CN113408098A - Optimal biphase code sequence waveform traversal search method based on graph search - Google Patents

Optimal biphase code sequence waveform traversal search method based on graph search Download PDF

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CN113408098A
CN113408098A CN202110476676.8A CN202110476676A CN113408098A CN 113408098 A CN113408098 A CN 113408098A CN 202110476676 A CN202110476676 A CN 202110476676A CN 113408098 A CN113408098 A CN 113408098A
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sequence
code
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宋媛媛
杨艺
任伟
陈新亮
龙腾
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Beijing Institute of Technology BIT
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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    • G06F17/18—Complex mathematical operations for evaluating statistical data, e.g. average values, frequency distributions, probability functions, regression analysis
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Abstract

The invention provides an optimal biphase code sequence waveform traversal searching method based on graph searching, and the time complexity can be obviously reduced by adopting an equivalent code pruning method from the maximum time delay. The invention applies the graph Search method to the traversal Search of the two-phase code, and the invention uses two methods which are most common in the graph Search, namely Depth-First Search (DFS) and Breadth-First Search (BFS), to carry out algorithm design on the traversal Search of the optimal two-phase code sequence. For the expansion mode of the graph, the initial search space is small from the maximum delay, but not from the minimum delay, otherwise, the initial search space is large; the adaptive strategy of the method is explained by considering the traversal search problem of the optimal biphase code sequence from the view point of graph search.

Description

Optimal biphase code sequence waveform traversal search method based on graph search
Technical Field
The invention relates to the technical field of radar waveform design, in particular to an optimal biphase code sequence waveform traversal searching method based on graph search.
Background
The method detects weak and small targets, improves signal isolation and is an important task in the field of radar detection. Considering this problem from the aspect of radar waveform design, it can be achieved by using an optimal biphase code sequence waveform of the lowest peak sidelobe level.
The search method for the optimal biphase code sequence waveform can be divided into a local search and a traversal search. The local search algorithm is low in time complexity and requires less time, but is easy to fall into a local optimal solution. To obtain all the optimal sequences, a traversal search must be used. The traditional exhaustive search algorithm can realize the traversal search of the optimal biphase code sequence, but the time complexity is too high, and along with the increase of the code length, a large amount of memory space is also needed in the search process, so that a pruning strategy needs to be searched to reduce the search space, the time complexity and the memory space.
The graph is used as a common data structure in the field of computers, algorithms (common algorithms such as a depth-first search algorithm and a breadth-first search algorithm) for searching the graph are applied to multiple fields such as image segmentation, external data storage and distributed computing, and graph searching can reduce a search space, reduce time complexity and reduce the requirement on a memory space.
However, the way in which the graph is expanded may affect the search space at the beginning; in addition, how to reduce complexity, namely, the pruning strategy problem also needs to be considered, and the existing graph search mode cannot be well applied to the traversal search of the optimal biphase code sequence waveform.
Disclosure of Invention
In view of this, the invention provides an optimal biphase code sequence waveform traversal search method based on graph search, and from the maximum time delay, the time complexity can be significantly reduced by adopting an equivalent code pruning method.
In order to achieve the above object, the optimal biphase code sequence waveform traversal search method based on graph search of the present invention, based on depth-first search, includes the following steps:
step one, setting a two-phase code sequence with a code element length of N as follows:
AN＝[a0 a1 ... aN-1] (1)
wherein a isn∈{-1，1}，n＝0，1，...，N-1；
The aperiodic autocorrelation function is defined as:
Figure BDA0003047304170000021
wherein tau is more than or equal to 0 and less than or equal to N-1;
take tau as N-1 and fill in a0And aN-1One case where | AACF (N-1) | ≦ optPSL is found as the first node P of the first level of the graph1Wherein, optPSL is the PSL value of the biphase code sequence with the optimal length;
step two, taking tau as N-2, and in step one, a0And aN-1Under constraint, fill in aN-τ-1And aτFinding a condition satisfying | AACF (N-2) | ≦ optPSL as the first child node P of the above node11；
Step three, decreasing tau to enable
Figure BDA0003047304170000022
Repeating the second step, judging whether the absolute value of AACF (tau) is less than or equal to optPSL every time, wherein,
Figure BDA0003047304170000031
time, judge
Figure BDA0003047304170000032
Up to the second of the figure
Figure BDA0003047304170000033
Filling a node in a layer;
step four, change
Figure BDA0003047304170000034
Filling the numerical value of the code element in the layer, and finding other possible nodes; then change the first
Figure BDA0003047304170000035
The layers fill in the values of the symbols until all possibilities have been tried, find a map of all sequences that satisfy PSL ≦ optPSL, and complete the search.
