Complete Complementary Codes Composed of Perfect Sequences and their Application to Mobile Communications

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1 Complete Complementary Codes Composed of Perfect Sequences and their Application to Mobile Communications Naoki SUEHIRO, and Toshiaki Imoto Graduate School of Systems and Information Engineering University of Tsukuba Tsukuba, , Japan Faculty of Engineering, Soka University Hachioji, Tokyo , Japan Abstract A complete complementary code composed of 4- phase perfect (orthogonal) sequences is shown The sequences compose a matrix whose cube is a unit matrix A method is also discussed, which has been proposed by Suehiro, etal for increasing the efficiency of the wireless frequency usage The method uses a pilot signal for measuring the multipath property as a complex number for each timeslot The data transmission signals transmitted together with the pilot signal using the same frequency and time without interference The signals with zero-autocorrelation zone and zerocrosscorrelation zone (ZCZ signals) are made based on the complete complementary code and used for above method I INTRODUCTION For future mobile systems, the frequency usage efficiency is the most inportant, because LSI will become small and fast, optical network will be grown up, and the demand for the wireless frequency band will be increased The wireless channel has many reflect pathes The reflected signals interfere the channel when the system doesn t know the multipath properties But, when the system knows the multipath properties, the reflected signal energies can be useful Because both of TDMA and FDMA use sine wave, the influence of multipath for them appears as the fading of the signal power in the same bit, where bit means the signal corresponds to one bit in data It is very difficult to divide the faded sine wave into original sine wave and the reflectpath sine waves On the other hand, because CDMA uses sequences of chips, where each chip has its address in the sequence, their may be some method to divide the original signal and the reflectpath signals In fact, when we use a pilot signal, both of whose frequency and time are the same as the signals which bears the data, we can divide the original signal and the multipath signals by using clever codes and modulation To realize it, a signal design method had been proposed[0] by using complete complementary code[3][8] Using the proposed method, we can design binary CDMA signals for wide shifts of without sidelobe nor co-channel interference In fact, the length of shifts without the influence of multipath is 2 n for any n Then, we proposed[] a method for increasing the information transmission rate based on a new method for estimation of multipath channel and convolution of spread time signals We used pilot signal and data-transmission signals When we use the complete complementary code for approximately synchronized CDMA systems without co-channel interference[0], we can assign the same time and the same frequency for both of the pilot signal and the data-transmission signals So, the estimation of the multipath channel can be exact estimation Because the estimation is exact and the correlation property of the complete complementary code is complete, even when the spreaded signal are comvoluted chip by chip into a multi-valued signal the detection of each bit is completely independent So, the very high rate of information transmission and the very high efficiency of frequency usage is realizable The aspected problem in the proposed system is that the proposed method for using the complete complementary codes is very complex So, we proposed another simple method for using the complete complementary codes as the zerocorrelation zone (ZCZ) sequences The BPSK implementation as an LSI chip has been done[4] A multicarrier CDMA architecture based on this method has been designed and simulated by computer simulation[3] The simulation result is successfull II COMPLETE COMPLEMENTARY CODE AND ZCZ SEQUENCE Suehiro[3][8] proposed the complete complementary code by expanding the concept of complementary series[] and even-shift orthogonal sequences[2] The complete complementary code is composed of several sets of auto-complementary codes, any two of which are cross-complementary codes In this paper, we introduce another method for making complete complementary codes proposed by Suehiro[5] In this method, the complete complementary codes are obtained from Hadamard matrices or polyphase matrices of H 3 = I, where I is the unit matrix

