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Description  |
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1. BACKGROUND OF THE INVENTION
This invention relates to an apparatus for generating orthogonal sequences
for code-modulating transmission signals for use in spread spectrum
communication systems, radar systems or the like.
2. DESCRIPTION OF THE PRIOR ART
Before describing the prior art, the mathematical properties of the
orthogonal sequences will first be described.
The term "sequence" used herein means time series of numerical values
a.sub.n shown in expression (1), below:
{a.sub.n }= . . . a.sub.n-1 a.sub.n a.sub.n+1 ( 1)
n is a factor representing the order of the sequences. a.sub.n is referred
to as a component of sequence and is a complex number. The sequence
{a.sub.n } is a periodic sequence in which there exists an integral number
of N satisfying expression (2), below:
a.sub.n+N =a.sub.n ( 2)
Thus, the sequence of expression (1) can be denoted by expression (3),
below:
{a.sub.n }= . . . a.sub.N-1 a.sub.0 a.sub.1 a.sub.2 . . . a.sub.N-1 a.sub.0
a.sub.1 ( 3)
In order to provide a quantum for describing the mathematical properties of
such a sequence, an autocorrelation function as defined by expression (4),
below, is often used:
##EQU1##
Wherein, * represents a complex conjugate. The reason why the
autocorrelation function is defined only within the range from m=0 to
m=N-1, as shown in expression (4), is that the sequence {a.sub.n } is a
periodic series and thus the autocorrelation function .rho..sub.(m) is a
periodic function. The period thereof is N and the same as that of the
sequence {a.sub.n }. Thus the function .rho..sub.(m) satisfies the
following expression:
.rho..sub.(m+N) =.rho..sub.(m) ( 5)
When such a sequence is applied to a practical system, it is necessary that
the autocorrelation function of expression (4) has such properties as
shown in FIG. 1, i.e., the function of autocorrelation has a sharp peak at
m=0 and takes a "considerably low value" in the remaining range of m (m=1,
. . . , N-1). The portion .rho..sub.(0) of the function at m=0 is referred
to as the lobe, and the other portion .rho..sub.(m) (m=1, . . . , N-1) of
the function than that at m=0 is referred to as the side lobe, and the
magnitude of the side lobe relative to the main lobe .rho..sub.(0) poses a
problem to be discussed. The magnitude of the side lobe which is a
"considerable low value" must thus satisfy relation (6), below:
.vertline..rho..sub.(m) .vertline.<<.vertline..rho..sub.(0)
.vertline..sub.m =1, . . . , N-1 (6)
With the satisfaction of relation (6), the sequence having zero magnitude
of the side lobe of the autocorrelation function, i.e., satisfying
expression (7), below, has excellent properties:
##EQU2##
The orthogonal sequence is defined as satisfying expression (7).
The usefulness of such an orthogonal sequence in its application to a
practical system will be described below with reference to FIG. 2.
FIG. 2 is a schematic representation of an example in which the orthogonal
sequence is applied to a radar system for detecting a target or targets.
In FIG. 2, numeral 1 refers to a local oscillator, 2 to a modulator, 3 to
an orthogonal sequence generator, 4 to a power amplifier, 5 to a low-noise
amplifier, 6 to a transmitting-antenna, 7 to a receiving-antenna, 8 to a
phase detector and 9 to a demodulator. Numerals 10a and 10b also refer to
targets A and B. Further, {a.sub.n } is a sequence, U a transmission
signal, S.sub.a a reflected signal produced by the reflection of the
transmission signal U on the target A (or 10a), S.sub.b a reflected signal
produced by the reflection of the transmission signal U on the target B
(or 10b), R a receiving signal, V a detection signal and Z a demondulated
signal.
For simplifying the description hereinbelow, all the mathematical
expressions of signals will be referred to by complex signals. Physically,
there is no such complex signal, but a real signal can correspond to the
real part of the complex signal, as expressed by Euler's formula, below:
e.sup.j.omega.t =cos .omega.t+j sin .omega.t (8)
(j: unit of complex number)
The local oscillator 1 generates a sinusoidal wave signal e.sup.j.omega.t
and transfers it to the modulator 2. On the other hand, the orthogonal
sequence generator 3 generates a sequence {a.sub.n } and transfers it to
the modulator 2 which, in turn, code modulates the sinusoidal wave signal
e.sup.j.omega.t with the sequence {a.sub.n } to output the transmission
signal U. The transmission signal U is amplified by the power amplifier 4
and then radiated through the transmitting-antenna 6 into the external
space. The transmission signal U radiated to the external space is partly
reflected by the target 10a and partly by the target 10b, thereby
producing the reflected signals S.sub.a and S.sub.b. The reflected signals
S.sub.a and S.sub.b are received by the receiving-antenna 7. The receiving
signal R is a compound signal of the reflected signals S.sub.a and S.sub.b
and has the following relation:
R=S.sub.a +S.sub.b ( 9)
The receiving signal R is amplified by the low-noise amplifier 5 and then
input into the phase detector 8. The latter acts to convert the receiving
signal R in RF (radio frequency) band to a detection signal V in VF (video
frequency) band by phase detecting the receiving signal R and then
transfer the detection signal V to the demodulator 9. The demodulator 9
receives the detection signal V and the sequence {a.sub.n } transferred
from the orthogonal sequence generator 3, demodulates the detection signal
V including a code-modulation signal and then outputs the demodulated
signal Z.
