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A new class of interlaced complementary codes based on components with unity peak sidelobes
ARTICLE IN PRESS
Signal Processing 88 (2008) 307–314 www.elsevier.com/locate/sigpro
A new class of interlaced complementary codes based on components with unity peak sidelobes$ Adly T. Fam, Indranil Sarkar Department of Electrical Engineering, The State University of New York at Buffalo, NY 14260, USA Received 22 March 2007; received in revised form 20 July 2007; accepted 3 August 2007 Available online 15 August 2007
Abstract A new class of biphase complementary code sets is proposed. Individual codes in each set have maximum sidelobes of unity magnitude. The individual codes could have gaps of zeros but they are interlaced together without gaps in the final scheme. A number of such codes, as presented in this paper, use Barker codes as building blocks with additional elements from fþ1; 1; 0g. The main drawback of regular complementary codes longer than 4 is that they have peak sidelobe magnitude greater than unity. In the presence of frequency selective fading or other factors, inexact sidelobe cancellation results in non-zero sidelobes at the output. This is minimized in the proposed schemes by using codes that have peak sidelobes of unity magnitude as the individual codes in the sets. The proposed codes are attractive due to their increased resistance to the effect of inexact sidelobe cancellation. A figure of merit is proposed to measure the frequency use efficiency of the proposed codes. r 2007 Elsevier B.V. All rights reserved. Keywords: Barker codes; Complementary codes; Pulse compression codes; Ternary complementary codes
1. Introduction Complementary codes or sequences were introduced by Golay [1] for use in radar astronomy. A complementary set is a set of finite sequences whose autocorrelation functions (ACFs), when added together, cancel out the sidelobes of each other. Any complementary set can be used to generate longer ones. The root sequences which are defined directly and not generated from shorter ones are $
Patent pending. USPTO Patent application #11672203.
Corresponding author. Tel.: +1 716 645 2422x2504;
fax: +1 716 645 3656. E-mail addresses: [email protected] (A.T. Fam), [email protected] (I. Sarkar).
also referred to as kernels. Golay [2] came up with kernels of binary complementary pairs of length 2, 10 and 26. Tseng and Liu [3] developed algorithms for recursive generation of complementary sequences. Sivaswamy [4] proposed some polyphase complementary sequences and gave kernels of complementary triplets of length 2 and 3. Frank [5] provided an excellent overview of polyphase complementary codes and proposed additional kernels for them. Biphase and polyphase complementary codes are attractive in radar, sonar and communications applications because they have zero sidelobes. This advantage is compromised in the presence of frequency selective fading. Unequal fading of the individual codes results in inexact cancellation of their sidelobes, thus degrading their performance.
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.08.003
ARTICLE IN PRESS A.T. Fam, I. Sarkar / Signal Processing 88 (2008) 307–314
308
In recent times, complementary codes have been used for pulse compression in ultrasound imaging systems [6,7]. In such applications, the codes are fired sequentially and hence any movement in the tissue being observed results in inexact sidelobe cancellation. This is manifested by artifacts in the images. Chiao and Hao [8] has presented a good overview of the usage of complementary codes in ultrasonic imaging and discussed several implementation issues. A logical prerequisite for a good complementary code set in any application is that the peak sidelobe of the individual codes are as small as possible. Discussions on this issue can be found in [5] as well as in [9]. This is satisfied by the new class of interlaced complementary codes (ICC) introduced next. Let us consider the class of codes that are constructed from a ternary alphabet D ¼ f1; þ1; 0g and have peak sidelobes of unity magnitude. We denote this class as T. The constituent codes of the ICC are members of T. Let {tn } be a code of length N in T. Its aperiodic ACF is given by At ðtÞ ¼
Nt1 X
tn tnþt ,
(1)
n¼0
where 0ptpN 1. The peak sidelobe constraint on the members of T specifies that max jAt ðtÞj ¼ 1. ta0
(2)
Barker codes are the only biphase (constructed from fþ1; 1g) codes that satisfy (2). Let the class of Barker codes be B. Obviously, B T. The constituent codes in the proposed ICC may contain Barker codes or Barker codes modified with additional elements from D. In this work, we present several schemes involving two or more codes from T that form complementary sets. The constituent codes with gaps of zeros are interlaced with other constituent codes to remove such gaps. Different frequencies are used for the different constituent codes resulting in a multifrequency code some components of which must be interlaced with each other. The final scheme could be purely time division multiplexed (TDM) or allow some amount of parallel transmission depending on the structure of the constituent codes. If the purely TDM scheme can be broken down into equal length blocks in a way such that each constituent code is contained only in one particular block, then these
equal length blocks can be transmitted simultaneously. This gives rise to a hybrid transmission scheme using both TDM and simultaneous transmission. The advantages of TDM are low peak to average power ratio resulting in a low probability of interception. Also, a purely TDM scheme requires only one transmitter with frequency hopping capabilities. However, using TDM alone results in longer codes that are more susceptible to multipath and eclipsing problems. The parallel transmission scheme results in shorter codes and hence is better suited to counter these problems. This comes at the cost of a higher peak to average power ratio, higher detectability and increased number of transmitters or a wideband transmitter. A hybrid scheme achieves a trade-off between these two transmission modalities. It should be noted that the proposed codes are related to the ternary complementary pairs (TCP) introduced by Gavish and Lempel [10] and subsequently investigated in details in [11–14]. The ICC sets share with the TCPs the property that the component codes are generated from fþ1; 1; 0g. However, the characteristics that make the ICC sets different are:
There could be more than two component codes in an ICC set. The component codes in an ICC set must have peak sidelobes of unity magnitude. The component codes are interlaced with each other without gaps such that the final transmission scheme is comprised of fþ1; 1g.
Due to these constraints, ICC sets are difficult to find and there are no known general method for their systematic generation. However, in spite of their heuristic nature, the proposed codes could be very important in various applications due to their desirable properties. 2. Interlaced complementary codes In this section we introduce several examples of the proposed ICC. As mentioned before, the constituent codes of the ICCs are constructed from D and hence may contain zeros. However, the final transmission scheme is free of zeros and contains only elements from fþ1; 1g. In these examples, we use Barker codes as building blocks with additional elements from D to obtain the constituent codes belonging to T. However, using Barker codes is not
ARTICLE IN PRESS A.T. Fam, I. Sarkar / Signal Processing 88 (2008) 307–314
The ACF R11 ðzÞ for the Barker code of length 11 is also shown in Table 1. R11 ðzÞ has 10 sidelobes on each side of the mainlobe of alternating 1’s and 0’s. Furthermore, it can also be observed that when they are aligned, 1 sidelobes of R11 ðzÞ coincides exactly with þ1 sidelobes of R13 ðzÞ. Thus, when added together, the sidelobes cancel out with the exception of one sidelobe each at either end of R13 ðzÞ. We also note that the Barker code of length 2, f1; 1g, produces two sidelobes of height 1. The code of length 2, with 11 zeros inserted between its two elements, is represented by B2 ðz12 Þ. The corresponding ACF R2 ðz12 Þ can be used to cancel out the two remaining sidelobes of R13 ðzÞ that R11 ðzÞ did not cancel. This is clearly shown in Table 1. It is easy to see that the transmission scheme should be such that the 11 gaps between the elements of B2 ðz12 Þ should be filled by code bits from some other code so that the final transmission scheme is free of gaps. We also observe that these gaps could be filled by placing B11 ðzÞ as it is in these gaps. The transmission scheme is therefore given by
a necessary condition for constructing codes belonging to T and consequently, the class of ICC. Let the z-transform of a Barker code of length N be denoted by BN ðzÞ. If ðk 1Þ zeros are placed between every two elements of the code, then the modified code is represented by BN ðzk Þ. The ACF of BN ðzÞ is given by RN ðzÞ ¼ BN ðzÞBN ðz1 Þ
(3)
from which we get RN ðzk Þ ¼ BN ðzk ÞBN ðzk Þ.
