Table I
Expected Error Signal for an m Sequence
x | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | E(S) |
---|
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 0)| = | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
E(St=0) = | 0 | + | 0 | + | 0 | + | 0 | + | 0 | + | 0 | + | 0 | = 0 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 1)| = | 0 | | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | |
E(St=1) = | 1 | + | 0 | + | 0 | + | 1 | + | 1 | + | 1 | + | 0 | = 4 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 2)| = | 0 | | 0 | | 1 | | 1 | | 1 | | 0 | | 1 | |
E(St=2) = | 1 | + | 1 | + | 0 | + | 1 | + | 0 | + | 0 | + | 1 | = 4 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 3)| = | 1 | | 0 | | 0 | | 1 | | 1 | | 1 | | 0 | |
E(St=3) = | 0 | + | 1 | + | 1 | + | 1 | + | 0 | + | 1 | + | 0 | = 4 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 4)| = | 0 | | 1 | | 0 | | 0 | | 1 | | 1 | | 1 | |
E(St=4) = | 1 | + | 0 | + | 1 | + | 0 | + | 0 | + | 1 | + | 1 | = 4 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 5)| = | 1 | | 0 | | 1 | | 0 | | 0 | | 1 | | 1 | |
E(St=5) = | 0 | + | 1 | + | 0 | + | 0 | + | 1 | + | 1 | + | 1 | = 4 |
∑|f(x) | 1 | | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | |
− f(x − 6)| = | 1 | | 1 | | 0 | | 1 | | 0 | | 0 | | 1 | |
E(St=6) = | 0 | + | 0 | + | 1 | + | 1 | + | 1 | + | 0 | + | 1 | = 4 |
|
Table II
An m′ Sequence with Two Related m Sequences
m sequence (n = 7): | 1 | 1 | 1 | 0 | 1 | 0 | 0 | |
m′ sequence (n = 14): | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | |
m sequence (n = 15): | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
Table III
Probabilities of Misalignment
t | E(S)m | E(S)m′ | zm | zm′ | pm | pm′ | (1 − pm) | (1 − pm′) | tpm | tpm′ |
---|
1 | 8 | 10 | 0.566 | 0.707 | 0.2857 | 0.2398 | 0.7143 | 0.7602 | 0.2857 | 0.2398 |
2 | 16 | 20 | 1.131 | 1.414 | 0.1290 | 0.0787 | 0.8710 | 0.9213 | 0.2580 | 0.1574 |
3 | 24 | 30 | 1.697 | 2.121 | 0.0449 | 0.0170 | 0.9551 | 0.9830 | 0.1347 | 0.0510 |
4 | 32 | 40 | 2.263 | 2.828 | 0.0118 | 0.0023 | 0.9882 | 0.9977 | 0.0472 | 0.0092 |
5 | 40 | 50 | 2.828 | 3.536 | 0.0023 | 0.0002 | 0.9977 | 0.9998 | 0.0115 | 0.0010 |
6 | 48 | 60 | 3.394 | 4.243 | 0.0004 | 0.0000 | 0.9996 | 1.0000 | 0.0024 | 0.0000 |
7 | 56 | 70 | 3.960 | 4.950 | 0.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
8 | 64 | 80 | 4.525 | 5.657 | 0.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
9 | 72 | 90 | 5.091 | 6.364 | 0.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
10 | 81 | 100 | 5.728 | 7.071 | 0.0000 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
Probability of alignment | | | | | 0.5856 | 0.6867 | | |
Probability of misalignment | | | | | 0.4144 | 0.3133 | | |
Mean error = ∑tp | | | | | | | 0.7395 | 0.4584 |
t: number of sample points by which a pattern is displaced from its aligned position.
E(
S)
m: error signal expected when the
m sequence is misaligned by
t.
zm: E(
S)
m, rendered in standard units by dividing it by the standard error of the difference between the error signal
St when the pattern is displaced by
t and the error signal
S0 when the pattern is aligned.
pm: the probability that
St will be less than
S0; therefore, the probability that the pattern will be misaligned by
t.
(1 −
pm): the probability that the pattern will
not be misaligned by the distance
t. The probability that the pattern will be correctly aligned (i.e.,
t = 0) is the joint probability that it will not be misaligned at each possible nonzero value of
t. Therefore, the probability of alignment is the product of (1 −
pm) for all values of
t.
The subscript
m′ indicates that the variable subscripted applied to an
m′ sequence instead of an
m sequence.
Table IV
Significant Expected Values of m and m′ Sequences
| m Sequence | m′ Sequence |
---|
|
---|
p = 2 | p > 2 |
---|
E(St=0) | 0 | 0 | 0 |
E(St=1) |
|
|
|
E(St=2) |
|
|
|
μ(′) = E(St≠0)min |
|
|
|
| |
|
|