The operation of the radar system shown in FIG. 2 will be described in more
detail by reference to FIGS. 3 and 4. FIG. 3 shows a timed relationship
between a transmission signal and a receiving signal, and FIG. 4 is a
vector diagram of the component a.sub.n of the sequence. In FIG. 3, U,
S.sub.a, S.sub.b and R designate explanatory waveforms of a code-modulated
transmission signal, a signal reflected by target A, a signal reflected by
target B and a receiving signal, respectively. As shown in FIG. 3, the
sinusoidal wave signal is subjected to code-modulation with the components
a.sub.0, a.sub.1, . . . , a.sub.N-1, which are selected one by one at
every period of time .tau., namely component a.sub.0 is used in the time
interval between t=0 and t=.tau. and component a.sub.1 is used in the time
interval between t=.tau. and t=2.tau., . . . , thereby producing a
code-modulated transmission signal. The mathematical expression U.sub.(t)
of the code-modulated transmission signal U is given by the following
expression:
##EQU3##
where, rect [t] is a rectangular function as defined by the following
expression:
##EQU4##
The rectangular function mathematically expresses the changeover of the
components of the sequence {a.sub.n }, and the modulation is expressed by
the product of the component a.sub.n and the sinusoidal wave signal
e.sup.j.omega.t. Since the modulation is expressed by the product, the
component a.sub.n can be definitely expressed by a vector as shown in FIG.
4. The amplitude .vertline.a.sub.n .vertline. of the component a.sub.n
shows the modulation of the amplitude of the sinusoidal wave signal and
the phase .phi..sub.n of the component a.sub.n shows the modulation of the
phase of the sinusoidal wave signal. Since the sequence {a.sub.n } is a
periodic series, the modulation to be applied to the sinusoidal wave
signal also has a periodicity as shown in FIG. 3. The periodicity T is
expressed by:
T=N.tau. (12)
Because the reflected signals are created by reflecting a part of the
transmission signal on the targets, the waveforms of the reflected signals
S.sub.a, S.sub.b are similar to the waveform of the transmission signal,
as shown in FIG. 3. The timing each of the reflected signals being
received by the receiving-antenna 7 is delayed by such a time as required
for the radio wave to propagate two times the slant range between the
radar system and each target. Such a time delay is indicated by t.sub.a
with respect to the reflected signal S.sub.a and also by t.sub.b with
respect to the reflected signal S.sub.b in FIG. 3. Thus, the mathematical
expressions S.sub.a (t), S.sub.b (t) of the reflected signals S.sub.a,
S.sub.b are expressed by the following expressions:
##EQU5##
where, .eta..sub.a, .eta..sub.b are constant values representing the
intensities of reflection of the radio waves on the targets A (or 10a) and
B (or 10b).
Since the receiving signal R is a compound signal formed by both the
reflected signals S.sub.a and S.sub.b as indicated by expression (9), its
mathematical expression R(t) is given as follows:
##EQU6##
The phase detector 8 detects the receiving signal R and this is
mathematically equivalent to the multiplication of e.sup.-j.omega.t. Thus,
the mathematical expression V(t) of the detection signal V is given as
follows:
##EQU7##
The correlation process of the detection signal V and the sequence {a.sub.n
} is performed in the demodulator 9. As a method of such a correlation
process, there are two types, an analog type and a digital type, but the
difference between both types is only that the demodulated signal Z
produced as a result of the process is outputted with an analog signal or
a ditigal signal.
The case of the correlation process being of the digital type will be
described below. In this case, the detection signal V transferred from the
phase detector 8 is first sampled and converted to a digital signal at the
demodulator 9. The sampling period is set as same as the time .tau. of the
components as shown in FIG. 3. The numeral expression of the detection
signal V(k.tau.) (k= . . . , -1, 0, 1, . . . ) which is converted to the
digital signal is given by the following expression based on expression
(16).
##EQU8##
Taking into consideration the rectangular function rect (t) taking 0 out of
the range of 0.ltoreq.t<1, as shown in expression (11), expression (17)
can simply be expressed by the following expression.
