Compression Ratio Expansion for Complementary Code Set Compressing a Signal to a Width of Several Sub-pulses

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1 Compression Ratio Expansion for Compementary Code Set Compressing a Signa to a Width of Severa Sub-puses Reiji Sato 2nd Research Center, Technica R and D Institute Japan Defense Agency Ikejiri, Setagaya-ku, Tokyo, , Japan Emai: reiji@jda-trdigojp Masanori Shinriki and Hiroshi Takase Nippon Institute of Technoogy 4-1 gakuendai, Miyashiro-cho,Saitama-ken , Japan Emai: shinriki@nitacjp and takase@nitacjp Abstract We present the compression ratio expansion methods for the compementary code set compressing a signa to a width of severa sub-puses We aso proposes the compementary code sets that consist of more that three individua codes and compress a signa to a width of severa sub-puses I Introduction The puse compression is one of the most important techniques for the radar since it coud improve range resoution and S/N at the same time However, there is a probem that unnecessary range sideobes, which have negative infuences on radar s detection performance, are present before and after the compressed puse It is therefore the most important issue to reduce range sideobes Adopting the compementary code set is one of the most effective methods to eiminate sideobes when the Dopper effect is sma; it, therefore, is used for weather radars and so on The compementary code set consists of a pair of individua codes, which have the property that the sum of their 2 auto-correation functions has no sideobe Nowadays, There are many attempts to appy it to the communication technoogy [1][2] In the fied of the puse compression technique, the coded puse compression method has exceent merits that using different codes each other can reduce mutua interference between neighbor radars The more usabe puse compression codes, we have, the more effective, such advantage of code moduation can be [3][4] On the other hand, the compression ratio expansion methods have been introduced for the compementary code set Furthermore, the compementary code sets composed of more than 3 individua codes have been proposed As a resut, it has become possibe to use the compementary code set at reativey many compression ratios and the freedom degree of the radar signa design becomes arge[5][6] On the other hand, in the fied of the ordinary coded puse compression using one code, a new puse compression code to compress a puse to a width of severa sub-puses was proposed and it was shown that it has some merits compared to the conventiona puse compression code [7] And then, a new compementary code set to compress a puse to a width of severa sub-puses, named Widey Puse-compressing Compementary Code Set (WPCCS), has aso been introduced and it has been reported that extremey more WPCCSs exist than the conventiona compementary code set in the region of reativey sma compression ratio Furthermore, other severa interesting properties about WPCCS has been reported First of a, there exist much more compete WPCCSs compared to the compete conventiona compementary code set A pair of compementary code set is defined to be compete if the sum of cross-correation functions between the mate codes of each compementary code set is aways zero for any time shift It is aso noted that, in the WPCCS, there are ong compex continuous networks connected by the reationship between 2 compementary code set that they construct a compete compementary code set jointy, whie there are ony isoated compete compementary codes in the conventiona This fact means that, using such compete WPCCSs on severa adjoining radars and synchronizing the puse transmission, it is possibe to suppress the mutua interference even on the same frequency Namey, the effective use of the frequency resource is possibe It has been aso reported that the autocorreation sideobe of an individua code for the WPCCS is smaer than that for the conventiona Using such WPCCSs, it is possibe to reduce the sideobe produced by the Dopper frequency Consequenty, it has been carified that the