2 A Definition of Complete Complemantary Code Suehiro[3][8] proposed the complete complementary code by expanding the concept of complementary series[] and even-shift orthogonal sequences[2] The complete complementary code is composed of several sets of auto-complementary codes, any two of which are cross-complementary codes We can make binary complete complementary codes of length 2 n for any n[3][8] For example, let A 0, A, B 0 and B be binary sequences as: A 0 =( ) A =(+ +++ ) B 0 =(+++ + ) B =( ) where + denotes, and denotes Let A 0 A 0, and A A be the autocorrelation function of A 0 and A Then A 0 A 0 + A A =(0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0) A couple of sequences is called an auto-complementary code, when the sum of autocorrelation function is 0 in every term except for the 0-shift term So, {A 0,A } is called an auto-complementary code Similarly, B 0 B 0 + B B =(0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0) So, {B 0,B } is also called an auto-complementary code On the other hand, A 0 B 0 + A B =(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) Equally, B 0 A 0 + B A =(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) Accordingly, {A 0,A } and {B 0,B } are called a set of crosscomplementary code, and {{A 0,A }, {B 0,B }} is called a set of complete complementary code B Hadamard or Polyphase Matrices of H 3 = I Let m and n be natural numbers, and N be N = m n Let also be i, i 0, i, j, j 0 and j as: i = i 0 N + i, j = j 0 N + j, where 0 i N 2, 0 i 0 N, 0 i N, 0 j N 2, 0 j 0 N, 0 j N In other words, (i 0,i ) is the N-nary notation of i Further, i 0 = i 00 m n + i 0 m n i 0(n ) m 0 and m-nary notation of i 0 is: i 0 = (i 00,i 0,,i 0(n ) ) Similarly, i, j 0 and j are the m-nary notation of i, j 0 and j Then, [Theorem ] When the (i, j)th (0 i N 2, 0 j N 2 ) element of an N 2 N 2 matrix H(i, j) is: H(i, j) = N exp {2π m (i j 0 + i 0 j + j 0 j )}, where is the inner product, then H 3 = I, where I is a unit matrix [Meaning of Theorem ] Let an N 2 N 2 matrix A be an m-nary dijit-inverse matrix For example, when m =3and n =, A = Next, let an N 2 N 2 matrix B be as: B(i, j) = N exp {2π m (i 0 j 0 + i j )} Then, B is a unitary matrix obtained by the Kronecker product of 2n DFT matrices of m m For example, when m =3 and n =, B = 3 w w 2 w w 2 w w 2 w 2 w w 2 w w 2 w w w w w 2 w 2 w 2 w w 2 w w 2 w 2 w w 2 w w w 2 w 2 w w 2 w 2 w 2 w w w w w 2 w 2 w w w 2 w 2 w w 2 w w w 2, where w = + 3 2, w 3 = Next, let N 2 N 2 matrix C be a diagonal matrix, whose diagonal elements are obtained from an N N matrix by ordering the rows The N N matrix is obtained as the Kronecker product of n DFT matrices of m m For example, when m =3and n =, C = O w w 2 O w 2 w

3 Thus, when m =3and n =, a periodic sequence whose autocorrelation function is zero (ABC) 3 for every term except for the period-multiple terms[4][7] w w 2 w 2 3 Whenever n =, all rows in the matrix H are perfect w w w 2 w 2 w sequences w 2 w w w 2 w w 2 w 2 w C Complete Complementary Code Derived from Poliphase = 3 w w 2 w w 2 w 2 w 2 w 2 Cubic Root of Unit Matrix w w 2 w 2 w w w w The rows of the polyphase cubic root of the unit matrix w 2 w w w 2 compose a set of complete complementary code For example, w w w w w w 2 w when m =4and n =, the rows of H are denoted as w 2 w w 2 w 2 w 2 w w 2 = I S 0,0 For the second example of m =4and n =, S,0 S 2,0 H = S 3, j k k j H = j k k j + S 0,, k + j + + j + k k + j + + j + k S 2,3 + j k k j S 3,3 + j k j k j k j + k k k k k + j k k j where + j k k j k j j + k j j j j k j j k, S 0,0 = ( j k k j), + + j + k k + j + + j + k k + j + + k j + j k + + k j j k + + S 3,3 =(+k jkkkk k + jjkjk) + k j j j j j k + j k j k j + k j + j k + + Then + k j k k k k k + j j k j k {{{S 0,0,S 0,,S 0,2,S 0,3 }, {S,0,S,,S,2,S,3 }, where k = j, and {S 2,0,S 2,,S 2,2,S 2,3 }, {S 3,0,S 3,,S 3,2,S 3,3 }} H 3 = I is a complete complementary code composed of 4-phase perfect sequences For the third example of m =2and n =2, H = H 3 = I In the first and second