##EQU9##
The demodulator 9 then performs such a correlation process operation as
shown in the following expression using the sampled detection signal V(k)
and the sequence {a.sub.n } transferred from the orthogonal sequence
generator 3, as shown in expression (19), to output the demodulated signal
Z(k):
##EQU10##
To explain in detail the demodulated signal Z(k) shown by expression (20),
expression (19) is substituted for by expression (20), whereby the
following expression is obtained:
##EQU11##
Comparing expression (21) with expression (4), the terms parenthesized by
[ ] in expression (21) are nothing but the autocorrelation function of the
sequence {a.sub.n }. Expression (21) can be rewritten using the
atuocorrelation function of the sequence to obtain expression (22), below:
##EQU12##
As shown by expression (22), the demodulated signals Z(k) is in the form
of adding the autocorrelation functions of the sequence {a.sub.n }.
FIG. 5 shows waveforms of the amplitude of the demodulated signal Z(k) for
explaining that the orthogonal sequence is useful. FIGS. 5(a) and (b) show
amplitude-waveforms of the demodulated signal Z(k) in the case of the
sequence being non-orthogonal, and FIG. 5(c) shows amplitude-waveforms in
the case of the sequence being orthogonal. In these figures, waveforms
Z.sub.a and Z.sub.b are main lobes of the autocorrelation function and
waveforms Y.sub.a and Y.sub.b are side lobes thereof. As seen in FIGS.
5(a) and (b), in the case of the sequence being non-orthogonal, it will be
understood that the main and side lobes interfere with one another. As
shown in FIG. 5(a), when there is no substantial difference in magnitude
between .eta..sub.a and .eta..sub.b (no remarkable difference between the
intensities of reflection of the radio waves on the targets 10a and 10b),
the interference between the main and side lobes is not so serious that
the reception of the reflected signals from both the targets can be
detected based on the demodulated signal. On the contrary, when there is a
substantial difference in magnitude between .eta..sub.a and .eta..sub.b
(large difference between the intensities of reflection of the radio waves
on the targets 10a and 10b), the interference between the main and side
lobes is a serious problem. As shown in FIG. 5(b ), for example, if the
main lobe Z.sub.b for the target 10b is covered with the side lobe Y.sub.a
for the target 10a, the reception of the reflected signals from both the
targets cannot be detected and the reception may be defined as if it would
be from only the target 10a. A situation where there is a large difference
in the intensities of reflection on two adjacent targets is often
experienced in the practical circumstances of the operation of radar
systems, typically when an airplane or the like is flying near a mountain,
for example.
Such problem can apparently be solved by using an orthogonal sequence for
the sequence, as shown in FIG. 5(c). With the orthogonal sequence, the
side lobe is zero and thus a small-magnitude signal of the main lobe is
not covered with any large-magnitude signal of the side lobe and the two
signals can be detected based on the demodulated signal no matter which
one of the signals has a larger magnitude than the other.
Although the description has been made while limiting the number of targets
to two, in the case of the simultaneous reception of reflected signals
from an N-number of targets, the detection can be similarly achieved by
using the orthogonal sequence, irrespective of the relative sizes of these
targets.
As conventional systems for generating orthogonal sequences having the
excellent properties as described above, a polyphase orthogonal sequence
has been known which is disclosed in "Phase Shift Pulse Code with Good
Periodic Correlation Properties" by R. Frank, IRE Trans., U.S.A.,
Information Theory, Vol. IT-8, published October, 1962.
FIG. 6 shows a flowchart of the algorithm for generating a conventional
polyphase orthogonal sequence. One period of the polyphase orthogonal
sequence is an orthogonal sequence having components (W.sup.K), each of
which is obtained in accordance with the algorithm. Namely, the value W is
first calculated at the step 11 as the following expression:
##EQU13##
wherein L is a natural number. The K-th power of the value W is then
obtained at the step 14 as the following expression:
##EQU14##
wherein K is an integer. The respective components are aligned with their
I and L increased one by one from I=0 and J=0 to I=L-1 and J=L-1, as shown
at the steps 12 through 18. This forms square L.times.L matrix as shown in
Table 1 and corresponds to sequential rows from the first line to the L-th
line.
TABLE 1
______________________________________
Line 1 W.sup.0, W.sup.0, . . . , W.sup.0
Line 2 W.sup.1, W.sup.2, . . . , W.sup.L-1
Line 3 W.sup.2, W.sup.4, . . . , W.sup.2(L-1)
.
.
Line L W.sup.L-1, W.sup.(L-1)2, . . . , W.sup.(L-1)(L-1)
______________________________________
Thus, the polyphase orthogonal sequence is given by the following
expression:
##EQU15##
As is apparent from expression (25), the period N of the polyphase
orthogonal sequence is determined only by L and given by the following
expression:
N=L.sup.2
(26)
The components of the polyphase orthogonal sequence will now be described
in detail.
As seen in expression (24), the amplitude of the component W.sup.K of the
polyphase orthogonal sequence is 1 irrespective of the value of K and
there is no need to amplifyingly modulate the sinusoidal wave signal in
code-modulation using the polyphase orthogonal sequence. On the other
hand, the phase of the component W.sup.K can be considered as follows.