WPCCS has severa interesting properties compared to the conventiona compementary code set However, there is ony one method that extends the compression ratio of the WPCCS So, the merits of the WPCCS is not sufficienty dispayed in the region of arge compression ratio [8] In this paper, we show that the compression ratio expansion method for the conventiona compementary code set, which was introduced in the artice [5], is aso appicabe for the WPCCS Further, we introduce that the WPCCS composed of more than 3 individua codes can be constructed by the same method for the conventiona compementary code set shown in the artice [6] It aso gives the compression ratio expansion method of the WPCCS simutaneousy As a resut, in the arge compression ratio region, the number of the WPCCS becomes

2 Report Documentation Page Form Approved OMB No Pubic reporting burden for the coection of information is estimated to average 1 hour per response, incuding the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and competing and reviewing the coection of information Send comments regarding this burden estimate or any other aspect of this coection of information, incuding suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arington VA Respondents shoud be aware that notwithstanding any other provision of aw, no person sha be subject to a penaty for faiing to compy with a coection of information if it does not dispay a currenty vaid OMB contro number 1 REPORT DATE 14 APR REPORT TYPE N/A 3 DATES COVERED - 4 TITLE AND SUBTITLE Compression Ratio Expansion for Compementary Code Set Compressing a Signa to a Width of Severa Sub-puses 5a CONTRACT NUMBER 5b GRANT NUMBER 5c PROGRAM ELEMENT NUMBER 6 AUTHOR(S) 5d PROJECT NUMBER 5e TASK NUMBER 5f WORK UNIT NUMBER 7 PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 2nd Research Center, Technica R and D Institute Japan Defense Agency Ikejiri, Setagaya-ku, Tokyo, , Japan 8 PERFORMING ORGANIZATION REPORT NUMBER 9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10 SPONSOR/MONITOR S ACRONYM(S) 12 DISTRIBUTION/AVAILABILITY STATEMENT Approved for pubic reease, distribution unimited 11 SPONSOR/MONITOR S REPORT NUMBER(S) 13 SUPPLEMENTARY NOTES See aso ADM001798, Proceedings of the Internationa Conference on Radar (RADAR 2003) Hed in Adeaide, Austraia on 3-5 September ABSTRACT 15 SUBJECT TERMS 16 SECURITY CLASSIFICATION OF: 17 LIMITATION OF ABSTRACT UU a REPORT b ABSTRACT c THIS PAGE 18 NUMBER OF PAGES 4 19a NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev 8-98) Prescribed by ANSI Std Z39-18

3 extremey arger and the freedom degree of the radar signa design becomes much arger, which make it possibe to use the important merits of the WPCCS moduation 5WDRWNUG YKFVJ II Definition The foowing briefy introduces the definitions and notations used in this paper The cross correation between individua code A and B is denoted by ψ AB, where τ is a time shift The conventiona compementary code set is defined by Eq(1) On the other hand, the WPCCS is defined by Eq(2) [8] Fig1 shows the compressed puse shape of the WPCCS The parameter in Eq (2), which is caed "compression parameter" from now on, represents the wideness of the compressed puse bottom The compression ratio approximatey corresponds to n/ in case of the WPCCS whie it is exacty n in case of the conventiona compementary code set, where n is code ength The compementary code set proposed at first by MJEGoay, which is composed of two individua codes, corresponds to the case when p is equa to 2 in Eq (1) After that, the compementary code set composed of more than 3 individua codes was proposed On the other hand, in the WPCCS of Eq (2), ony one case when p is equa to 2 in Eq (2) was investigated before The case when p is more than 3 in Eq (2) is proposed in the section 4 Fig 1 Compressed Puse Shape ψ AiA i = ψ AiA i = { 0, k = 0 pn, k =0 { 0, k nonzero 0 < k < pn, k =0 III Property of WPCCS In this section, the severa interesting properties of the WPCCS reported before is introduced First of a, it shoud be noted that extremey more WPCCSs exist than the