309
(4)
The following steps are involved in the construction of valid interlaced codes free of any gaps. The challenge is to have the individual codes with unity peak sidelobes fit together such that the final scheme is composed of fþ1; 1g without any gaps of zeros. (1) Two or more constituent codes should be chosen from T. (2) The desired number of zeros is inserted between the elements of the constituent codes such that the sidelobes of the ACFs cancel each other out. (3) Finally, the constituent codes have to be interlaced with each other such that there are no zeros in the final transmission scheme. The constituent codes may have to be delayed with respect to each other so that they fit together without any zeros. (4) Each constituent code is transmitted after modulation by a particular frequency. The number of frequencies used is therefore equal to the number of constituent codes. (5) Appropriate delays would also be needed at the receiver end for each constituent code, so that the ACFs (or the output of the matched filters) are aligned properly before addition.
TðzÞ ¼ B13 ðzÞ þ z13 B2 ðz12 Þ þ z14 B11 ðzÞ.
(5)
Thus, in this transmission scheme B13 ðzÞ is transmitted first. The first bit of B2 ðzÞ is transmitted next followed by the entire B11 ðzÞ. Finally the remaining bit of B2 ðz12 Þ is transmitted resulting in transmission of all the codes in an interlaced fashion and free of any gaps. Table 2 depicts the transmission scheme graphically. The columns t1 –t26 depict the time slots whereas f 1 –f 3 depict the three different frequencies used to modulate B13 ðzÞ, B2 ðz12 Þ and B11 ðzÞ, respectively. Since no gaps (or 0’s) are allowed in the transmission scheme and only one code bit can be transmitted in a given time slot, each column in the table should have only a single 1 or 1. Also, no columns can have all 0’s. Similar tables can be constructed for all the transmission schemes described in this paper. There are some obvious alternative schemes to the one mentioned above. Each of the B13 ðzÞ, B11 ðzÞ and B2 ðz12 Þ as well as the entire transmission scheme can be flipped to produce 16 obvious alternative
2.1. ICC set I involving B13 ðzÞ; B11 ðzÞ and B2 ðz12 Þ The ACF R13 ðzÞ of the Barker code of length 13 is shown in Table 1. It can be seen that on each side of the mainlobe, there are 12 sidelobes of alternating þ1’s and 0’s. Table 1 Aligned autocorrelation profiles of codes used in ICC set I R13 ðzÞ R11 ðzÞ R2 ðz12 Þ
1
0
1
0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
13 11 2
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0
1
0
1
ARTICLE IN PRESS A.T. Fam, I. Sarkar / Signal Processing 88 (2008) 307–314
310 Table 2 Transmission scheme for ICC-I t2
t6
t7
t8
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0
1 1 1 0 0 0 0 0 0
t1 f1 f2 f3
t3
t4
t5
t9
t10 t11
t12 t13
1 1 0 0 0 0
Table 3 Hybrid transmission scheme for ICC-I t1 f1 f2 f3
t2
t3
t4
t5
t6 t7 t8
t9
t10 t11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
t12 t13
1 1 0 0 1 1
1 1 0
schemes. It is also to be noted that B13 ðzÞ comprises the first 13 bits of the scheme. The combination of B11 ðzÞ and B2 ðz12 Þ comprise the remaining 13 bits. Hence, B13 ðzÞ and the combination of B11 ðzÞ and B2 ðz12 Þ can be transmitted simultaneously to reduce the overall length of the complementary code. This transmission scheme contains both simultaneous and TDM transmissions. This hybrid scheme is depicted in Table 3. The parallel components for this case are given as T P1 ðzÞ ¼ B13 ðzÞ,
(6)
T P2 ðzÞ ¼ B2 ðz12 Þ þ z1 B11 ðzÞ.
(7)
All ICC sets are constructed in a way that they can always be transmitted in a sequential bit by bit fashion. However, some amount of parallel transmission is possible in certain code sets depending on their structure. ICC set I is an example of such a code. 2.2. ICC set II involving B13 ðz2 Þ; B11 ðz2 Þ and B2 ðz24 Þ If we put one gap in between the elements of B13 ðzÞ and B11 ðzÞ, we come up with two modified Barker codes B13 ðz2 Þ and B11 ðz2 Þ, respectively. The corresponding ACFs are now R13 ðz2 Þ and R11 ðz2 Þ, respectively. When lined up with one another, all the sidelobes will cancel each other except the ones at the extreme ends of R13 ðz2 Þ. We note that in this case, we will require R2 ðz24 Þ to cancel these two remaining sidelobes. Hence, we need to transmit B2 ðz24 Þ in the transmission scheme.
t14 t15
1 0 0 1 0 0
t16
t17
t18
0 0 0 0 0 0 0 0 1 1 1 1
t19 t20 t21 0 0 1
0 0 1
t22 t23 t24
0 0 0 0 1 1
0 0 1
t25 t26
0 0 0 0 1 1
0 1 0
In the transmission scheme, 11 of the 12 gaps created in B13 ðz2 Þ are filled by the elements of B11 ðz2 Þ. The one remaining gap just before the last element of B13 ðz2 Þ needs to be filled somehow using the code bits of B2 ðz24 Þ. We observe that if we place the first bit of B2 ðz24 Þ just before the first element of B13 ðz2 Þ, the last bit of B2 ðz24 Þ occupies the gap to be filled. Thus, the gap-free transmission scheme is given by TðzÞ ¼ B2 ðz24 Þ þ z1 B13 ðz2 Þ þ z2 B11 ðz2 Þ.