Using integers p and q, the integer K can be expressed as:
K=pL+q (27a)
wherein
0.ltoreq.q<L (27b)
Therefore, expression (24) can be modified as follows:
##EQU16##
Expression (28) shows that the phase of the component W.sup.K is
equivalent to
##EQU17##
in accordance with expression (27a). The integer q can take all the
integers from 0 to L-1 as shown in expression (27b), and thus the phase of
W.sup.K takes an L number of values at a pitch of
##EQU18##
from 0 to
##EQU19##
As an example, the vector diagram of the components W.sup.K in case of L=8
is shown in FIG. 7. As shown in FIG. 7, the phase of the components of the
polyphase orthogonal sequence in the case of L=8 takes the 8-number of
values at a pitch of .pi./4. In a case that the polyphase orthogonal
sequence is used, therefore, it is necessary to phase modulate the
sinusoidal wave signal using the L-number of phases.
FIG. 8 shows an example of configuration of the modulator 2 shown in FIG. 2
wherein the sinusoidal wave signal e.sup.j.omega.t is code-modulated by
applying such polyphase orthogonal sequence as mentioned above. In FIG. 8,
19a, 19b, . . . , 19c and 19d are phase shifters, 20 a controller, and 21
a switch. The phase shifters 19a, 19b, . . . , 19c and 19d act to advance
the phase of the sinusoidal wave signal transferred from the local
oscillator 1 shown in FIG. 2 by an amount of
##EQU20##
respectively. The switch 21 is operated to select one of the shifters to
which the sinusoidal wave signal is transferred for every .tau. time and
this operation responds to a command signal C transferred from the
controller 20. The controller 20 calculates the phase of the components
W.sup.K of the polyphase orthogonal sequence to produce the command signal
C corresponding to the integer value q. If q=0, for example, the
controller 20 produces a command signal C with which the switch 21
connects its terminal I to its terminal O, and if q=L-1, it produces
another command signal C to operate the switch 21 for connection between
its terminal I and D.
In this manner, when a code-modulation is performed with a polyphase
orthogonal sequence, it is necessary to change the L-number of the phases
and thus the (L-1)-number of phase shifters are required.
As described above, the conventional techniques require a plurality of
phase shifters to be provided in the code-modulator, when the sinusoidal
wave signal is code-modulated by using the polyphase orthogonal sequence
in a radar system, for example. The number of the phase shifters depends
upon the period N of the polyphase orthogonal sequence and is equal to
##EQU21##
Therefore, there is the problem that when the code-modulation is performed
by using a polyphase orthogonal sequence the hardware of the modulator
becomes overly large.
SUMMARY OF THE INVENTION
The present invention was made to solve the problems described above, and
it is an object of this invention to provide an apparatus for generating
an orthogonal sequence the components of which take only two kinds of
complex number.
According to the present invention, such a purpose is achieved by an
apparatus for generating an orthogonal sequence which comprises an
M-sequence generator for generating an M-sequence the components of which
are 0 and 1 and the period is N and substitution means
(component-substituting unit) connected to the output stage of the
M-sequence generator for substituting the components of the M-sequence
input thereto in response to their value of 0, 1, wherein if the component
of the M-sequence is 0 it is substituted for by A.sub.o
e.sup.j.phi..sbsp.o, where A.sub.o is a positive real number, and if the
component is 1 it is substituted for by A.sub.1 e.sup.j.phi..sbsp.1, where
A.sub.1 is a positive real number, and the substitution being performed to
satisfy such condition that a trigonometric function f.sub.1 (.phi..sub.1
-.phi..sub.o) having a phase of (.phi..sub.1 -.phi..sub.o) is a ratio of
two functions f.sub.2 which is a quadratic function of A.sub.1 /A.sub.o
with a coefficient of a linear function of the period N of the M-sequence
and f.sub.3 which is a linear function of A.sub.1 /A.sub.o with a
coefficient of a linear function of N, whereby the orthogonal sequence is
generated.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram showing a waveform of an autocorrelation function;
FIG. 2 is a schematic representation showing a radar arrangement for
exemplifying the system to which the orthogonal sequence generator is
applicable;
FIGS. 3(a)-(d) are diagrams showing the timing of transmission and
receiving signals of the radar system;
FIG. 4 is a vector diagram of a component of the sequence;
FIGS. 5(a)-(c) are diagrams showing waveforms of a demodulated signal;
FIG. 6 is a flowchart showing the algorithm for forming a polyphase
orthogonal sequence;
FIG. 7 is a vector diagram of a component of the polyphase orthogonal
sequence;
FIG. 8 is a detailed block diagram of a conventional modulator;
FIG. 9 is a diagram showing an arrangement of an orthogonal sequence
generator utilizing the orthogonal sequence generating system in
accordance with the present invention;
FIG. 10 is a flowchart showing the operation of a component-substituting
unit shown in FIG. 9;
FIGS. 11 and 12 are vector diagrams of the components of the orthogonal
sequence generated from embodiments of the present invention; and
FIGS. 13 and 14 are detailed block diagrams of a modulator in accordance
with the present invention.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention will now be described by way of an embodiment with
reference to the drawings. In FIG. 9, numeral 24 denotes an M-sequence
generator comprising linear feedback shift register (herein appropriately
referred to as linear feedback shift register), 22a, 22b, 22c, 22d delay
elements, 23 an exclusive OR operation unit, and 25 a
component-substituting unit operable as a means for substituting a complex
number for the value of the component of the M-sequence, the unit being
comprised of a microcomputer, etc.