example, all rows are perfect sequences A perfect sequence (an orthogonal sequence) is III CHANNEL ESTIMATION FOR APPROXIMATELY SYNCHRONIZED CDMA SYSTEMS BY USING A PILOT SIGNAL COMPOSED OF AN AUTO- COMPLEMENTARY CODE IN A SET OF COMPLETE COMPLEMENTARY CODE In this section, A 0 =(+++ ), A =(+ ++), B 0 = (+ + +) and B =(+ ) Let a(,a 0,A 0,A 0,A 0, ) 0 + a(,a,a,a,a, ) T be a pilot signal for estimating a multipath channel, where the length of the signal is 8 + T, because the length of A 0 or A is 4, and () T denotes the delay of T The signal should be modulated by the carrier of frequency f 0, and transmitted When a receiver received the transmitted pilot signal, the receiver demodulates the received signal by carrier of frequency f 0, and input the signal demodulated by the frequency f 0 into the matched filter for (A 0 A 0 ), and input the signal delayed by the delay factor T into the matched filter for (A A ) Then,

4 the addition of the output of two matched filters is: a(,a 0,A 0,A 0,A 0, ) T A 0,A 0 + a(,a,a,a,a, ) 2T A 0,A 0 + a(,a 0,A 0,A 0,A 0, ) 0 A,A + a(,a,a,a,a, ) T A,A = a(x,x,x,,x,x,x) 0 + a(x,,x,0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0,x,,x) T + a(x,x,x,,x,x,x) 2T Let also be multipath channel represented as (p 0,p,p 2,p 3 ), where p k is the complex multipath factor, and the largest time delay of multipaths is 3 Then, the output of the receiver filters through the multipath channel is: 6a() 0 +6a(x,,x,0,p 0, 0, 0, 0,p 0, 0, 0, 0,p 0, 0,x,,x,0, 0, 0) T +6a() 2T +6a() 0 +6a(0,x,,x,0,p, 0, 0, 0,p, 0, 0, 0,p, 0,x,,x,0, 0) T +6a() 2T +6a() 0 +6a(0, 0,x,,x,0,p 2, 0, 0, 0,p 2, 0, 0, 0,p 2, 0,x,,x,0) T +6a() 2T +6a() 0 +6a(0, 0, 0,x,,x,0,p 3, 0, 0, 0,p 3, 0, 0, 0,p 3, 0,x,,x) T +6a() 2T =6a() 0 +6a(x,,x,p 2,p 3,p 0,p,p 2,p 3,p 0,p,x,,x) T +6a() 2T, where each x denotes some value Thus, the receiver can obtain the multipath channel property (p 0,p,p 2,p 3 ) easily Using a pilot signal, the receiver can know multipath channel property IV HIGH RATE INFORMATION TRANSMISSION BASED ON CONVOLUTION OF SIGNALS In this section, we propose a high rate information transmission method Even if the channels are multipath channels, the method is available, because the receiver can know the multipath channel property by the method written in th previous section Now, the transmitter transmits data (b 0,b,b 2,b 3 ) by using another auto-complementary code (B 0,B )=(++ +, + ) in the same complete complementary code of the previous section The transmitted signal is a kind of convolutional signal: b 0 (,B 0,B 0,B 0,B 0,, 0, 0, 0) 0 + b 0 (,B,B,B,B,, 0, 0, 0) T + b (0,,B 0,B 0,B 0,B 0,, 0, 0) 0 + b (0,,B,B,B,B,, 0, 0) T + b 2 (0, 0,,B 0,B 0,B 0,B 0,, 0) 0 + b 2 (0, 0,,B,B,B,B,, 0) T + b 3 (0, 0, 0,,B 0,B 0,B 0,B 0, ) 0 + b 3 (0, 0, 0,,B,B,B,B, ) T This data signal is transmitted at the same time as the pilot signal of the previous section The data signal and the pilot signal share the same frequency band together So, the estimated multipath property is very reliable for the data detection When the receiver received the transmitted data signal, the receiver demodulates the received data signal by carrier of frequency f 0, and input the signal demodulated by the delay 0 into the matched filter for (B 0 B 0 ), and input the signal demodulated by the delay T into the matched filter for (B B ) Then, the addition of the output of two matched filters is: 6b 0 () 0 +6b 0 (x,,x,p 2,p 3,p 0,p,p 2,p 3,p 0,p,x,,x,0, 0, 0) T +6b 0 () 2T +6b () 0 +6b (0,x,,x,p 2,p 3,p 0,p,p 2,p 3,p 0,p,x,,x,0, 0) T +6b () 2T +6b 2 () 0 +6b 2 (0, 0,x,,x,p 2,p 3,p 0,p,p 2,p 3,p 0,p,x,,x,0) T +6b 2 () 2T +6b 3 () 0 +6b 3 (0, 0, 0,x,,x,p 2,p 3,p 0,p,p 2,p 3,p 0,p,x,,x) T +6b 3 () 2T = 6() 0 +6(x,,q,q 2,q 3,q 4,q 5,x,,x) T +6() 2T Because q 5 = q, we obtain the following equations: p b 0 + p 0 b + p 3 b 2 + p 2 b 3 = q p 2 b 0 + p b + p 0 b 2 + p 3 b 3 = q 2 p 3 b 0 + p 2 b + p b 2 + p 0 b 3 = q 3 p 0 b 0 + p 3 b + p 2 b 2 + p b 3 = q 4 Because (p 0,p,p 2,p 3 ) is obtained by the pilot signal detection described in the previous section, and (q,q 2,q 3,q 4 ) is obtained by the data signaldetection described in this section, the data (b 0,b,b 2,b 3 ) can be obtained by solving the equations V DISCUSSION The LSI chip[4] uses the complete complementary code of