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Two­level and Three­level
Autocorrelation functions and their statistics
The prototypical multipl
i circuit
Each product between the immediate delayed signals executed
a multiplier unit, result accumulated each
t
= nDt) provide ACF
of input signal. The immediate and delayed input signals each have sign
magnitude bit; input numbers can designated and +1. The multiplier
makes distinction between and
it recognizes only the three levels and
possible products such levels also and which the output
multiplier circuit biases up and
2 avoid occurrence negative
numbers
in products. The overflow from clocked two­bit accumulator drives (32­
ripple counter further accumulation.
nature logic employed the multiplication/accumulation operations can
appreciated reference figure. The accumulator's LSB and next held
flip­flops respectively. When clocked, these flipflops toggle (change state)
if their inputs (HI).
When the biased product
a
2, ff2 toggle. (Note that G3 will
then both low (LO) toggle).
When the biased product
a
1, ff1 toggle. Thus G3 every
other time the biased product
is
a provide carry from the first
second accumulator stage. (Note that G3 cannot both together). the
two­level case, both magnitudes always high, never toggled.
Finally, when biased product neither flip
is allowed toggle.

MULTIPLICATION TABLE
2­level biased 3­level biased (3­)
V
|
V
1
_________________ ______________________
1
|
2
1
1
2
0
­1
|
0
0
0
0
1
1
1
0
2
________________________________________________________________________________________________________________
biased output multiplier circuit
is
in each case one larger than would
without bias. Accordingly, every accumulation carries extra total,
which readily adjusted off­line subtracting number clock cycles
integration from readout value
of accumulation (taking care remember
least significant bits resident
Two and Three­level digitization
digitizer samples the instantaneous input voltage rates 100 MHz. The
input voltage gaussian amplitude distribution with value
probability that any given sample lies between and
V
+ volts given
F
(
)
=
2
(
)
1
/
V
2
/
2
(
)
2­ 3­level digitizer converts each voltage sample into
a number,
V
3
, which
considered dimensionless. two­level digitizer converts every positive voltage
sample into
a and every negative voltage into
A three­level digitizer converts
voltages above
,
a threshold voltage, into +1, converts voltages below
into Voltages between and are converted
a zero.
Distortion
coarse quantization (2­level 3­level sampling) used correlator distorts the
measured ACF, because signal Gaussian random process, this distortion
be corrected. relations between measured 2­bit and 3­bit ACFs,
2
3
,
undistorted normalized ACF, can derived from Price's theorem and
(e.g. Hagen Farley Radio Science 775, 1973)

2
t
V
(
)
r
2
d
x
x
2
0
2 arcsin
(
)
3
t
)
V
(
)
r
3
1
d
-
x
2
Ð
/
(
)
2
1
Ô
Õ
Ã
/
2
1
Ô
Õ
Ã
È
Ê
É
Û
Ø
Ç (3),
where
a
is digitizer threshold level the rms signal voltage. These relations
plotted
in figure below.
In practise except zero have
correlations (10%
or less) usual spectral applications, and both curves quite
linear the corresponding regions interest.
values
of
r
2 and
r when
r
=
1 (which usually only arises case "zero
lag") obtained calculating the expectation the square digitizer
output. the 2­level case, where
V always equal ­1,
2
2
= Thus, since
r
2 (1)
º
1, zero contains power information, which therefore
obtained separately from
a power counter 2­bit correlator. On other hand
in
3­level case
V
3
2
is except when input voltage the dead zone between
and The expectation
3
2 therefore evaluated using Gaussian
distribution
of input voltage,
r
(
)
=
-
2 exp
2
/
2
2
(
)
0
Ð
2
reconcile this expression with above, general expression
3 make
following substitution (3):
y
/
2
/
1
)
Then
r
(
)
=
/
2
(
)
y
(
d
y
/
2
2
¥
/
y
2
/
2
definite integral found
in §3.363 Table Integrals, Series,
Products Gradshteyn
& Ryzhik (Academic Press, 1965) and indeed
identicalACF Stat istics
While distortion be corrected, there price pay 3­level sampling;
more samples must be averaged than the multibit case the same post­integration
variance.
Multibit Sampling lags)
the multibit case, scatter zero (Power) given by
“
“
m
)
/
m
2
where
m represents multibit
N number samples used form
estimate. For other lags, where the correlation small, variance given
“
“
r
r
)
2
r
=
1
/
Npresentation follows used John Hagen appendix our 2048­lag
correlator (p100ff). continuing,
it useful have three prior relations
in mind:­
variance,
2 binomially distributed variable, for which probability one
of only possible states and that the alternate,
q after trials
2
p
/
N
It also convenient
to remember that the variance scaled parameter such
r,
when related another parameter
r through
r
2
=
2
alternate way looking here
is
k
=
dr
/ dr, whereon becomes
r
2
=
d
(
)
2
Thirdly, multibit relation between correlation lag,
r
, true correlation,r
,
is (Price's theorem, Table Hagen and Farley Radio Science 775, 1973
r
=
2
where the variance input "gaussian noise". Hence from
2
4
r
2 (10).
It convenient for purpose write the alternate form
r
=
2
d
r
2
(
)
{
}
)
,
whereon variance estimate
2 from
(
S
2
;
(
)
=
r
2
/
d
2
(
{
}
2 (11).
Two­level Sampling (all lags)
two­level sampling zero exactly no variance (and
information). When small, however, we have from (2), (6),
r
2
2
1
;
r
(
)
=
2
/
;
r
2
2
2
=
.Thus two­level sampling costs factor
of
in integration time over multibit
sampling.
Three­leve
l Sampl (except zero lag)
Consider the case where
r The variance
r
is then given
r
3
2
=
-
r
N
-
1
2
/
(
) exp
x
(
)
/
¥
Ð
[
] (12).
obtain expression
d
3
/
d using linear approximation
r
3
r
/
(
)
-
/
(
)
2
)
whereon
r
(
)
2
=
r
3
2
r
(
)
-
2 from
) becomes
r
(
)
2
=
/
4
(
)
- exp
(
)
2
(
)
/
2
(
2
/
2
(
)
d
x
/
Ð
[
2
function, multiplied ratio
r
3
(
)
2
2 plotted below. has
a minimum
value 1.53 a/s 0.6. Thus three­level sampling requires
a factor 1.53 more
integration time than multibit sampling when correlation small.
Three leve
l Power Est imate (zero ag)
have seen above
in discussion three­level ACF that sensitivity
maximized small correlations when
» 0.6 [where a/s
threshold digitizer voltage
(
)
to rms
(
s signal]. But deal now with

