TWO BIT CORRELAIOR
by B. G. Clark
In one bit correlation one records only the sign -- + or -. In
two bit correlation, we have a sign and a magnitude
bit, giving four
possible states. Let us denote the digitization of a variable A by [A]
and denote the four possible states by
+L if
[A] =
H
+S if H
>0
-S if 0>>-H
-L if -H >
where H is the second clipping level. We now have the freedom to define a
multiplication table [A] * [B], which, because of the symmetries of the
problem may, without loss of generality, be taken to be
r.A.1
[A] * [B] =
-L -S +S +L
-L
n m -m -n
-S
m I -1 -m
+S -m -1 1
+L
-n
-m
m
n
if A and B are equal independent gaussion noises from the receivers, and
C is the noise from the source,
x2
1
ProblX<A<X+ dXl= Prob{X<B<X+ dX}
x2
1
Prob {X<C<X+ dX}= 1/271. a
2a2
1
e
The correlator output is < [A+C] * [B+C] >.
dX
= v2Tr
a
e
dx
2
Employing all the symmetries of the problem,
[A+C] * [B+C] > = 2n [ Prob
{A-vc›nr\B+c>11}
- Prob {A+C>Hr)13+C<-H}
+4m [ProbIA+C>Hr1H>B+C >0 }
- ProblA+C>Hr10>B+C> -111]
+ 2 [ Prob{H>A+C> OnH>B+C> 0}
Prob{H>A+C> 0
n0
>B+C> -H
Co
•
Prob{X<C<X+ dX} [nProb{A>H-X}
2
X=
( Prob{B> H-X } -Prob {B>H+X })
+ 2mProb {A>H-X} (Prob {H-X>B> -X }
- Prob{H+X>B>X}) + Prob{H-X>A> -X }
( Prob{H-X>B> -X} - Prob1H+X>B>X1) ]
=2
11Tr-- a
Erf
2J
+ Erf
H + X
H
-Erf
a
( 2Erf (
H - X
a
+(Erf
-Erf
a
H+G X
(
1
/2Try
t
2
2
dt
)
) )
a
Erf
H+X
a
-X
where
Erf (y) =
11. -
1
+ 2m
- Erf
I
n-Erf
[
)
)i
a
+Erf
)
) +Erf
H - X
( 2 Erf
a
If we approximate a low signal-to-noise ratio, a «a then for all X of
1
interest X<<a and we can expand the error functions
C2
-2
1
Er f (C + y)
E r f (C) + y
e
then
1
< [A+C] * [B+C] > = 2
1/27 a
H2
+m
2X 22
e
( 1 - e
ITCY
2
Tr
2
1
a2
2
20.
2
2
1
2
r
x
n 770_2
1
H2
2a 2
)
H2
2-,c7 2
)
e
(1
x
-
dX
a
n + 1 - 2m) e
[
+ (2m - 2) e
H2
2-672-
-
+11
The noise on the correlator output is
N 2 = • ( [A] * [B] )
2
+ 2m 2 Prob f
•
Prob
I
A
!
> H r) IB1
At < H
r) 1BI <
(1-2Erf( a ) )
+ 4 ( Erf
> H (1
n2 Prob
>
2
2
I B1
H}
H1
+ 4m 2 ( 1 - 2 E r f
a
Er f
For continuous multiplication, the signal-to-noise ratio is simply
a
a
2
1
2
so the degradation produced by the quantization is
2
D= ( S/N ) / ( S c /N c ) =
[
Tr
(4n 2 + 4 - 8m 2 ) (Er f
(n + 1 - 2m) e
[
)
2
1- (4m
H2
a2
H2
4n 2 ) Er
2
+1]
+ (2m - 2) e
n2 ]
a
David Sun has programmed this equation and solves for the optimum (H/a)
given m and n.
Below are two tables giving, respectively, D and (H/a).
2
3
4
6
8
12
16
20
24
1
.787
.838
.855
.858
.851
.839
.832
.828
.825
2
.745
.800
.842
.875
.878
.865
.853
.845
.839
.769
.802
.855
.869
.880
.870
.860
.852
.771
.821
.857
.881
.880
.872
.864
.790
.830
.872
.881
.879
.873
.765
.803
.856
.877
.881
.878
.779
.837
.867
.878
.880
.759
.818
.854
.872
.878
INP-
3
4
5
6
D
7
8
m.\\n
2
3
4
6
8
12
16
20
24
1
.701
.782
.821
.828
.796
.740
.707
.687
.674
2
1.371
1.049
.977
.967
.948
.864
.800
.759
.733
1.410
1.199
1.036
1.006
.953
.885
.830
.792
1.385
1.122
1.036
.989
.942
.891
.847
1.223
1.077
1.002
.970
.932
.892
1.305
1.135
1.012
.981
.955
.924
1.197
1.028
.987
.967
.944
1.251
1.053
.993
.972
.955
3
4
5
6
7
8
Wu
The optimum signal-to-noise ratio occurs at
m = 4, n = 14, H/a = .967, D = .882.
5
However, the convenience of having n and m be powers of two would recommend
m = 4, n = 16, H/ a = .952, D = .880
at a negligible loss in signal-to-noise ratio.
The signal-to-noise ratio of the two bit correlator is thus about
.88 that of a continuous multiplier, as contrasted with .64 for a one bit
correlator.
June 14, 1966