(8)
Table 4 shows the graphical representation of the transmission scheme. Once again, t1 –t26 represent the time slots and f 1 –f 3 represent the frequencies used to modulate B2 ðz24 Þ, B13 ðz2 Þ and B11 ðz2 Þ, respectively. Once again, reversal of the transmission order of the individual codes or the entire scheme produce several equivalent ICC sets. Unlike the ICC set I, simultaneous transmission is not possible for this set since the purely TDM scheme shown in Table 4 cannot be divided into equal length blocks containing complete constituent codes. 2.3. ICC set III involving B13 ðzÞ; B11 ðzÞ; B7 ðz2 Þ and B5 ðz2 Þ This set is based on Barker codes of length 13, 11, 7 and 5. We have shown the ACFs R13 ðzÞ and R11 ðzÞ in Table 1. When these two ACFs are added together, the sidelobes cancel each other except the two positive sidelobes at the two extremes of R13 ðzÞ. In this set, we seek to cancel out these sidelobes with the ACF of suitably modified length 7 Barker code. We use B7 ðz2 Þ such that the resultant ACF, R7 ðz2 Þ cancels the two residual sidelobes of R13 ðzÞ but introduces two new negative sidelobes. We observe that these negative sidelobes can easily be cancelled out by the ACF of the length 5 Barker code when one zero is inserted between each of its elements. Using similar notations as before, this modified length 5 Barker code and the corresponding ACF are represented as B5 ðz2 Þ and R5 ðz2 Þ, respectively.
ARTICLE IN PRESS A.T. Fam, I. Sarkar / Signal Processing 88 (2008) 307–314
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Table 4 Transmission scheme for ICC-II
f1 f2 f3
t1
t2
t3
t4
t5
t6
t7
t8
t9
1 0 0
0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1
t10
t11 t12 t13 t14 t15
0 0 1 0 0 1
0 1 0
0 0 1
0 1 0
t16
t17 t18
0 0 0 0 1 0 1 0 1
t19 t20 t21
0 0 1 0 0 1
0 1 0
t22
t23 t24 t25
0 0 0 0 1 0 1 0 1
0 1 0
0 0 0 0
1 1 0
t26
1 0 0 1 0 0
Table 5 Aligned autocorrelation profiles of codes used in ICC set III R13 ðzÞ R11 ðzÞ R7 ðz2 Þ R5 ðz2 Þ
1
0
1
0
1 1 0
0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
0 0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
When these ACFs, i.e. R13 ðzÞ, R11 ðzÞ, R7 ðz2 Þ and R5 ðz2 Þ are summed together, the sidelobes cancel each other as shown in Table 5. Next, we propose a TDM and interlaced scheme for gap-free transmission of this code set. For this code set, we transmit B13 ðzÞ and B11 ðzÞ back to back. Then we send B7 ðz2 Þ and B5 ðz2 Þ interlaced with each other. We observe that this results in a gap just before the last bit. To avoid the gap, we flip this entire transmission scheme and the first bit of the flipped scheme is made to occupy the aforementioned gap. The final transmission scheme is given by TðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ þ z24 B7 ðz2 Þ þ z25 B5 ðz2 Þ þ z35 B7 ðz2 Þ þ z38 B5 ðz2 Þ þ z48 B11 ðzÞ þ z59 B13 ðzÞ.
ð9Þ
The long code that results can be shortened significantly by using a hybrid transmission scheme. We observe that the TDM transmission scheme given by (9) can be broken into three parallel components each of length 24. The parallel components are given by T P1 ðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ,
(10)
T P2 ðzÞ ¼ z24 B7 ðz2 Þ þ z1 B5 ðz2 Þ þ z11 B7 ðz2 Þ þ z14 B5 ðz2 Þ, T P3 ðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ.
ð11Þ (12)
The hybrid transmission scheme is shown in Table 6. Since each constituent code is used twice in this scheme, a mainlobe of height 72 is achieved.
0 0 0 0
13 11 7 5
0 0 0 0
1 1 0 0
0 0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
1 1 1 1
0 0 0
0
1
0
1
2.4. ICC set IV involving modified B13 ðzÞ; B11 ðzÞ and B2 ðzÞ In this set, we start with B13 ðzÞ. In the time domain the Barker code BN ðzÞ is denoted as bN ðnÞ. It should be noted that the time reversed or flipped version of this code is denoted by bN ðN nÞ while the corresponding z-transform is given by zN BN ðz1 Þ. Starting with b13 ðnÞ and appending a 1 and thirteen 0’s at the beginning, the first constituent code is constructed as c1 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b13 ðnÞ.
(13)
13
Next, the Barker code of length 13 is flipped and thirteen 0’s and a 1 are appended at the end. Thus, the second constituent code looks like c2 ðnÞ ¼ ½b13 ð13 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1.
(14)
13
It is easy to see that c1 ðnÞ and c2 ðnÞ will interlace with each other with the b13 ð13 nÞ of c2 ðnÞ occupying the positions of the thirteen 0’s in c1 ðnÞ. To counter the sidelobes due to these two codes, we employ the Barker codes of lengths 11 and 2 using each of them twice. The length 11 and length 2 codes are modified to produce the following constituent codes: c3 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b11 ðnÞ,
(15)
c4 ðnÞ ¼ ½b11 ð11 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(16)
14
14
ARTICLE IN PRESS A.T. Fam, I. Sarkar / Signal Processing 88 (2008) 307–314
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Table 6 Hybrid transmission scheme for ICC-III t1
t2
t3
t4
t5
f1 f2
1 0
1 0
1 0
1 0
1 0
f3 f4 f5 f6
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
f7 f8
1 0
1 0
1 0
1 0
t6
t7
t8
t9
t10
t11
t12
t13
t14
t15
t16
t17
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 1
0 1
0 1
0 1
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
1 0 0 0
0 0 1 0
1 0 0 0
0 0 1 0
0 0 0 1
0 0 1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 1
0 1
0 1
c5 ðnÞ ¼ c6 ðnÞ ¼ ½1 |fflfflfflfflffl 0 0ffl{zfflfflfflfflffl ffl0} 1,
(17)
11
t20
t21
t22
t23
t24
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
Add a 1 before at least ðN 1Þ zeros preceding one of the code blocks thus yielding
TðzÞ ¼ C 1 ðzÞ þ z1 C 2 ðzÞ þ z28 C 3 ðzÞ þ z30 C 4 ðzÞ ð19Þ
This ICC set uses six different frequencies to produce a mainlobe of height 56. A parallel scheme is also possible for this scheme apart from the sequential scheme described by Eq. (19). The combination of c1 ðnÞ and c2 ðnÞ is completely contained in the first 28 bits of the sequential transmission scheme. On the other hand the combination of c3 ðnÞ, c4 ðnÞ, c5 ðnÞ and c6 ðnÞ is completely contained in the remaining 28 bits of sequential transmission scheme. Hence, the two combination schemes can be transmitted in parallel.