Wherein in the step 5, a is fixed0a1When the value is 00, by judging aN-2aN-1And bN-2bN-1To determine which equivalent code the sequence may be obtained through, the correspondence is shown in the following table:
Figure BDA0003047304170000036
wherein R represents a para-sequence ANFlip each bit symbol: r (a)n)＝aN-1-n；
N represents a pair sequence ANEach bit symbol of (a) is inverted: n (a)n)＝-an；
S represents a pair sequence ANPerforms symbol alternation for each bit symbol: s (a)n)＝(-1)nan；
The two-phase code sequence obtained by the above operation for one two-phase code sequence is called an equivalent code.
Wherein, the judgment of equivalent codes can only be carried out at the last layer of the graph, namely
Figure BDA0003047304170000037
Is carried out.
The invention also provides an optimal biphase code sequence waveform traversal searching method based on graph search, which is based on breadth-first search and comprises the following steps:
step one, setting a two-phase code sequence with a code element length of N as follows:
AN＝[a0 a1 ... aN-1] (1)
wherein a isn∈{-1，1}，n＝0，1，...，N-1；
The aperiodic autocorrelation function is defined as:
Figure BDA0003047304170000041
wherein tau is more than or equal to 0 and less than or equal to N-1;
take tau as N-1 and fill in a0And aN-1Possible value combination is carried out, all conditions that | AACF (N-1) | is less than or equal to optPSL are found out, equivalent code nodes are removed, and the rest nodes are used as nodes of the first layer of the graph;
step two, taking tau as N-2, and searching the second layer node: filling all nodes reserved in the first layer with all a satisfying | AACF (N-2) | < optPSLN-τ-1And aτEliminating equivalent code nodes, and taking the rest nodes as nodes of the second layer of the graph;
step three, repeating step two 0 until
Figure BDA0003047304170000042
And all nodes are listed, or all branches are cut off to complete the search.
Wherein, in the third step, a is fixed0a1When the value is 00, by judging aN-2aN-1And bN-2bN-1To determine which equivalent code the sequence may be obtained through, the correspondence is shown in the following table:
Figure BDA0003047304170000043
Figure BDA0003047304170000051
wherein R represents a para-sequence ANFlip each bit symbol: r (a)n)＝aN-1-n；
N represents a pair sequence ANEach bit symbol of (a) is inverted: n (a)n)＝-an；
S represents a pair sequence ANPerforms symbol alternation for each bit symbol: s (a)n)＝(-1)nan；
The two-phase code sequence obtained by the above operation for one two-phase code sequence is called an equivalent code.
Has the advantages that:
the invention applies the graph Search method to the traversal Search of the two-phase code, and the invention uses two methods which are most common in the graph Search, namely Depth-First Search (DFS) and Breadth-First Search (BFS), to carry out algorithm design on the traversal Search of the optimal two-phase code sequence. For the expansion mode of the graph, the initial search space is small from the maximum delay, but not from the minimum delay, otherwise, the initial search space is large; the adaptive strategy of the method is explained by considering the traversal search problem of the optimal biphase code sequence from the view point of graph search.
The invention provides an improved equivalent code pruning strategy by taking a minimum Peak Sidelobe Level (PSL) and an equivalent code as a search pruning strategy and based on the definition of the equivalent code. In the pruning strategy, a proposition of the equivalent code is obtained by utilizing the concepts of the PSL and the equivalent code, and the time complexity of an equivalent code pruning method is lower than that of the existing method.
Drawings
Fig. 1 is a schematic diagram of depth-first search when the symbol length N is 6 and the optPSL is 2 according to the present invention;
fig. 2 is a schematic diagram of breadth-first search when the symbol length N is 6 and the optPSL is 2 according to the present invention;
fig. 3(a) is a search time comparison graph of the depth-first search method and the breadth-first search method when the PSL of the present invention is 2.
