Equivalent of single and dual SSRGs
 Property 10: Let f1 ( x) and f 2 ( x) be two characteristic
polynomials. Then any sequence that is the mod 2 sum of
sequences generated by f 1 ( x) and f 2 ( x) can be generated by
the SSRG having characteristic polynomial f1 ( x) f 2 ( x) .
For example,
f ( x)  1  x  x 2  x 3  x 4  x 5  x 6  (1  x 2  x 3 )(1  x  x 3 )
Figure shows the diagram with two shift registers is equivalent to the
single shifter.
S0
1
S1
2
S2
3
S3
Output
Sequence
S0
1
S1
2
S2
3
S3
教育部網路通訊人才培育先導型計畫
S0
1
S1
2
S2
3
S3
4
S4
5
S5
Wireless Communication Technologies 2.4.2
6
S6
1
Pseudo-random sequences (PN sequence)
 Random sequences can be generated by making independent
samples of a zero-mean noise process possessing a
symmetrical density function.
 PN sequences, on the other hand, are generated by shift
register generators.
f ( x , x ,..., x )
1
S0
Example for a PN
sequence (n = 5):
1
S1
2
S2
2
m
m
Sm
Output
Sequence
Clock
1011101100011111001101001000010
教育部網路通訊人才培育先導型計畫
Wireless Communication Technologies 2.4.2
2
Postulates of randomness [following Golomb]
 P-1: In every sequence period, the number of +1’s does not
differ from the number of –1’s by more than 1.
 P-2: For every sequence period, half the runs (of all 1’s or
all –1’s) have length 1, one-fourth have length 2, one-eight
have length 3, etc., as long as the number of runs equals 1.
 P-3: The autocorrelation function R (m ) is binary valued; that
is
1 m  0
1 p
R(m)   bn bn m  
p n1
c 0 | m | p
where bn  1 2an , bn  {1} , and an  {0,1} .
In fact, we will desire c  1 , so that the sequence “looks”
white. A sequence of that satisfies Postulates 1-3 will be
called a pseudonoise (PN), or pseudorandom, sequence.
教育部網路通訊人才培育先導型計畫
Wireless Communication Technologies 2.4.2
3
Example 2.4-2 M-sequence for three postulates
 Consider as an example the output sequence
{an }  1,1,1,0,1,0,0, or {bn }  1,1,1, 1,1, 1, 1,
 We see that there are four –1’s and three +1’s in the period of
{bn } , which satisfies Postutulate 1.
 Of the total of four runs, one-half have length 1 and onefourth have length 2. Notice also there are two runs of +1’s
and two runs of –1’s.
 The autocorrelation function R (m) is
R ( m) 
7
1
bn bn m

7 n1
教育部網路通訊人才培育先導型計畫
1 m  0

 1

0 | m | p

 7
1 1 1 -1 1 -1 -1
R (m)
Wireless Communication Technologies 2.4.2
4
Properties of linear SRG Sequences
 Property 11: The randomness postulate-1, is satisfied for all
maximal length sequence.
1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0
-
=1
 Property 12: The run property postulate-2, holds for all
maximal length shift register sequences.
1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0
8/16
2/16
4/16
教育部網路通訊人才培育先導型計畫
1/16
1/16
Wireless Communication Technologies 2.4.2
5
Properties of linear SRG Sequences
 Property 13: The two-level autocorrelation property holds for
all maximal length sequence.
 We see that a maximal length sequence satisfies all three
postulates of randomness and therefore is a pseudonoise
sequence.
 Consequently, all our comments about PN sequences are
equally applicable to maximal length sequences.
教育部網路通訊人才培育先導型計畫
Wireless Communication Technologies 2.4.2
6