(20)
XðN1Þ
(18)
Also, ci ðnÞ 2 T, 8 i. The ACFs produced by c3 ðnÞ; c4 ðnÞ; c5 ðnÞ and c6 ðnÞ cancel out the sidelobes due to c1 ðnÞ and c2 ðnÞ. All these codes can be interlaced with each other perfectly and the transmission scheme in the z-domain notations is given as
þ z29 C 5 ðzÞ þ z42 C 6 ðzÞ.
t19
0ffl{zfflfflfflfflffl ffl0} bN ðnÞ. c1 ðnÞ ¼ ½1 0|fflfflfflfflffl
which corresponds to C 5 ðzÞ ¼ C 6 ðzÞ ¼ B2 ðz12 Þ.
t18
Add a 1 before at least N 1 zeros preceding the other code and flip the final product. This gives 0ffl{zfflfflfflfflffl ffl0} 1. c2 ðnÞ ¼ ½bN ðN nÞ 0|fflfflfflfflffl
(21)
XðN1Þ
The insertion of 1’s and 1’s should be done only if it is possible to accommodate them in a scheme without gaps.
Eqs. (13)–(16) makes use of this concept. In this example we extend the concept to ICC set III to show how the mainlobe can be enhanced further without using more frequencies. The constituent codes for these sets are formed as follows: c1 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b13 ðnÞ,
(22)
c2 ðnÞ ¼ ½b13 ð13 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(23)
c3 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b11 ðnÞ,
(24)
c4 ðnÞ ¼ ½b11 ð11 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(25)
c5 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b5 ðn " 2Þ,
(26)
64
64
2.5. ICC set V obtained by modifying set III
13
Some ICC sets could be modified to obtain higher mainlobes using the same number of frequencies. Consider the Barker code of length N. If we add to it a 1 or 1 before or after N or more zeros, the resultant ACF will still have unity peak sidelobe. Hence, if a code contains two copies of any code block of length N from T, the following trick can be applied to enhance the mainlobe further.
13
14
where bN ðn " mÞ is a modified version of bN ðnÞ having ðm 1Þ zeros between every two successive
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0ffl{zfflfflfflfflffl ffl0} 1, c6 ðnÞ ¼ ½b5 ð5 nÞ 0 |fflfflfflfflffl
(27)
imum N possible with the given number of frequencies. These open questions are of both theoretical and practical significance.
0ffl{zfflfflfflfflffl ffl0} b7 ðn " 2Þ, c7 ðnÞ ¼ ½1 0 |fflfflfflfflffl
(28)
4. Comparison with orthogonal matrix codes
0ffl{zfflfflfflfflffl ffl0} 1. c8 ðnÞ ¼ ½b7 ð7 nÞ 0 |fflfflfflfflffl
(29)
bits 14
12
12
It can be shown that these codes could be interlaced with each other yielding a gap-free transmission scheme which in the z-domain notations is given by TðzÞ ¼ C 1 ðzÞ þ z1 C 3 ðzÞ þ z2 C 2 ðzÞ þ z26 C 7 ðzÞ þ z27 C 5 ðzÞ þ z28 C 8 ðzÞ þ z29 C 6 ðzÞ þ z54 C 4 ðzÞ.
ð30Þ
Using this ICC set, we can achieve a mainlobe height of 80 using the same eight frequencies used in ICC set III. 3. Frequency diversity efficiency We define the frequency diversity efficiency (FDE) as the ratio of the height of the mainlobe H m and the number of frequencies N f . The FDE is given by Z¼
Hm . Nf
(31)
The FDE for the different schemes proposed in this paper are summarized in Table 7. Iterative application of the idea mentioned for ICC set V would result in higher Z provided it could be done without gaps. Let SN ¼ fC N g be the set of all ICC codes of length N. Let Z^N be the maximum Z achieved by any code in SN . It is our conjecture that lim Z^ N ¼ 1.
(32)
N!1
It is an important topic of further research to prove or disprove this conjecture. Also, given a certain number of frequencies, it is important to find the maximum achievable Z or equivalently, the maxTable 7 FDE for the proposed schemes Scheme
I
II
III
IV
V
Hm Nf Z
26 3 8.67
26 3 8.67
72 8 9
56 6 9.33
80 8 10
Orthogonal matrices could be used to generate codes with no sidelobes. This could be done either with biphase elements, as discussed elegantly in [15], or polyphase elements as in the DFT matrix [5]. In such cases, the matrix elements of a N N matrix are transmitted row-wise using N frequencies resulting in a mainlobe of N 2 . The conjugate transpose or the Hermitian of the transmitted matrix forms its matched filter in such cases. This approach is more susceptible to frequency selective fading effects since each inner-product contribution to the output is affected by all frequencies. It is also more vulnerable to jamming at any of the frequencies. The codes proposed in this paper results in more graceful degradation in the presence of fading and jamming effects since each element code has unity peak sidelobe and depends only on one frequency. Also, since coherence is easier to maintain at each frequency but more difficult to maintain simultaneously at all frequencies, the ICC codes are more resistant to partial loss in coherence. The five examples of ICC sets in this paper have larger Z’s than the nearest matrix codes greater or equal in length. 5. Conclusions In this work, a new class of interlaced complementary codes is introduced. The constituent codes are composed of elements from D ¼ fþ1; 1; 0g and have a peak sidelobe magnitude of unity. Some of the individual components have gaps of zeros, but when interlaced with other components, the gaps disappear. The unity peak sidelobe level of the constituent codes is the least possible and results in superior performance in the presence of frequency selective fading. The frequency diversity efficiency is measured by Z which is defined as the code length per frequency. Some of the proposed codes could be particularly suitable in countering terrain bounce jamming and eclipsing problems in addition to the regular complementary code applications. A possible countermeasure to terrain bounce jamming is to utilize the longer delay of the bounced return than the direct one. By searching in a small time window preceding the bounced return, the much smaller
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direct return could be identified. For this approach to work, zero sidelobes are required to avoid confusion between the sidelobes of the bounced return and the mainlobe of the direct one. Also, resistance to fading is required to avoid any significant re-appearance of the sidelobes. Both these requirements are satisfied by the proposed class of codes. The complete characterization of these codes is a topic for further research. Acknowledgments The authors would like to thank the anonymous reviewers for their helpful comments and pointing out some important references especially in the area of ternary complementary codes. References [1] M.J.E. Golay, Multislit spectrometry, J. Opt. Soc. Amer. 39 (1949) 437. [2] M.J.E. Golay, Complementary series, IRE Trans. Inform. Theory IT-7 (April 1961) 82–87. [3] C.-C. Tseng, C.L. Liu, Complementary sets of sequences, IEEE Trans. Inform. Theory IT-8 (September 1972) 644–652. [4] R. Sivaswamy, Multiphase complementary codes, IEEE Trans. Inform. Theory IT-24 (5) (March 1973) 214–218.
