predict the RF electrical performance of an Analog-to-Digital 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, signal-to-noise 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 RF Cafe Calculator Workbook v7.8 (download for free) - RF Cafeoffset 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.

Download RF Cafe Calculator Workbook to have the hard work done for you.

Full Scale Voltage & Power

Power of a sinewave is calculated based on the root-mean-square (rms) value of the full-scale voltage. VFS(rms) calculated from the peak (pk) input voltage is:

ADC full-scale voltage from Vpk (Vrms) - RF Cafe

Using the peak-to-peak voltage (pk-pk):

ADC full-scale voltage from Vpk-pk (Vrms) - RF Cafe

Full-scale input power in units of milliwatts (mW) based on full-scale peak-to-peak input voltage is:

ADC full-scale power in mW units - RF Cafe

Full-scale input power in units of dBm is:

ADC full-scale power in dBm units - RF Cafe

Quantization Levels of an Analog-to-Digital Converter (ADC)

The value of a 1-bit (LSB, aka "q" level) voltage step anywhere between 0 and N bits for an N-bit ADC is:

ADC LSB voltage of 1 bit (mV pk-pk) - RF Cafe

ADC LSB voltage of 1 bit (mV rms) - RF Cafe

The value of an n-bit voltage step anywhere between 0 and N bits for an N-bit ADC is:

ADC voltage of n LSB bits (mV pk-pk) - RF Cafe

ADC voltage of n LSB bits (mV rms) - RF Cafe

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 "n2 - n1" 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:

ADC power of n bits (mW) - RF Cafe

ADC power of n bits (dBm) - RF Cafe

The difference in Pn1(dBm) and Pn2(dBm) is expressed in units of dB as follows:

ADC power step in decibels - RF Cafe

Signal-to-Noise Ratio (SNR) of an Analog-to-Digital Converter (ADC)

Most sources give the ideal quantization-based signal-to-noise 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.

ADC Signal-to-Noise Ratio equation - RF Cafe

ADC SNR Due to Aperture Jitter - RF CafeAnother 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 finput_signal*taperture_jitter > 1/2π.

ADC SNR equation due to aperture jitter - RF Cafe

It might be best to use the worst case of either SNRclock_jitter or SNRquant for system budget planning. Datasheets often provide SNR information, which should be used instead of any generalized equations. 

Noise Figure of an Analog-to-Digital 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

TI ADS4149 ADC specifications - RF CafeAn example for the ADS4149 from the aforementioned TI paper (page 4) helps to clarify the application.

fsample_rate = 250 Msps

N = 16 bits

Vpk-pk = 2 V

SNRfull_scale = 71.9 dB

kTBT=290K,B=1_Hz = -174 dBm

R = 200 Ω

Related Pages on RF Cafe

- A/D Converter Calculations for RF Applications

- A/D & D/A Converter Vendors

 

 

Updated August 16, 2019

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