The delay elements 22a-22d store numerical values having 0 or 1. The
transfer of the numerical values of those delay elements has a period and
the numerical values are transferred at every 1 clock in the direction of
the arrow shown in FIG. 9. The numerical value b.sub.n is outputted as a
component of the M-sequence to the exterior and transferred to the
component-substituting unit 25. The exclusive OR operation unit 23
calculates the exclusive OR of the numerical values b.sub.n and d.sub.n
transferred from the delay elements 22a and 22d and transfers the
resultant numerical value e.sub.n to the delay element 22a. The definition
of the exclusive OR is shown in Table 2.
TABLE 2
______________________________________
Output Input
b.sub.n d.sub.n
e.sub.n
______________________________________
0 0 0
0 1 1
1 0 1
1 1 0
______________________________________
In such a linear feedback shift register, the sequence created by providing
at least one delay element with an initial value of, or other than
providing each of the delay elements with initial value of 0 is a known
M-sequence. The M-sequence is of the maximum period among the sequences
created from the linear feedback shift register. In the embodiment of FIG.
9, the delay elements in the linear feedback shift register are arranged
at four stages, but they can generally be extended at k-stages. However,
it is necessary to satisfy some limited combination of the feedback tap
position in order to create the M-sequence with the k-stage linear
feedback shift register. Such combinations have already been obtained. One
example of the combinations is shown in Table 3.
TABLE 3
______________________________________
Stage Code length Longest connection tap
______________________________________
2.degree.
3 [2.1]
3.degree.
7 [3.1]
4 15 [4.1]
5.degree.
31 [5.2] [5.4.3.2] [5.4.2.1]
6 63 [6.1] [6.5.2.1] [6.5.3.2]
7.degree.
127 [7.1] [7.3] [7.3.2.1] [7.4.3.2]
[7.6.4.2] [7.6.3.1] [7.6.5.2]
[7.6.5.4.2.1] [7.5.4.3.2.1]
8 255 [8.4.3.2] [8.6.5.3] [8.6.5.2.] [8.5.3.1]
[8.6.5.1] [8.7.6.1] [8.7.6.5.2.1]
[8.6.4.3.2.1]
9 511 [9.4] [9.6.4.3] [9.8.5.4] [9.8.4.1]
[9.5.3.2] [9.8.6.5] [9.8.7.2]
[9.6.5.4.2.1] [9.7.6.4.3.1] [9.8.7.6.5.3]
10 1023 [10.3] [10.8.3.2] [10.4.3.1] [10.8.5.1]
[10.8.5.4] [10.9.4.1] [10.8.4.3]
[10.5.3.2] [10.5.2.1] [10.9.4.2]
11 2047 [11.1] [11.8.5.2] [11.7.3.2] [11.5.3.5]
[11.10.3.2] [11.6.5.1] [11.5.3.1]
[11.9.4.1] [11.8.6.2] [11.9.8.3]
12 4095 [12.6.4.1] [12.9.3.2] [12.11.10.5.2.1]
[12.11.6.4.1] [12.11.9.7.6.5]
[12.11.9.5.3.1] [12.11.9.8.7.4]
[12.11.9.7.6.5] [12.9.8.3.2.1]
[12.10.9.8.6.2]
13.degree.
8191 [13.4.3.1] [13.10.9.7.5.4]
[13.11.8.7.4.1] [13.12.8.7.6.5]
[13.9.8.7.5.1] [13.12.6.5.4.3]
[13.12.11.9.5.3] [13.12.11.5.2.1]
[13.12.9.8.4.2] [13.8.7.4.3.2]
14 16.383 [14.12.2.1] [14.13.4.2] [14.13.11.9]
[14.10.6.1] [14.11.6.1] [14.12.11.1]
[14.6.4.2] [14.11.9.6.5.2]
[14.13.6.5.3.1] [14.13.12.8.4.1]
[14.8.7.6.4.2] [14.10.6.5.4.1]
[14.13.12.7.6.3] [14.13.11.10.8.3]
15 32.767 [15.13.10.9] [15.13.10.1] [15.14.9.2]
[15.1] [15.9.4.1] [15.12.3.1] [15.10.5.4]
[ 15.10.5.4.3.2] [15.11.7.6.2.1]
[15.7.6.3.2.1] [15.10.9.8.5.3]
[15.12.5.4.3.2] [15.10.9.7.5.3]
[15.13.12.10] [15.13.10.2] [15.12.9.1]
[15.14.12.2] [15.13.9.6] [15.7.4.1]
[15.4] [15.13.7.4]
16 65.535 [16.12.3.1] [16.12.9.6] [16.9.4.3]
[16.12.7.2] [16.10.7.6] [16.15.7.2]
[16.9.5.2] [16.13.9.6] [16.15.4.2]
[16.15.9.4]
17.degree.