zero lag where correlation
is total. Let
r
3 denote 3­level zero lag,
x
multibit voltage,
V digitized value, whereon
r
3 exp
-
2
/
2
(
)
/
2
;
“
r
3
N
-
1
3 (14),
from estimates. zero lag
º
1
,
2 directly estimated, (9),
r
3
/
“
r
3 (15).
there only two possible outcomes
(1 from each evaluation
V
3
2
, the
variance
of
“
r
3 that binomial variate, which case, from (7),
3
2
3
r
(
)
N
-
1 (16).
derivative
3 with respect
2 obtained differentiating the term (14)
integrating parts,
is
d
3
/
d
2
=
2
/
2
(
)
/
2
2
[
] (17),
variance
of
2 obtained substituting (16)
& (17)
S
2
;
“
S
)
-
4
2
r
3
r
(
) exp
/
(
)
2
[
]
N
/
)
2
[
] (18).
Maximum (power) sensitivity occurs when 1.5 which 0.13
“
S
)
-
4
=
- (19),
which requires nearly 150% much integration time the multibit case
achieve the same precision. However when 0.6, our usual choice
non­zero sensitivity, we have
“
S
)
-
4
= (20),
which requires factor more integration time than the multibit case (e.g.
power counters) achieve same precision.

Setting the digitizer threshold three­level sampling
Despite the lower variance multibit power counter data (which
is
in
case only available the current correlator
at
a bandwidth 50 MHz), digitizer
threshold value
is best directly utilising zero reading, which does not
require invocation any scaling factors. When the threshold voltage
a
is set
equal 0.6 the average unbiased multiplier product
r
= exp
-
2
/
(
0
.
¥
Ð
d
x
/
= 5485
Since the average bias unity (to enable correlation products be positive), the
level setting procedure adjusts the
s coupled input signal relative
to
digitizer threshold voltage attenuating This achieved making short,
preliminary integrations and adjusting the attenuation until the (biased) zero
reading 1.5485 times mean bias. The required accuracy »20%.
It should noted though that from time time incidence
of during
attennuation­setting integration results
in inappropriate less than optimal) settings
of attenuation
of individual subcorrelators with procedure.
Oversampling
Coarse quantization
a signal results
a signal somewhat greater
bandwidth. Consequently the Nyquist quantized signal greater
twice the original bandwidth,
if we sample faster expect some statistical
improvement
in correlation function estimates. This improvement virtually
complete
if sample
at times the original bandwidth. case integration
factor 2.46 2­level sampling relative multibit sampling
is decreased
to
1.82 factor 1.53 for 3­level sampling decreased 1.26. The
disadvantage oversampling that only data from every other lag used
frequency resolution
is half.