length 64, which is obtained the Hadamard matrix of H 3 = I The Hadamard matrix is obtained as m =2and n =3 When we use a complementary code composed of 6 autocomplementary codes, where each auto-complementary code is composed of 6 binary sequences of length 256, we can assign auto-complementary code to pilot signals and 5 autocomplementary codes to information transmission signals In this case, the information transmission ratio represented by chip/bit is about (3/k) chip/bit, where k is the QAM factor When 2 k points in the signal domein can be detected separately, each information bit needs 3 k chips So, when k =4, each information bit needs 033 chips k depends on the ratio between signal power and heat noise power There are huge variation of complete complementary code[3][8] In fact, the variation of complete complementary code depends on the variation of the Hadamard matrices or unitary matrices, which are used for the making of the complete complementary code So, we can assign different

5 complete complementary codes for each cell Because we can use very long sequence for composing a complete complementary code without decreasing the information transmission rate, the problem of inter-cell interference can be solved If we use complete complementary code of short length, the amplifier doesn t need large dinamic range, then the QAM method increases the information transmission rate farthermore In this case, the limit of the information transmission rate depends on the ratio between the received signal power, and the heat noise of the receiver and inter-cell interference ACKNOWLEDGEMENTS The authors wish to thank to Prof Mitsutoshi Hatori, Dr Shinya Matsufuji Prof Noriyoshi Kuroyanagi, Prof Makoto Nakamura and Dr Hideyuki Torii for their discussion This research is supported by International Communications Foundation REFERENCES [] MJEGolay, Complementary series, IRE Trans on Information Theory, vol IT-7, pp82-87, April 96 [2] YTaki, HMiyakawa, MHatori and SNamba, Even-shift orthogonal sequences, IEEE Trans on IT, volit-5, pp , 969 [3] NSuehiro, Complete complementary code composed of N-multipleshift orthogonal sequences, IECE (presently IEICE) Trans, volj65-a, pp (in Japanese), Dec 982 [4] NSuehiro, Modulatable orthogonal sequences, Technical Report of IECE (presently IEICE), vol IT-83-8, pp9-5 (in Japanese), May 983 [5] NSuehiro, Complete complementary codes derived from cubic roots of unit matrices, IECE (presently IEICE) Trans, volj66-a, pp486-49(in Japanese), June 983 [6] NSuehiro and MHatori, Polyphase periodic sequences without crosscorrelation and their application to asynchronous SSMA systems, IEICE Trans, volj68-a (in Japanese), Oct 985 [7] NSuehiro and MHatori, Modulatable orthogonal sequences and their application to SSMA systems, IEEE Trans on Information Theory, volit-34, Jan 988 [8] NSuehiro and MHatori, N-shift cross-orthogonal sequences, IEEE Trans on Information Theory, volit-34, pp43-46, Jan 988 [9] NSuehiro, A signal design without co-channel interference for approximately synchronized CDMA systems, IEEE Journal of Selected Areas in Communications, vol2, pp837-84, June 994 [0] NSuehiro and NKuroyanagi, Binary signal design for approximately synchronized CDMA systems without detection sidelobe nor co-channel interference using auto- and cross-complementary codes Proceedings of ICUPC 98, Oct 998 [] NSuehiro, NKuroyanagi and SMatsufuji, High rate information transmission based on multipath estimation and signal convolution in approximetely synchronized CDMA systems without co-channel interference, Proceedings of WPMC 99, pp36-367, Sept 999 [2] NSuehiro, NKuroyanagi, TImoto and SMatsufuji, Very efficient frequency usage system using convolutional spread time signals based on complete complementary code, Proceedings of PIMRC 2000, Sept 2000 [3] H-HChen, J-FYeh and NSuehiro, A multicarrier CDMA architecture based on orthogonal complementary codes for new generation of wideband wireless communications, IEEE Communications Magazine, vol39, No0, pp26-35, Oct 200 [4] JChen, NSuehiro and ZFan, Baseband signal processing and its VLSI implementation for a new CDMA system based on zero-correlationzone sequences, Proceedings of 2nd Joint Symposium on Opto- and Microelectronics Devices and Circuits, March, 2002