[5] R.L. Frank, Polyphase complementary codes, IEEE Trans. Inform. Theory IT-26 (6) (October 1980) 641–647. [6] T. Misaridis, J.A. Jensen, Use of modulated excitation signals in medical ultrasound. Part I: basic concepts and expected benefits, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 177–191. [7] T. Misaridis, J.A. Jensen, Use of modulated excitation signals in medical ultrasound. Part II: design and performance for medical imaging applications, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 192–207. [8] R.Y. Chiao, X. Hao, Coded excitation for diagnostic ultrasound: a system developer’s perspective, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 160–170. [9] J.O. Mutton, Advanced pulse compression techniques, in: Proceedings of the IEEE NAECON, May 1975, pp.141–148. [10] A. Gavish, A. Lempel, On ternary complementary sequences, IEEE Trans. Inform. Theory 40 (2) (March 1994) 522–526. [11] R. Craigen, C. Koukouvinos, A theory of ternary complementary pairs, J. Combin. Theory Ser. A 96 (2001) 358–375. [12] M. Gysin, J. Seberry, On ternary complementary pairs, Austral. J. Combin. 23 (2001) 153–170. [13] R. Craigen, Boolean and ternary complementary pairs, J. Combin.Theory Ser. A 104 (2003) 1–16. [14] R. Craigen, S. Georgiou, W. Gibson, C. Koukouvinos, Further explorations into ternary complementary pairs, J. Combin. Theory Ser. A 113 (2006) 952–965. [15] F.F. Kretschemer Jr., K. Gerlach, Low sidelobe radar waveforms derived from orthogonal matrices, IEEE Trans. Aerospace Electronics Systems 27 (1) (January 1991) 92–101.
Signal Processing 88 (2008) 307–314 www.elsevier.com/locate/sigpro
A new class of interlaced complementary codes based on components with unity peak sidelobes$ Adly T. Fam, Indranil Sarkar Department of Electrical Engineering, The State University of New York at Buffalo, NY 14260, USA Received 22 March 2007; received in revised form 20 July 2007; accepted 3 August 2007 Available online 15 August 2007
Abstract A new class of biphase complementary code sets is proposed. Individual codes in each set have maximum sidelobes of unity magnitude. The individual codes could have gaps of zeros but they are interlaced together without gaps in the final scheme. A number of such codes, as presented in this paper, use Barker codes as building blocks with additional elements from fþ1; 1; 0g. The main drawback of regular complementary codes longer than 4 is that they have peak sidelobe magnitude greater than unity. In the presence of frequency selective fading or other factors, inexact sidelobe cancellation results in non-zero sidelobes at the output. This is minimized in the proposed schemes by using codes that have peak sidelobes of unity magnitude as the individual codes in the sets. The proposed codes are attractive due to their increased resistance to the effect of inexact sidelobe cancellation. A figure of merit is proposed to measure the frequency use efficiency of the proposed codes. r 2007 Elsevier B.V. All rights reserved. Keywords: Barker codes; Complementary codes; Pulse compression codes; Ternary complementary codes
1. Introduction Complementary codes or sequences were introduced by Golay [1] for use in radar astronomy. A complementary set is a set of finite sequences whose autocorrelation functions (ACFs), when added together, cancel out the sidelobes of each other. Any complementary set can be used to generate longer ones. The root sequences which are defined directly and not generated from shorter ones are $
Patent pending. USPTO Patent application #11672203.
Corresponding author. Tel.: +1 716 645 2422x2504;
fax: +1 716 645 3656. E-mail addresses: [email protected] (A.T. Fam), [email protected] (I. Sarkar).
also referred to as kernels. Golay [2] came up with kernels of binary complementary pairs of length 2, 10 and 26. Tseng and Liu [3] developed algorithms for recursive generation of complementary sequences. Sivaswamy [4] proposed some polyphase complementary sequences and gave kernels of complementary triplets of length 2 and 3. Frank [5] provided an excellent overview of polyphase complementary codes and proposed additional kernels for them. Biphase and polyphase complementary codes are attractive in radar, sonar and communications applications because they have zero sidelobes. This advantage is compromised in the presence of frequency selective fading. Unequal fading of the individual codes results in inexact cancellation of their sidelobes, thus degrading their performance.
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.08.003
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308
In recent times, complementary codes have been used for pulse compression in ultrasound imaging systems [6,7]. In such applications, the codes are fired sequentially and hence any movement in the tissue being observed results in inexact sidelobe cancellation. This is manifested by artifacts in the images. Chiao and Hao [8] has presented a good overview of the usage of complementary codes in ultrasonic imaging and discussed several implementation issues. A logical prerequisite for a good complementary code set in any application is that the peak sidelobe of the individual codes are as small as possible. Discussions on this issue can be found in [5] as well as in [9]. This is satisfied by the new class of interlaced complementary codes (ICC) introduced next. Let us consider the class of codes that are constructed from a ternary alphabet D ¼ f1; þ1; 0g and have peak sidelobes of unity magnitude. We denote this class as T. The constituent codes of the ICC are members of T. Let {tn } be a code of length N in T. Its aperiodic ACF is given by At ðtÞ ¼
Nt1 X
tn tnþt ,
(1)
n¼0
where 0ptpN 1. The peak sidelobe constraint on the members of T specifies that max jAt ðtÞj ¼ 1. ta0
(2)
Barker codes are the only biphase (constructed from fþ1; 1g) codes that satisfy (2). Let the class of Barker codes be B. Obviously, B T. The constituent codes in the proposed ICC may contain Barker codes or Barker codes modified with additional elements from D. In this work, we present several schemes involving two or more codes from T that form complementary sets. The constituent codes with gaps of zeros are interlaced with other constituent codes to remove such gaps. Different frequencies are used for the different constituent codes resulting in a multifrequency code some components of which must be interlaced with each other. The final scheme could be purely time division multiplexed (TDM) or allow some amount of parallel transmission depending on the structure of the constituent codes. If the purely TDM scheme can be broken down into equal length blocks in a way such that each constituent code is contained only in one particular block, then these
equal length blocks can be transmitted simultaneously. This gives rise to a hybrid transmission scheme using both TDM and simultaneous transmission. The advantages of TDM are low peak to average power ratio resulting in a low probability of interception. Also, a purely TDM scheme requires only one transmitter with frequency hopping capabilities. However, using TDM alone results in longer codes that are more susceptible to multipath and eclipsing problems. The parallel transmission scheme results in shorter codes and hence is better suited to counter these problems. This comes at the cost of a higher peak to average power ratio, higher detectability and increased number of transmitters or a wideband transmitter. A hybrid scheme achieves a trade-off between these two transmission modalities. It should be noted that the proposed codes are related to the ternary complementary pairs (TCP) introduced by Gavish and Lempel [10] and subsequently investigated in details in [11–14]. The ICC sets share with the TCPs the property that the component codes are generated from fþ1; 1; 0g. However, the characteristics that make the ICC sets different are:
There could be more than two component codes in an ICC set. The component codes in an ICC set must have peak sidelobes of unity magnitude. The component codes are interlaced with each other without gaps such that the final transmission scheme is comprised of fþ1; 1g.
