Analog‐toAnalog
to Digital Conversion
Revisited in the Frequency
qy Domain
CHIKADA, Yoshihiro
National Astronomical Observatory
off JapanJ

Can it be possible to make
A lAnalog‐to‐Digital
Di i l CConversion
inoiseless?
Yes or no ?

Analog‐to‐Digital conversion (ADC) is
comprised of two steps.
• One is sampling• And the other is quantization.

quantizerADC
samplerA quantizer is a nonnon‐linear
linear circuit• It has stair case input‐output characteristics.
outputinput

quantizer

Let us view quantizationin the time‐domain• OutputO t t iis splitlit iinto
t didiscretet llevelsl separatedt db
by 1 LSBLSB.
– LSB = least significant bitoutput
1 LSBinput

quantizer

1 LSB

Quantization causes error

Quantization
i icauses error.

Error cause quantization noise• Noise power = variance of error εQuantization
i icauses error ε.

Quantization noise power = 1/12 LSB2
• If
– The qquantization stepp width is small enoughgcompared to the input amplitude– And the quantizer has large enough number ofsteps to cover the input amplitude

Anyway,
Quantization noise power ≠ 0• Can it be possible to makeAnalog‐to‐Digital Conversion noiseless?
• To
T answer that,th t consider
id otherth examples
l off
non‐linear circuits.

Typical Frequency Converter with aMixer Diode
Mixer diodeRFIF
RF BPFIF BPFLO• A mixer diode is surely non‐linearnon‐linear.
• It introduces harmonics and inter‐modulations.
• But they are blocked by the IF Band‐Pass Filter!
• And we get distortion
distortion‐less/noise‐lessless/noise less IF.
IFTypicalReduction
NoiselessinAnalog‐toNoise
Analog‐toDigitalDigitalConverterConverterquantizer
AnalogAl
input

Input filter( )Outputfilter

DigitalDi
i loutput

LO or some known
ksignall• A quantizer is surely non‐linear
non‐linear.
• It introduces harmonics and inter‐modulations.
• But they will be blocked by the Output Filter!
• And we get distortion
distortion‐less/noise‐lessless/noise less Digital output
with increased word‐length.

Is there any predecessor eliminating
distortion by a non‐linear effect?
• Yes, therehis.
i It was patented
d 70 years ago.
• “AC biasing in the magnetic audio
recorder”• By adding CW (‘LO’) of several tens of kHz, thepatent reduced the distortion which wascaused by non‐linearity of magnetic material.
• SONY utilized it to energize its voyage to a
distinguished company.

To Reduce Noise inAnalog‐to Digital Converter• Think in the frequency domain.
• To do so…
• Analytical treatment todistinguishgharmonics andinter‐modulations is needed.

Break down an N‐level quantizer intoa sum of N‐1 2‐level quantizers.
Shifted Unit
Step function
Shifted Unit
Step function

Sum ofO
Outputs

Shifted UnitStep function• Then we only need to analyze a 2‐level quantizer.

iei.e.,• Output of an n‐level quantizer is as follows.

Assume thatthe signals are ‘pink noise’.

Thishi ‘‘pink
i k noise’
i ’
assumption is very
natural tomacroscopic
objects where
many independent
oscillators are inthe sight.

Consider a time series a(t)of a ‘pink noise’ A(ω),a(t) obeys
obe s the Normal Distribution.
Distrib tion
• ∵） a(t) is Fourier Transform of A(ω), i.e., ateachh t it is a sum off phase*A(ω).
h * ( )
• Remembere e be tthat
at PDF(a+b)(a b) iss coconvolution
o ut o oofPDF(a) and PDF(b).
• TheTh CCentralt l LiLimit
it Theorem
Thconfirmsfiththeconclusion.

• After lengthy calculations:– Breaking down an N‐level quantizer into asum of N‐1 2‐level quantizers,– Considering a Unit Step function as a limitof Error function when steepness →∞,– Using Taylor expansion of Error function,– Assuming ‘pink
pink noise’
noise inputs,–…

The correlation function of the output
or the quantization correction is:whereImportant thing is:• We have succeeded to analytically derive theoutput correlation function for any ADC as asum over:– K: harmonics order– κ : inter‐modulation orderExample of 2‐level
2 level quantizer spectrum• Baseband
b d inputicauses many hharmonics
i anddinter‐modulations in the signal band.

Example of 2‐level
2 level quantizer spectrum• Non‐baseband
b b d iinput reducesdhharmonics
i anddinter‐modulations in the signal band.

Frequency domain approach has
succeeded in deeper understandings• The correction is conventionally understoodg of jjoint
via time domain as integral
probabilities between the digitized signals.
• We have analytically showed this correction
in general form via frequency domain
approach.
hAn Introduction:
Small‐bit ADC in radio astronomy• In radio
di astronomy,• to have receivingg bandwidth as wide aspossible,• ADCs with very small number of quantizationlevels (usually 2‐8 levels, i.e., 1‐3 bits) havebeen usedused.
• Therefore it is very important to developmeans tot minimizei i i noisesiand/ord/ spectrumtdeformation caused by an ADC.

The Van Vleck correctionor the quantization correction• Correct– the observed digitizedgcorrelation coefficient– to the analog oneone.

• For an example, in a two
l l quantizer case, thelevel
hformer is proportional
to arcsine of the latter.
– A Nyquist‐sampled baseband case

Sudden rise in the correction curve
whenh ththe correlationl ti coefficientffi i tapproaches to unity is:• Caused by thet e higherg e κ aanddKoorderde teterms.
sSudden rise in the correction curve
whenh theh correlationl i coefficient
ffi i
approaches to unity is:• Caused by thet e higher
g e κ aanddKoorderde teterms.
sSummary• Via frequency domain:• A way to realize ‘noiseless/reduced‐noise’
noiseless/reduced noise ADC
was demonstrated.
• AnalyticalA l i l expression
i off ADC output correlationl ifunction was given.
• The expression was given as a sum overharmonics order and inter‐modulation orderorder.
• The expression is very convenient for filtering.