Intuitively, if the signal-to-noise ratio were unity fora single bit of the four-bit word, the output signal-to-noise ratio would be 2/1 (the bits of the signal add directly, but the noise adds as the
8
This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and
Table 1 Correlation Coefficients: (a) Four-Bit Sequences; (b) One-Bit Sequences; (c) Two-BitSequences. GROUP 2
\0011 0101
SIGNAL - fl( t )
88;*3$$$8 3 0% 2 0 0 0 0
GROUP 1
0000+4
00
0
0 -4+2+2+2 -2+2 -2 -2 -2
0+4 00 0
0 -4 0 c2+2 -2+2 -2+2 -2 -2 0+2 -2 0 -2+2 0+2 -2+2+2 -2 -2+2 - 2
0+4 0
0 -4 0 0 0 0 0
- Qcv
s?
g
0110 1001 0 1010 1100 1111
00
0+4, -40 -4+4
+2+2 -2 -2 -2+2 -2 -2+2+2+2 -2-2+2+2 -2
0
c
E
0
0 -4 0 00
0+4
0 -2+2 -2 0 -2 -2
+2
a
e2 aa
'
0 -4 0
00
0+4 0
+2 -2+2 -2+2+2+2 -2+2+2+20 00 0
-4 0
0
0+4 -2 -2 -2-2 t 4
0001+2+2+2 -2
+2 -2 -2
0
00
0
0-4
"0N
\
0010+2+2 -2+2 -2 0100+2 -2+2+2 -23
+2 -2 -2 -2+2 -2
0+4
0-4
0 0 0 0 00h
0
0+4
0
0-400
00+201I
a 0111 -2+2+2+2 -2 -2 -2
+2 0
0 0
0+4-40-4+4
0 0 0
E
1000
+2 -2 -2 -2+2+2+2 -2+2
0
0
101
mfl(t)
0
0+2 -2 0 0 -2+2
I
1011 -2+2 -2 - 2+2+2 -2 1101 -2 -2+2 -2
0
0-4
00
0+4
O 1111
-2 0
0+2
+2 -2+2+2+2+2
I
0-4
0
00
0+4
1110 -2 -2 -2+2 -2
+e
-4
0
0
0
0
0+4
square root of the sum of the squares). The signal-to-noise ratio has doubled and the information rate has reduced by only one-fourth. This is a good improvement; unfortunately, it cannot be realized. True, the signal-to-noise ratio out of the correlator is improved over the bit-by-bit detection, but the outputs of the correlators must be compared to determine the largest coefficient. It is probable that noise in another correlator willbe greater than the coefficient in the correct correlator. This probability increases as the numb
er of correlators increases. The situation is discussed in detail later. The group code method still provides a substantial improvement in the signal-to-noise ratio over bit-by-bit detection. The number of correlators in group 1 can be reduced from eight to four if the - 4 volt output also is utilized. Since each correlator responds only to its matched signal and its complement, a single correlator can be used to detect two sequences. The correlator with the greatest absolute magnitude is selected as the one having the best probability of containing the signal. The sign of the coefficient determines whether the signal was in the first or second four sequences of a group. A code of this type is said to be bi-orthogonal since negative as well as positive cross-correlation coefficients are utilized. When only the positive outputs are used the code is orthogonal.
9
This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and
In the preceding example the prearranged order the four-bit sequences was most important. of A few fundamental rules have been developed to generate codes with bit lengths of n= 2k, where k= 0, 1, 2, 3, These rules are amenable to computer programming but are somewhatdifferent from those found in the literature. Consider first the correlation table for a one-bit word as in Table l(b). Only one correlator is required if t h e bi-orthogonal properties of the signal are utilized. This table and the correlation table for the two-bit sequences of Table l(c) can be written by inspection. The first word of the four-bit table is generated by combining the first word, 00, of the two-bit table with itself to form 0000. The next word is made by combining the first word, 00, with its complement, 11, to form 0011. The same process is carried out with the second word of the two-bit table to form the next two words. The complements of these four words are now taken in reverse order; this generates the code of group 1. The group 2 code is started by combining the first word of the two-bit table with the second. The next word is obtained by combining the first word of the two-bit table with the complement of the second. For the third and fourth words the twobit second word is combined with the first word and then with its complement. The next four words are the complements of the newly generated words in reverse order.
.
The eight-bit correlation table has 256 sequences which form a matrix of 65,536 correlation coefficients. A digital computer program for generating the 8-bit table w a s written by utilizing the four-bit table and the rules for forming the table from the preceding paragraph. The program was written in FAP (Fortran Assembly Program) s o that the bit patterns of the signals f, ( t ) and stored waveforms f,( t ) could be cyclically varied. That is, the first bit of all the eight-bit sequences was moved to the last-bit position anda new s e t of correlation coefficients computed. The program is arranged so that a print-out contains the correlation between the first sixteen signals and