Due to these constraints, ICC sets are difficult to find and there are no known general method for their systematic generation. However, in spite of their heuristic nature, the proposed codes could be very important in various applications due to their desirable properties. 2. Interlaced complementary codes In this section we introduce several examples of the proposed ICC. As mentioned before, the constituent codes of the ICCs are constructed from D and hence may contain zeros. However, the final transmission scheme is free of zeros and contains only elements from fþ1; 1g. In these examples, we use Barker codes as building blocks with additional elements from D to obtain the constituent codes belonging to T. However, using Barker codes is not
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The ACF R11 ðzÞ for the Barker code of length 11 is also shown in Table 1. R11 ðzÞ has 10 sidelobes on each side of the mainlobe of alternating 1’s and 0’s. Furthermore, it can also be observed that when they are aligned, 1 sidelobes of R11 ðzÞ coincides exactly with þ1 sidelobes of R13 ðzÞ. Thus, when added together, the sidelobes cancel out with the exception of one sidelobe each at either end of R13 ðzÞ. We also note that the Barker code of length 2, f1; 1g, produces two sidelobes of height 1. The code of length 2, with 11 zeros inserted between its two elements, is represented by B2 ðz12 Þ. The corresponding ACF R2 ðz12 Þ can be used to cancel out the two remaining sidelobes of R13 ðzÞ that R11 ðzÞ did not cancel. This is clearly shown in Table 1. It is easy to see that the transmission scheme should be such that the 11 gaps between the elements of B2 ðz12 Þ should be filled by code bits from some other code so that the final transmission scheme is free of gaps. We also observe that these gaps could be filled by placing B11 ðzÞ as it is in these gaps. The transmission scheme is therefore given by
a necessary condition for constructing codes belonging to T and consequently, the class of ICC. Let the z-transform of a Barker code of length N be denoted by BN ðzÞ. If ðk 1Þ zeros are placed between every two elements of the code, then the modified code is represented by BN ðzk Þ. The ACF of BN ðzÞ is given by RN ðzÞ ¼ BN ðzÞBN ðz1 Þ
(3)
from which we get RN ðzk Þ ¼ BN ðzk ÞBN ðzk Þ.
309
(4)
The following steps are involved in the construction of valid interlaced codes free of any gaps. The challenge is to have the individual codes with unity peak sidelobes fit together such that the final scheme is composed of fþ1; 1g without any gaps of zeros. (1) Two or more constituent codes should be chosen from T. (2) The desired number of zeros is inserted between the elements of the constituent codes such that the sidelobes of the ACFs cancel each other out. (3) Finally, the constituent codes have to be interlaced with each other such that there are no zeros in the final transmission scheme. The constituent codes may have to be delayed with respect to each other so that they fit together without any zeros. (4) Each constituent code is transmitted after modulation by a particular frequency. The number of frequencies used is therefore equal to the number of constituent codes. (5) Appropriate delays would also be needed at the receiver end for each constituent code, so that the ACFs (or the output of the matched filters) are aligned properly before addition.
TðzÞ ¼ B13 ðzÞ þ z13 B2 ðz12 Þ þ z14 B11 ðzÞ.
(5)
Thus, in this transmission scheme B13 ðzÞ is transmitted first. The first bit of B2 ðzÞ is transmitted next followed by the entire B11 ðzÞ. Finally the remaining bit of B2 ðz12 Þ is transmitted resulting in transmission of all the codes in an interlaced fashion and free of any gaps. Table 2 depicts the transmission scheme graphically. The columns t1 –t26 depict the time slots whereas f 1 –f 3 depict the three different frequencies used to modulate B13 ðzÞ, B2 ðz12 Þ and B11 ðzÞ, respectively. Since no gaps (or 0’s) are allowed in the transmission scheme and only one code bit can be transmitted in a given time slot, each column in the table should have only a single 1 or 1. Also, no columns can have all 0’s. Similar tables can be constructed for all the transmission schemes described in this paper. There are some obvious alternative schemes to the one mentioned above. Each of the B13 ðzÞ, B11 ðzÞ and B2 ðz12 Þ as well as the entire transmission scheme can be flipped to produce 16 obvious alternative
2.1. ICC set I involving B13 ðzÞ; B11 ðzÞ and B2 ðz12 Þ The ACF R13 ðzÞ of the Barker code of length 13 is shown in Table 1. It can be seen that on each side of the mainlobe, there are 12 sidelobes of alternating þ1’s and 0’s. Table 1 Aligned autocorrelation profiles of codes used in ICC set I R13 ðzÞ R11 ðzÞ R2 ðz12 Þ
1
0
1
0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
13 11 2
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0 0 0
1 1 0
0
1
0
1
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310 Table 2 Transmission scheme for ICC-I t2
t6
t7
t8
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0
1 1 1 0 0 0 0 0 0
t1 f1 f2 f3
t3
t4
t5
t9
t10 t11
t12 t13
1 1 0 0 0 0
Table 3 Hybrid transmission scheme for ICC-I t1 f1 f2 f3
t2
t3
t4
t5
t6 t7 t8
t9
t10 t11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
t12 t13
1 1 0 0 1 1
1 1 0
schemes. It is also to be noted that B13 ðzÞ comprises the first 13 bits of the scheme. The combination of B11 ðzÞ and B2 ðz12 Þ comprise the remaining 13 bits. Hence, B13 ðzÞ and the combination of B11 ðzÞ and B2 ðz12 Þ can be transmitted simultaneously to reduce the overall length of the complementary code. This transmission scheme contains both simultaneous and TDM transmissions. This hybrid scheme is depicted in Table 3. The parallel components for this case are given as T P1 ðzÞ ¼ B13 ðzÞ,
(6)
T P2 ðzÞ ¼ B2 ðz12 Þ þ z1 B11 ðzÞ.
(7)
All ICC sets are constructed in a way that they can always be transmitted in a sequential bit by bit fashion. However, some amount of parallel transmission is possible in certain code sets depending on their structure. ICC set I is an example of such a code. 2.2. ICC set II involving B13 ðz2 Þ; B11 ðz2 Þ and B2 ðz24 Þ If we put one gap in between the elements of B13 ðzÞ and B11 ðzÞ, we come up with two modified Barker codes B13 ðz2 Þ and B11 ðz2 Þ, respectively. The corresponding ACFs are now R13 ðz2 Þ and R11 ðz2 Þ, respectively. When lined up with one another, all the sidelobes will cancel each other except the ones at the extreme ends of R13 ðz2 Þ. We note that in this case, we will require R2 ðz24 Þ to cancel these two remaining sidelobes. Hence, we need to transmit B2 ðz24 Þ in the transmission scheme.
t14 t15
1 0 0 1 0 0
t16
t17
t18
0 0 0 0 0 0 0 0 1 1 1 1
t19 t20 t21 0 0 1
0 0 1
t22 t23 t24
0 0 0 0 1 1
0 0 1
t25 t26
0 0 0 0 1 1
0 1 0
In the transmission scheme, 11 of the 12 gaps created in B13 ðz2 Þ are filled by the elements of B11 ðz2 Þ. The one remaining gap just before the last element of B13 ðz2 Þ needs to be filled somehow using the code bits of B2 ðz24 Þ. We observe that if we place the first bit of B2 ðz24 Þ just before the first element of B13 ðz2 Þ, the last bit of B2 ðz24 Þ occupies the gap to be filled. Thus, the gap-free transmission scheme is given by TðzÞ ¼ B2 ðz24 Þ þ z1 B13 ðz2 Þ þ z2 B11 ðz2 Þ.