131.071 [17.3] [17.3.2.1] [17.7.4.3] [17.16.3.1]
[17.12.6.3.2.1] [17.8.7.6.4.3]
[17.11.8.6.4.2.] [17.9.8.6.4.1]
[17.16.14.10.3.2] [17.12.11.8.5.2]
18 262.143 [18.7] 18.10.7.5] [18.13.11.9.8.7.6.3]
[18.17.16.15.10.9.8.7]
[18.15.12.11.9.8.7.6]
19.degree.
524.287 [19.5.2.1] [19.13.8.5.4.3]
[19.12.10.9.7.3] [19.17.15.14.13.12.6.1]
[19.17.15.14.13.9.8.4.2.1]
[19.16.13.11.10.9.4.1] [19.9.8.7.6.3]
[19.16.15.13.12.9.5.4.2.1]
[19.18.15.14.11.10.8.5.3.2]
[19.18.17.16.12.7.6.5.3.1]
20 1.048.575 [20.3] [20.9.5.3] [20.19.4.3]
[20.11.8.6.3.2] [20.17.14.10.7.4.3.2]
21 2.097.151 [21.2] [21.14.7.2] [21.13.5.2]
[21.14.7.6.3.2] [21.8.7.4.3.2]
[21.10.6.4.3.2] [21.15.10.9.5.4.3.2]
[21.14.12.7.6.4.3.2]
[21.20.19.18.5.4.3.2]
22 4.194.303 [22.1] [22.9.5.1] [22.20.18.16.6.4.2.1]
[22.19.16.13.10.7.4.1] [22.17.9.7.2.1]
[22.17.13.12.8.7.2.1]
[22.14.13.12.7.3.2.1]
23 8.388.607 [23.5] [23.17.11.5] [23.5.4.1]
[23.12.5.4] [23.21.7.5] [23.16.13.6.5.3]
[23.11.10.7.6.5] [23.15.10.9.7.5.4.3]
[23.17.11.9.8.5.4.1]
[23.18.16.13.11.8.5.2]
24 16.777.215 [24.7.2] [24.4.3.1]
[24.22.20.18.16.14.11.9.8.7.5.4]
[24.21.19.18.17.16.15.14.13.10.9.5.4.1]
25 33.554.431 [25.3] [25.3.2.1] [25.20.5.3] [25.12.4.3]
[25.17.10.3.2.1] [25.23.21.19.9.7.5.3]
[25.18.12.11.6.5.4] [25.20.16.11.5.3.2.1]
[25.12.11.8.7.6.4.3]
26 67.108.863 [26.6.2.1]
[26.22.21.16.12.11.10.8.5.4.3.1]
27 134.217.727 [27.5.2.1] [27.18.11.10.9.5.4.3]
28 268.435.455 [28.3] [28.13.11.9.5.3] [28.22.11.10.4.3]
[28.24.20.16.12.8.4.3.2.1]
29 536.870.911 [29.2] [29.20.11.2] [29.13.7.2]
[29.21.5.2] [29.26.5.2] [29.19.16.6.3.2]
[29.18.14.6.3.2]
30 1.073.741.823
[30.23.2.1] [30.6.4.1]
[30.24.20.16.14.13.11.7.2.1]
31.degree.
2.147.483.647
[31.29.21.17] [31.28.19.15] [31.3]
[31.3.2.1] [31.13.8.3] [31.21.12.3.2.1]
[31.20.18.7.5.3] [31.30.29.25]
[31.28.24.10] [31.20.15.5.4.3]
[31.16.8.4.3.2]
32 4.294.967.295
[32.22.2.1] [32.7.5.3.2.1]
[32.28.19.18.16.14.11.10.9.6.5.1]
33 8.589.934.591
[33.13] [33.22.13.11] [33.26.14.10]
[33.6.4.1] [33.22.16.13.11.8]
61.degree.
2.305.843.009
[61.5.2.1]
213.693.951
89.degree.
618.970.019.
[89.6.5.3]
642.690.137.
449.662.112
______________________________________
In Table 3, the number s of the connection taps are corresponding to those
of the delay elements, for example, in FIG. 9 the delay element 22a is
number 1, the delay element 22b is number 2, the delay element 22c is
number 3 and the delay element 22d is number 4.