Fig. 3(b) is a search time comparison graph of the depth-first search method and the breadth-first search method when the PSL of the present invention is 3.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The invention provides an optimal biphase code sequence waveform traversal searching method based on graph search, which comprises the following principle analysis:
let the two-phase code sequence with symbol length N be:
AN＝[a0 a1 ... aN-1] (1)
wherein a isnE { -1, 1}, N { -0, 1., N-1, followed by 0 for-1. The aperiodic autocorrelation function is defined as:
Figure BDA0003047304170000061
wherein tau is more than or equal to 0 and less than or equal to N-1, the PSL of the sequence is calculated in the following mode:
Figure BDA0003047304170000071
namely, it is
Figure BDA0003047304170000072
As can be seen from the definition of AACF, only a needs to be determined0And aN-1AACF (N-1) is known, and AACF (1) is known only by determining each digit in the sequence. Therefore, from the viewpoint of reducing the search space, it is necessary to construct a map from τ — N-1 in order to make the search space sufficiently small in the initial stage of the search.
According to the related concept of the equivalent code, a proposition about the equivalent code is provided, and an improved equivalent code pruning strategy is provided according to the proposition, which comprises the following steps:
according to the calculation formula of PSL, the sequence A is obtainedNEach bit symbol of (a) is operated as follows, the PSL value of which remains unchanged: (1) flip (reverse): r (a)n)＝aN-1-n(ii) a (2) Negation (Negative): n (a)n)＝-an(ii) a (3) Sign alternation (Alternating-sign): s (a)n)＝(-1)nanThe two-phase code sequence obtained by the above operation for one two-phase code sequence is called an equivalent code, and these sequences and the original sequence form an equivalent code set.
Proposition: an equivalent code set is closed and contains at most 1 two-phase code sequence and the other 7 equivalent code sequences. By F1→F2→F3Indicating sequential F of a two-phase code sequence1、F2、F3And when the code length N is an even number, the 7 equivalent codes are respectively as follows: (1) r; (2) n; (3) s; (4) n → S, S → N; (5) r → N and N → R; (6) r → S, S → N → R, S → R → N, N → S → R; (7) s → R, R → N → S, R → S → N, N → R → S; when the code length N is odd, the 7 equivalent codes are: (1) r; (2) n; (3) s; (4) n → S, S → N; (5) r → N and N- → R; (6) r → S and S → R; (7) s → N → R, S → R- → N, N → S → R, R- → N → S, R- → S- → N, N → R → S, wherein a plurality of expressions in each sequence number represent the same two-phase code sequence.
If a0a1And when the two-phase code sequence is obtained by the traversal search, all the two-phase code sequences do not contain equivalent codes obtained by the operations (2), (3) and (4). Under this constraint, if a biphase code sequence A is knownN＝[a0 a1 ... aN-1]And an equivalent code sequence B to be detectedN＝[b0 b1 ... bN-1]Can be determined by judging aN-2aN-1And bN-2bN-1To determine which equivalent code the sequence may be obtained through, and the correspondence is shown in the following table.
Figure BDA0003047304170000081
Therefore, when the last two bits of the symbol of the sequence to be detected are determined, the possible equivalent code operation can be purposely determined according to the above table, thereby reducing the time required for determining the equivalent code.
The invention provides algorithm steps of depth-first search and breadth-first search in graph search applied to optimal biphase code traversal search based on PSL and an improved equivalent code pruning strategy.
Specifically, the optimal biphase code sequence traversal search algorithm based on depth-first search comprises the following steps:
step one, taking tau as N-1 and filling a0And aN-1Finding a condition satisfying | AACF (N-1) | ≦ optPSL (optPSL is the PSL value of the biphase code sequence with the optimal length, and if unknown in advance, setting a numerical value) as the first node P of the first layer of the graph1；
Step two, taking tau as N-2, and in step one, a0And aN-1Under constraint, fill in aN-τ-1And aτFinding a condition satisfying | AACF (N-2) | ≦ optPSL as the first child node P of the above node11；
Step three, decreasing tau to enable
Figure BDA0003047304170000091
Repeating the second step, judging whether the absolute value of AACF (tau) is less than or equal to optPSL every time, wherein,
Figure BDA0003047304170000092
need to judge
Figure BDA0003047304170000093
Up to the second of the figure
Figure BDA0003047304170000094
Filling a node in a layer;
step four, change
Figure BDA0003047304170000095
Filling the numerical value of the code element in the layer, and finding other possible nodes; then change the first
Figure BDA0003047304170000096
The layers fill in the values of the symbols until all possibilities are tried, finding a map of all sequences that satisfy PSL ≦ optPSL.