(8)
Table 4 shows the graphical representation of the transmission scheme. Once again, t1 –t26 represent the time slots and f 1 –f 3 represent the frequencies used to modulate B2 ðz24 Þ, B13 ðz2 Þ and B11 ðz2 Þ, respectively. Once again, reversal of the transmission order of the individual codes or the entire scheme produce several equivalent ICC sets. Unlike the ICC set I, simultaneous transmission is not possible for this set since the purely TDM scheme shown in Table 4 cannot be divided into equal length blocks containing complete constituent codes. 2.3. ICC set III involving B13 ðzÞ; B11 ðzÞ; B7 ðz2 Þ and B5 ðz2 Þ This set is based on Barker codes of length 13, 11, 7 and 5. We have shown the ACFs R13 ðzÞ and R11 ðzÞ in Table 1. When these two ACFs are added together, the sidelobes cancel each other except the two positive sidelobes at the two extremes of R13 ðzÞ. In this set, we seek to cancel out these sidelobes with the ACF of suitably modified length 7 Barker code. We use B7 ðz2 Þ such that the resultant ACF, R7 ðz2 Þ cancels the two residual sidelobes of R13 ðzÞ but introduces two new negative sidelobes. We observe that these negative sidelobes can easily be cancelled out by the ACF of the length 5 Barker code when one zero is inserted between each of its elements. Using similar notations as before, this modified length 5 Barker code and the corresponding ACF are represented as B5 ðz2 Þ and R5 ðz2 Þ, respectively.
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Table 4 Transmission scheme for ICC-II
f1 f2 f3
t1
t2
t3
t4
t5
t6
t7
t8
t9
1 0 0
0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1
t10
t11 t12 t13 t14 t15
0 0 1 0 0 1
0 1 0
0 0 1
0 1 0
t16
t17 t18
0 0 0 0 1 0 1 0 1
t19 t20 t21
0 0 1 0 0 1
0 1 0
t22
t23 t24 t25
0 0 0 0 1 0 1 0 1
0 1 0
0 0 0 0
1 1 0
t26
1 0 0 1 0 0
Table 5 Aligned autocorrelation profiles of codes used in ICC set III R13 ðzÞ R11 ðzÞ R7 ðz2 Þ R5 ðz2 Þ
1
0
1
0
1 1 0
0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
0 0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
When these ACFs, i.e. R13 ðzÞ, R11 ðzÞ, R7 ðz2 Þ and R5 ðz2 Þ are summed together, the sidelobes cancel each other as shown in Table 5. Next, we propose a TDM and interlaced scheme for gap-free transmission of this code set. For this code set, we transmit B13 ðzÞ and B11 ðzÞ back to back. Then we send B7 ðz2 Þ and B5 ðz2 Þ interlaced with each other. We observe that this results in a gap just before the last bit. To avoid the gap, we flip this entire transmission scheme and the first bit of the flipped scheme is made to occupy the aforementioned gap. The final transmission scheme is given by TðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ þ z24 B7 ðz2 Þ þ z25 B5 ðz2 Þ þ z35 B7 ðz2 Þ þ z38 B5 ðz2 Þ þ z48 B11 ðzÞ þ z59 B13 ðzÞ.
ð9Þ
The long code that results can be shortened significantly by using a hybrid transmission scheme. We observe that the TDM transmission scheme given by (9) can be broken into three parallel components each of length 24. The parallel components are given by T P1 ðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ,
(10)
T P2 ðzÞ ¼ z24 B7 ðz2 Þ þ z1 B5 ðz2 Þ þ z11 B7 ðz2 Þ þ z14 B5 ðz2 Þ, T P3 ðzÞ ¼ B13 ðzÞ þ z13 B11 ðzÞ.
ð11Þ (12)
The hybrid transmission scheme is shown in Table 6. Since each constituent code is used twice in this scheme, a mainlobe of height 72 is achieved.
0 0 0 0
13 11 7 5
0 0 0 0
1 1 0 0
0 0 0 0
1 1 1 1
0 0 0 0
1 1 0 0
1 1 1 1
0 0 0
0
1
0
1
2.4. ICC set IV involving modified B13 ðzÞ; B11 ðzÞ and B2 ðzÞ In this set, we start with B13 ðzÞ. In the time domain the Barker code BN ðzÞ is denoted as bN ðnÞ. It should be noted that the time reversed or flipped version of this code is denoted by bN ðN nÞ while the corresponding z-transform is given by zN BN ðz1 Þ. Starting with b13 ðnÞ and appending a 1 and thirteen 0’s at the beginning, the first constituent code is constructed as c1 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b13 ðnÞ.
(13)
13
Next, the Barker code of length 13 is flipped and thirteen 0’s and a 1 are appended at the end. Thus, the second constituent code looks like c2 ðnÞ ¼ ½b13 ð13 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1.
(14)
13
It is easy to see that c1 ðnÞ and c2 ðnÞ will interlace with each other with the b13 ð13 nÞ of c2 ðnÞ occupying the positions of the thirteen 0’s in c1 ðnÞ. To counter the sidelobes due to these two codes, we employ the Barker codes of lengths 11 and 2 using each of them twice. The length 11 and length 2 codes are modified to produce the following constituent codes: c3 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b11 ðnÞ,
(15)
c4 ðnÞ ¼ ½b11 ð11 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(16)
14
14
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312
Table 6 Hybrid transmission scheme for ICC-III t1
t2
t3
t4
t5
f1 f2
1 0
1 0
1 0
1 0
1 0
f3 f4 f5 f6
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
f7 f8
1 0
1 0
1 0
1 0
t6
t7
t8
t9
t10
t11
t12
t13
t14
t15
t16
t17
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 1
0 1
0 1
0 1
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
1 0 0 0
0 1 0 0
1 0 0 0
0 0 1 0
1 0 0 0
0 0 1 0
0 0 0 1
0 0 1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
1 0
0 1
0 1
0 1
c5 ðnÞ ¼ c6 ðnÞ ¼ ½1 |fflfflfflfflffl 0 0ffl{zfflfflfflfflffl ffl0} 1,
(17)
11
t20
t21
t22
t23
t24
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 0 0 1
0 0 1 0
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
Add a 1 before at least ðN 1Þ zeros preceding one of the code blocks thus yielding
TðzÞ ¼ C 1 ðzÞ þ z1 C 2 ðzÞ þ z28 C 3 ðzÞ þ z30 C 4 ðzÞ ð19Þ
This ICC set uses six different frequencies to produce a mainlobe of height 56. A parallel scheme is also possible for this scheme apart from the sequential scheme described by Eq. (19). The combination of c1 ðnÞ and c2 ðnÞ is completely contained in the first 28 bits of the sequential transmission scheme. On the other hand the combination of c3 ðnÞ, c4 ðnÞ, c5 ðnÞ and c6 ðnÞ is completely contained in the remaining 28 bits of sequential transmission scheme. Hence, the two combination schemes can be transmitted in parallel.