The period N of the M-sequence created by the feedback shift register the
tap combination of which is shown in Table 3 is determined by the number
of stage k of the linear feedback shift register and given by the
following expression:
N=2.sup.k -1 (29)
In the case of four stage as shown in FIG. 9, the period of the M-code
sequence is 15. Thus, that figure shows the case of k=4 and N=15.
The component-substituting unit 25 is then inputted with the M-sequence
{b.sub.n } created by the linear feedback shift register 24, thereby
substituting the component a.sub.n for the component b.sub.n.
The operation of the component-substituting unit 25 will then be described
by referring to the flowchart shown in FIG. 10. The components b.sub.n
successively transferred from the linear feedback shift register 24 are
input to the component-substituting unit 25. The values of the components
are checked at step 27. If b.sub.n =0, then step 29 is executed, and if
b.sub.n =1, then step 28 is executed. By the execution of step 28, the
value of the component a.sub.n is set at the following complex number and
output to the exterior at step 30:
a.sub.n =A.sub.1 e.sup.j.phi..sbsp.1 (30)
On the other hand, by the execution of step 29, the value of the component
a.sub.n is set at the following complex number and output to the exterior
at step 30.
a.sub.n =A.sub.0 e.sup.j.phi..sbsp.0 (31)
At this time, the parameters A.sub.1, .phi..sub.1 in expression (31) and
the parameters A.sub.0, .phi..sub.0 in expression (30) are determined to
satisfy the following expression:
##EQU22##
As shown in expression (32), the phase difference .phi..sub.1 -.phi..sub.0
and the amplitude ratio A.sub.1 /A.sub.0 are important, and in normal
operation the following expression can be satisfied.
.phi..sub.0 =0 (33a)
A.sub.0 =1, or A.sub.0 =A.sub.1 =1 (33b)
It will be described below that the sequence {a.sub.n } created in this
manner are of an orthogonal sequence.
In the embodiment of the present invention, the autocorrelation function of
the sequence {a.sub.n } out of the range of m=0 is expressed by expression
(34) in accordance with expression (4):
##EQU23##
where, .alpha.: number of combinations of components for which a.sub.n and
a.sub.n+m are both A.sub.0 e.sup.j.phi..sbsp.0,
.beta.: number of combination of components for which a.sub.n is A.sub.0
e.sup.j.phi..sbsp.0 and a.sub.n+m is A.sub.1 e.sup.j.phi..sbsp.1,
.gamma.: number of combination of components for which a.sub.n is A.sub.1
e.sup.j.phi..sbsp.1 and a.sub.n+m is A.sub.0 e.sup.j.phi..sbsp.0,
.delta.: number of combination of components for which a.sub.n and
a.sub.n+m are both A.sub.1 e.sup.j.phi..sbsp.1.
Since the sequence {a.sub.n } is obtained by substituting the components of
the M-sequence {b.sub.n }, .alpha., .beta., .gamma. and .delta. can be
considered as follows:
.alpha.: number of combination of components for which b.sub.n and
b.sub.n+m are both 0,
.beta.: number of combination of components for which b.sub.n is 0 and
b.sub.n+m is 1,
.gamma.: number of combination of components for which b.sub.n is 1 and
b.sub.n+m is 0,
.delta.: number of combination of components for which b.sub.n and
b.sub.n+m are both 1.
.alpha., .beta., .gamma. and .delta. can take a constant value irrespective
of the value of m in the view of the properties of the M-sequence. The
properties of the M-sequence necessary for obtaining the values of
.alpha., .beta., .gamma. and .delta. will be described below and those are
also disclosed in "CODE THEORY" (Third Edition) by Miyakawa, Iwadare, et
al., published from Shoko-do on July 20, 1976.
(1) The period N of M-sequence is 2.sup.k -1 (N=2.sup.k -1).
(2) In one period of M-sequence, (2.sup.k-1 -1) number of 0 and (2.sup.k-1)
number of 1 are contained.
(3) Vector B.sub.i (i=0, . . . , N-1) is defined by using components
{b.sub.0, b.sub.1, . . . , b.sub.N-1 } in one period of M-sequence
{b.sub.n } as shown below:
##EQU24##
The Hamming distance d.sub.H (B.sub.i, B.sub.j) of the above vectors
B.sub.i and B.sub.j can be obtained and shown by expression (36),
irrespective of the combination of B.sub.i and B.sub.j :
d.sub.H (B.sub.i, B.sub.j)=2.sup.k-1 (i.noteq.j) (36)
Here, "Hamming distance d.sub.H " is defined as expression (37a) when two
vectors are expressed by A=(a.sub.0, a.sub.1, . . . , a.sub.N-1) and
B=(b.sub.0, b.sub.1, . . . , b.sub.N-1).
##EQU25##
The values of .alpha., .beta., .gamma. and .delta. can be obtained by
comparing the components of vectors B.sub.0 and B.sub.m (m=1, 2, . . . ,
N-1).