conventiona compementary code sets in the region of reativey sma compression ratio Furthermore, there exist much more compete WPCCSs compared to the compete compementary code set of the conventiona A pair of compementary code set is defined to be compete if the sum of cross-correation functions between the mate codes of each compementary code set is aways zero for any time shift, ie a pair of compementary code set (or a pair of WPCCS), {A i ;1 i p} and {B i ;1 i p} is defined to be compete when it satisfies Eq (3) It is aso an interesting characteristics that, in the WPCCS, there are ong compex continuous networks connected by the reationship between 2 compementary code set that they construct a compete compementary code set jointy as shown in Fig2, whie there are ony isoated compete compementary codes in the conventiona This fact means that, using such WPCCSs on severa adjoining radars and synchronizing the puse transmission, it is possibe to suppress the mutua interference even on the same frequency Namey, the effective use of the frequency resource is possibe It has been aso shown that the auto-correation sideobe of (1) (2) Compementary code set Constructing compete compementary code set jointy Fig 2 Compressed Puse Shape an individua code for the WPCCS is smaer than that for the conventiona Then, it is possibe to reduce the sideobe produced by the Dopper frequency [8] Consequenty, it has been carified that the WPCCS has severa advantages compared to the conventiona compementary code set However, there is ony one method that extends the compression ratio of the WPCCS So, the merits of the WPCCS is not sufficienty dispayed in the region of arge compression ratio ψ AiB i =0 (3) IV Compression Ratio Expansion Method In this section, it is shown that the compression ratio expansion method for the conventiona compementary code set proposed in the artice [5] is appicabe aso for the WPCCS Firsty, et s define some notations A B represents the code with ength nm defined by Eq(4) when the code A with ength n and code B with ength m are given C = A B =((b 1 A)(b 2 A) (b m A)) (4)

4 Theorem 1 When the binary code set {A 1,A 2 } is the compementary code set, ie, satisfies Eq (1) and its ength is n, further the code set {B 1,B 2 } is the WPCCS, ie, satisfies Eq (2) and its ength and compression parameter are respectivey m and Then, the code set {C 1,C 2 } given by Eq(5) is the WPCCS whose ength and compression parameter is respectivey 2nm and Where, à is the code obtained by arranging the code eements of A inversey {C 1,C 2 } = {B A1 1 BÃ2 2,BA2 1 B Ã1 2 } (5) Proof Let define k and r integers and et s defined as s = k mod m, ie k = rm + s, then, ψ C1C 1 +ψ C2C 2 (r)+ψã1ã 1 +ψ B1B 1 ( m + s){ψ A1A 1 (r +1)+ψ A2A 2 (r +1)} ( m + (r (r +1)} s){ψã2ã 2 +1)+ψÃ1à 1 (s) ( n + r)+ψ {ψã2a 1 ( Ã1)A ( n + r)} 2 ( m + s) ( n + r +1)+ψ {ψã2a 1 ( Ã1)A ( n + r +1)} 2 (r)+ψã1ã 1 (s) ( n + r) ( n + r)} {ψã2a 1 ψã1a 2 ( m + s) ( n + r +1) ( n + r +1)} {ψã2a 1 ψã1a 2 (r)+ψã1ã { 1 2n{ψB1B = 1 (s) (s)} k <m 0 k m Eq( 6) indicates that the code set {C 1,C 2 } aso satisfies Eq (2) and its compression parameter is if the code set {B 1,B 2 } satisfies Eq (2) and its compression parameter is Proof End The concrete exampe is shown beow Let A 1 = (0, 0),A 2 = (0, 1) and B 1 = (0, 1/2, 0),B 2 = (0, 1/2, 1), Then {A 1,A 2 } satisfies Eq (1) and {B 1,B 2 } satisfies Eq (2) {C 1,C 2 } is obtained from {A 1,A 2 } and {B 1,B 2 } by Eq(5) as foows C 1 =(0, 1/2, 0, 0, 1/2, 0, 1, 3/2, 0, 0, 1/2, 1) C 2 =(0, 1/2, 0, 1, 3/2, 1, 1, 3/2, 0, 1, 3/2, 0) Then, (C 1,C 2 ) satisfies Eq (2) Fig3(a) shows the compressed puse shape of (C 1,C 2 ) (6) V Compementary Code Set Composedof Severa Codes In the WPCCS, ony one case when p = 2 in Eq (2) was investigated before The case when p is more than 3 in Eq (2) is proposed in this section Namey, It is shown that the WPCCS composed of more than 3 individua codes can be constructed by the method, which was proposed in the artice [2], to construct the conventiona compementary code set composed of more than 3 individua codes It is aso shown