It should be noted that a is fixed when the depth-first search is used to perform the traversal search on the optimal biphase code sequence0a100, and the judgment of equivalent code can only be at the last layer of the graph, namely
Figure BDA0003047304170000097
And (4) performing the search, namely, reducing the search space by using the constraint condition of the equivalent code.
The optimal biphase code sequence traversal search algorithm based on breadth-first search comprises the following steps:
step one, taking tau as N-1 and filling a0And aN-1And (4) possible value combination is carried out, all conditions that | AACF (N-1) | is less than or equal to optPSL are found out, equivalent code nodes are removed, and the rest nodes are used as nodes of the first layer of the graph.
And step two, taking tau as N-2, and searching the second layer node. Filling all nodes reserved in the first layer with all a satisfying | AACF (N-2) | < optPSLN-τ-1And aτAnd (4) possible value combination, eliminating equivalent code nodes, and taking the remaining nodes as nodes of the second layer of the graph.
Step three, repeating step two 0 until
Figure BDA0003047304170000101
And lists all nodes or prunes all branches, at which point the tree has been filled.
Similarly, a is also fixed when performing a traversal search for the optimal biphase code sequence using breadth-first search0a1＝00。
Let us assume that the symbol length N is 6 and optPSL is 2, fix a0a1＝00，A simulation example applying the present invention is given for depth-first search, and the implementation process thereof is analyzed and explained, specifically as follows:
step one, taking tau-N-1-5 and filling a50, when | AACF (5) | 1 ≦ optPSL, node a0a500 as the first node P of the first layer of the graph1；
Step two, taking tau as N-2 as 4, at a0a5Under the constraint of 00, fill in a40, when | AACF (4) | 2 ≦ optPSL, node a will be selected1a400 as the first child node P of P111；
Step three, taking tau as N-3 as 3, at this time
Figure BDA0003047304170000102
Need to find the satisfaction
Figure BDA0003047304170000103
The value combination of (1). At a0a5＝00、a1a4Under the constraint of 00, fill in a2a3When 00, | AACF (1) | 3 > optPSL; modification of a2a3When the requirement is satisfied, the node is taken as P11First child node P of111。
Step four, keeping a0a5＝00、a1a4Continuing to modify a for constraint 002a3The value combination of (1). When a is2a3When 10, PSL satisfies the requirement, but the node is P111R operation equivalent code nodes; when a is2a3When the node is 11, the node is taken as P11Second child node P of112。
Step five, keeping a0a5Constraint of 00, modify a1a4As P0112Continue to look for a2a3The value combination of (a) can be found out2a3When the two nodes satisfy the requirements of 00 and 01, the two nodes are respectively taken as P121And P122。
Step six, modifying a0a5Repeating steps two to five with the constraint of 01, the search result shown in fig. 1 can be obtained, where the value in o represents the value of the symbol of the current node, and the value in □ represents the access sequence of the current node.
Let us assume that the symbol length N is 6 and optPSL is 2, fix a0a100, a simulation example applying the method is given by breadth-first search, and the implementation process is analyzed and explained, and the specific steps are as follows:
step one, taking tau as N-1 as 5, when a5When | AACF (5) | ≦ optPSL is true when |, or 1, the node a is set0a5Node P as the first layer of the graph, 00 and 01, respectively1And P2；
Step two, taking tau as N-2 as 4, at a0a5Under the constraint of 00, fill in a4When | AACF (4) | ≦ optPSL is true when 0 or 1, the node a is set1a4As nodes P, 00 and 011Is a child node P11And P12(ii) a Similarly, in a0a5Under the constraint of 01, fill in a4When 0 or 1, | AACF (4) | ≦ optPSL holds, but a4When 0, the node is P12R of (2) operate equivalent code nodes, so that only node a is used1a4As node P ═ 012Is a child node P21；
Step three, taking tau as N-3 as 3, at this time
Figure BDA0003047304170000111
Need to find the satisfaction
Figure BDA0003047304170000112
The value combination of (1). Are respectively represented by P11、P12And P21For the father node, all child nodes are found by using the PSL and the equivalent code as pruning strategies, and the search result shown in fig. 2 can be obtained, wherein the intra-o value represents the value of the code element of the current node, and the intra-o value represents the access sequence of the current node.
The search times of the two methods recorded at PSL ═ 2 and PSL ═ 3, respectively, can obtain the results shown in fig. 3, and the time complexity of the two statistically obtained methods is shown in the following table, which shows that the complexity is lower than that of the conventional search method.