First, the sum of .alpha. and .delta. is the number of components for which
.epsilon..sub.n of expression (37b) is 0, however replacing vectors A and
B with vectors B.sub.0 and B.sub.m. Since the Hamming distance of B.sub.0
and B.sub.m is 2.sup.k-1, irrespective of the value of m, as shown by
expression (36), the sum is represented as follows:
.alpha.+.delta.=N-2.sup.k-1 =2.sup.k -1-2.sup.k-1 =2.sup.k-1 -1 (38)
.beta. and .gamma. will be considered below.
Again, the number of combination .beta. of components in the case of
b.sub.n =0 for vector B.sub.0 and b.sub.n+m =1 for vector B.sub.m can be
expressed by follows:
.beta.=2.sup.k-2 +d (39)
Then, the number .beta. of combination .gamma. of components in the case of
b.sub.n =1 for vector B.sub.0 and b.sub.n+m =0 for vector B.sub.m can also
be expressed by:
.gamma.=2.sup.k-2 -d (40)
This is because the Hamming distance of vectors B.sub.0 and B.sub.m can be
represented as follows:
d.sub.H (B.sub.0, B.sub.m)=.beta.+.gamma.=2.sup.k-1.
On the other hand, due to the properties of M-sequence, the number of
0-value components of vector B.sub.0 is 2.sup.k-1 -1, and the number of
1-value components is 2.sup.k-1, and thus the number p.sub.0 of the
remaining 0-value components of vector B.sub.0 and the number p.sub.1 of
the remaining 1-value components thereof are given by the following
expressions:
p.sub.0 =2.sup.k-1 -1-.beta.=2.sup.k-2 -1-d (41a)
p.sub.1 =2.sup.k-1 -.gamma.=2.sup.k-2 +d (41b)
Similarly, the number q.sub.0 of the remaining 0-value components of vector
B.sub.m and the number q.sub.1 of the remaining 1-value components thereof
are given by the following expressions:
q.sub.0 =2.sup.k-1 -1-.gamma.=2.sup.k-2 -1+d (42a)
q.sub.1 =2.sup.k-1 -.beta.=2.sup.k-2 -d (42b)
However, since both p.sub.0 and q.sub.0 represent the numbers of
combination of the components in the case of b.sub.n =0 for vector B.sub.0
and b.sub.n+m =0 for vectors B.sub.m, the following condition must be
satisfied:
p.sub.0 =q.sub.0 (43a)
Similarly, both p.sub.1 and q.sub.1 both represent the numbers of
combination of the components in the case of b.sub.n =1 for vector B.sub.0
and b.sub.n+m =1 for vector B.sub.m, and therefore the following condition
must be satisfied:
p.sub.1 =q.sub.1 (43b)
From expressions (43a) and (43b), d=0, and thus .beta. and .gamma. are
equal to each other and are given by the following expression,
irrespective of the value of m:
.beta.=.gamma.=2.sup.k-2 (45)
Since the number of 0-value components and the number of 1-value components
of vector B.sub.0 (and vector B.sub.m) are (2.sup.k-1 -1) and 2.sup.k-1
respectively, as mentioned above, .alpha., .beta., .gamma. and .delta.
must be satisfy the following relations:
.alpha.+.gamma.=.alpha.+.beta.=2.sup.k-1 -1 (46a)
.beta.+.delta.=.gamma.+.delta.=2.sup.k-1 (46b)
By substituting expression (45) for expressions (46a) and (46b), the
following relations are obtained:
.alpha.=2.sup.k-2 -1 (47)
.delta.=2.sup.k-2 (48)
.alpha. and .delta. obtained by expressions (47) and (48) satisfy
expression (38) and accordingly, there is no contradiction.
.beta. and .gamma. obtained by expression (45), .alpha. by expression (47)
and .delta. by expression (48) substitute for expression (34) and then the
autocorrelation function of the sequence {a.sub.n } of the present
embodiment can be written as follows:
##EQU26##
Since the condition under which the sequence {a.sub.n } is an orthogonal
sequence is .rho..sub.(m) =0, (m=1, 2, . . . , N-1), as mentioned above
concerning the expression (7), expression (32) can be obtained from
expressions (49) and (29).
From the foregoing, it is understood that the sequence generated in the
above-described embodiment is an orthogonal sequence.
FIG. 11 is a vector diagram of the component of the orthogonal sequence of
the embodiment of this invention (however, A.sub.0 =1, and thus the ratio
A=A.sub.1 /A.sub.0 =A.sub.1). In the orthogonal sequence of the embodiment
of this invention, as shown in FIG. 11, the components can take two
values, 1 and Ae.sup.j.phi., and thus the amplitudes thereof are 1 and A.
Moreover, the case of A.sub.0 =A.sub.1 =1, the components take 1 and
e.sup.j.phi., and the amplitudes are all 1, as shown in FIG. 12.
Shown in FIG. 13 is an exemplary arrangemen | | |