that the method gives the compression ratio expansion way of the WPCCS simutaneousy First of a, et s define some notations When the code set X = {A i ;1 i p} and the matrix with q rows and p coumns shown in Eq (7) are given, HX represents the code set whose eement codes are each rows of the matrix shown in the right side of Eq(8) Where h k =1 Y = HX = h 11 h 12 h 1p h 21 h 22 h 2p H = h q1 h q2 h qp (h 11 A 1 )(h 12 A 2 ) (h 1p A p ) (h 21 A 1 )(h 22 A 2 ) (h 2p A p ) (h q1 A 1 )(h q2 A 2 ) (h qp A p ) Theorem 2 When X = {A i ;1 i p} is the WPCCS whose compression parameter is, and a coumns of the matrix with q rows and p coumns shown in Eq (7) are orthogona each other Then, the code set Y given by Eq(8) is aso the WPCCS whose compression parameter is Proof First of a, Let s investigate 2 matrices shown beow h 1r A r h 1s A s h 2r A r and h 2s A s h qr A r h qs A s The sum of the q cross correations between two codes which are respectivey the same row of the right matrix and eft matrix is easiy cacuated as foows ψ (hira r)(h isa s) = h ir h isψ ArA s = q δ rs ψ ArA s Therefore, when each row of Y given by Eq (8) is represented by B i Namey, B i =(h i1 A 1 )(h i2 A 2 ) (h ip A p ) Then, Eq (7) (8)

5 (9) is obtained ψ BiB i = q ψ ArA r (9) Proof End Especiay, it is noted that the WPCCS composed of more than 3 individua codes is obtained from that composed of 2 individua codes when p =2, 3 q in Eq (8) Moreover, the theorem 2 gives the compression ratio expansion method for the WPCCS Especiay, it gives the compression ratio expansion method for the WPCCS composed of 2 individua codes when p =2,q =2 The concrete exampe is shown beow Let A 1 = (0, 1/2, 0),A 2 =(0, 1/2, 1) and H = 1 i 1 i ,Y = B 1 B 2 B 3 B 4 Then, from Eq (8), B1 =(0, 1/2, 0, 1/2, 1, 3/2) B2 =(0, 1/2, 0, 3/2, 0, 1/2) B3 =(1, 3/2, 1, 0, 1/2, 1) B4 =(1, 3/2, 1, 1, 3/2, 0) (B1,B2,B3,B4) satisfies Eq (2) Fig3(b) shows the compressed puse shape of (B1,B2,B3,B4) VI Concusion We have proposed the compression ratio expansion method for the WPCCS Further, the paper introduced that the WPCCS composed of more than 3 individua codes can be constructed It aso gives the compression ratio expansion method of the WPCCS at the same time There was ony one compression ratio expansion method for the WPCCS before and it makes ony one WPCCS of arger compression ratio from the one with smaer compression ratio So the number of the WPCCS didn t increase though the compression ratio increased before However, we have given severa compression ratio expansion methods for the WPCCS Further, they can construct severa WPCCSs with arger compression ratio from the one with smaer compression ratio As a resut, in the arge compression ratio region, the number of the WPCCS becomes extremey arger and the freedom degree of the radar signa design becomes much arger, which make it possibe to use the important merits of the WPCCS moduation The remaining probem is to investigate whether other merits of the WPCCS obtained in the region of sma compression ratio can be aso obtained in the region of the arger compression ratio Fig 3 Compressed Puse Shape of Exampe References [1] Editor in Chief, Skonik, M I (1990) Radar Handbook, second edition McGraw-Hi Inc, 1990 [2] Richard DJ van Nee (1996) OFDM codes for peak-to average power reduction and error correction, Proc IEEE Gobecom, pp , Nov 1996 [3] Skonik, M I (1980) Introduction to Radar Systems, second edition McGraw-Hi Inc, pp ,1980 [4] FENathanson (1991) Radar Design Principes, second edition Scitech Pubishing Inc, pp564,1991 [5] MJEGoay (1961) Compementary series, IRE Trans Information Theory, voit-7 pp82-87, 1961 [6] CCTseng and CLLiu (1972) Compementary Sets of Sequences, IEEE Trans Information Theory Vo IT-18, No 5, pp ,1972 [7] Sato, R, and Shinriki, M (1999) Coded Puse Compression with Reduced Bandwidth, Jour IEICE Transactions on Communications, voe82-b, No7, pp , 1999 [8] H Takase, RSato and MShinriki (2002) New 4-Phase Compementary Codes Compressing a Puse to a Width of Severa Sub-puses, Trans IEICE of Japan (in Japanese) voj85-b, No8, pp , 2002