Algorithm PSL＝2 PSL＝3
DFS O(c1·1.44N) O(c3·1.64N)
BFS O(c2·1.33N) O(c4·1.49N)
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (5)

1. A graph search-based optimal biphase code sequence waveform traversal search method is characterized by comprising the following steps of based on depth-first search:
step one, setting a two-phase code sequence with a code element length of N as follows:
AN＝[a0 a1 ... aN-1] (1)
wherein a isn∈{-1，1}，n＝0，1，...，N-1；
The aperiodic autocorrelation function is defined as:
Figure FDA0003047304160000011
wherein tau is more than or equal to 0 and less than or equal to N-1;
take tau as N-1 and fill in a0And aN-1One case where | AACF (N-1) | ≦ optPSL is found as the first node P of the first level of the graph1Wherein, optPSL is the PSL value of the biphase code sequence with the optimal length;
step two, taking tau as N-2, and in step one, a0And aN-1Under constraint, fill in aN-τ-1And aτFinding a condition satisfying | AACF (N-2) | ≦ optPSL as the first child node P of the above node11；
Step three, decreasing tau to enable
Figure FDA0003047304160000012
Repeating the second step, judging whether the absolute value of AACF (tau) is less than or equal to optPSL every time, wherein,
Figure FDA0003047304160000013
time, judge
Figure FDA0003047304160000016
Up to the second of the figure
Figure FDA0003047304160000014
Filling a node in a layer;
step four, change
Figure FDA0003047304160000015
Filling the numerical value of the code element in the layer, and finding other possible nodes; then change the first
Figure FDA0003047304160000021
The layers fill in the values of the symbols until all possibilities have been tried, find a map of all sequences that satisfy PSL ≦ optPSL, and complete the search.
2. The graph search-based optimal biphase code sequence waveform traversal search method according to claim 1, wherein in the step 5, a is fixed0a1When the value is 00, by judging aN-2aN-1And bN-2bN-1To determine which equivalent code the sequence may be obtained through, the correspondence is shown in the following table:
Figure FDA0003047304160000022
wherein R represents a para-sequence ANFlip each bit symbol: r (a)n)＝aN-1-n；
N represents a pair sequence ANEach bit symbol of (a) is inverted: n (a)n)＝-an；
S represents a pair sequence ANPerforms symbol alternation for each bit symbol: s (a)n)＝(-1)nan；
The two-phase code sequence obtained by the above operation for one two-phase code sequence is called an equivalent code.
3. The method of claim 2, wherein the determination of equivalent codes is only performed at the last layer of the graph, i.e. the optimal biphase code sequence waveform traversal search method based on graph search
Figure FDA0003047304160000023
Is carried out.
4. A graph search-based optimal diphase code sequence waveform traversal search method is characterized by comprising the following steps of based on breadth-first search:
step one, setting a two-phase code sequence with a code element length of N as follows:
AN＝[a0 a1 … aN-1] (1)
wherein a isn∈{-1，1}，n＝0，1，...，N-1；
The aperiodic autocorrelation function is defined as:
Figure FDA0003047304160000031
wherein tau is more than or equal to 0 and less than or equal to N-1;
take tau as N-1 and fill in a0And aN-1Possible value combination is carried out, all conditions that | AACF (N-1) | is less than or equal to optPSL are found out, equivalent code nodes are removed, and the rest nodes are used as nodes of the first layer of the graph;
step two, taking tau as N-2, and searching the second layer node: filling all nodes reserved in the first layer with all a satisfying | AACF (N-2) | < optPSLN-τ-1And aτEliminating equivalent code nodes, and taking the rest nodes as nodes of the second layer of the graph;
step three, repeating step two 0 until
Figure FDA0003047304160000032
And all nodes are listed, or all branches are cut off to complete the search.
5. The graph search-based optimal biphase code sequence waveform traversal search method according to claim 4, wherein a is fixed in the third step0a1When the value is 00, by judging aN-2aN-1And bN-2bN-1To determine which equivalent code the sequence may be obtained through, the correspondence is shown in the following table:
Figure FDA0003047304160000041
wherein R represents a para-sequence ANFlip each bit symbol: r (a)n)＝aN-1-n；
N represents a pair sequence ANEach bit symbol of (a) is inverted: n (a)n)＝-an；
S represents a pair sequence ANPerforms symbol alternation for each bit symbol: s (a)n)＝(-1)nan；
The two-phase code sequence obtained by the above operation for one two-phase code sequence is called an equivalent code.
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