(20)
XðN1Þ
(18)
Also, ci ðnÞ 2 T, 8 i. The ACFs produced by c3 ðnÞ; c4 ðnÞ; c5 ðnÞ and c6 ðnÞ cancel out the sidelobes due to c1 ðnÞ and c2 ðnÞ. All these codes can be interlaced with each other perfectly and the transmission scheme in the z-domain notations is given as
þ z29 C 5 ðzÞ þ z42 C 6 ðzÞ.
t19
0ffl{zfflfflfflfflffl ffl0} bN ðnÞ. c1 ðnÞ ¼ ½1 0|fflfflfflfflffl
which corresponds to C 5 ðzÞ ¼ C 6 ðzÞ ¼ B2 ðz12 Þ.
t18
Add a 1 before at least N 1 zeros preceding the other code and flip the final product. This gives 0ffl{zfflfflfflfflffl ffl0} 1. c2 ðnÞ ¼ ½bN ðN nÞ 0|fflfflfflfflffl
(21)
XðN1Þ
The insertion of 1’s and 1’s should be done only if it is possible to accommodate them in a scheme without gaps.
Eqs. (13)–(16) makes use of this concept. In this example we extend the concept to ICC set III to show how the mainlobe can be enhanced further without using more frequencies. The constituent codes for these sets are formed as follows: c1 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b13 ðnÞ,
(22)
c2 ðnÞ ¼ ½b13 ð13 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(23)
c3 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b11 ðnÞ,
(24)
c4 ðnÞ ¼ ½b11 ð11 nÞ 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} 1,
(25)
c5 ðnÞ ¼ ½1 0|fflfflfflfflffl 0ffl{zfflfflfflfflffl ffl0} b5 ðn " 2Þ,
(26)
64
64
2.5. ICC set V obtained by modifying set III
13
Some ICC sets could be modified to obtain higher mainlobes using the same number of frequencies. Consider the Barker code of length N. If we add to it a 1 or 1 before or after N or more zeros, the resultant ACF will still have unity peak sidelobe. Hence, if a code contains two copies of any code block of length N from T, the following trick can be applied to enhance the mainlobe further.
13
14
where bN ðn " mÞ is a modified version of bN ðnÞ having ðm 1Þ zeros between every two successive
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313
0ffl{zfflfflfflfflffl ffl0} 1, c6 ðnÞ ¼ ½b5 ð5 nÞ 0 |fflfflfflfflffl
(27)
imum N possible with the given number of frequencies. These open questions are of both theoretical and practical significance.
0ffl{zfflfflfflfflffl ffl0} b7 ðn " 2Þ, c7 ðnÞ ¼ ½1 0 |fflfflfflfflffl
(28)
4. Comparison with orthogonal matrix codes
0ffl{zfflfflfflfflffl ffl0} 1. c8 ðnÞ ¼ ½b7 ð7 nÞ 0 |fflfflfflfflffl
(29)
bits 14
12
12
It can be shown that these codes could be interlaced with each other yielding a gap-free transmission scheme which in the z-domain notations is given by TðzÞ ¼ C 1 ðzÞ þ z1 C 3 ðzÞ þ z2 C 2 ðzÞ þ z26 C 7 ðzÞ þ z27 C 5 ðzÞ þ z28 C 8 ðzÞ þ z29 C 6 ðzÞ þ z54 C 4 ðzÞ.
ð30Þ
Using this ICC set, we can achieve a mainlobe height of 80 using the same eight frequencies used in ICC set III. 3. Frequency diversity efficiency We define the frequency diversity efficiency (FDE) as the ratio of the height of the mainlobe H m and the number of frequencies N f . The FDE is given by Z¼
Hm . Nf
(31)
The FDE for the different schemes proposed in this paper are summarized in Table 7. Iterative application of the idea mentioned for ICC set V would result in higher Z provided it could be done without gaps. Let SN ¼ fC N g be the set of all ICC codes of length N. Let Z^N be the maximum Z achieved by any code in SN . It is our conjecture that lim Z^ N ¼ 1.
(32)
N!1
It is an important topic of further research to prove or disprove this conjecture. Also, given a certain number of frequencies, it is important to find the maximum achievable Z or equivalently, the maxTable 7 FDE for the proposed schemes Scheme
I
II
III
IV
V
Hm Nf Z
26 3 8.67
26 3 8.67
72 8 9
56 6 9.33
80 8 10
Orthogonal matrices could be used to generate codes with no sidelobes. This could be done either with biphase elements, as discussed elegantly in [15], or polyphase elements as in the DFT matrix [5]. In such cases, the matrix elements of a N N matrix are transmitted row-wise using N frequencies resulting in a mainlobe of N 2 . The conjugate transpose or the Hermitian of the transmitted matrix forms its matched filter in such cases. This approach is more susceptible to frequency selective fading effects since each inner-product contribution to the output is affected by all frequencies. It is also more vulnerable to jamming at any of the frequencies. The codes proposed in this paper results in more graceful degradation in the presence of fading and jamming effects since each element code has unity peak sidelobe and depends only on one frequency. Also, since coherence is easier to maintain at each frequency but more difficult to maintain simultaneously at all frequencies, the ICC codes are more resistant to partial loss in coherence. The five examples of ICC sets in this paper have larger Z’s than the nearest matrix codes greater or equal in length. 5. Conclusions In this work, a new class of interlaced complementary codes is introduced. The constituent codes are composed of elements from D ¼ fþ1; 1; 0g and have a peak sidelobe magnitude of unity. Some of the individual components have gaps of zeros, but when interlaced with other components, the gaps disappear. The unity peak sidelobe level of the constituent codes is the least possible and results in superior performance in the presence of frequency selective fading. The frequency diversity efficiency is measured by Z which is defined as the code length per frequency. Some of the proposed codes could be particularly suitable in countering terrain bounce jamming and eclipsing problems in addition to the regular complementary code applications. A possible countermeasure to terrain bounce jamming is to utilize the longer delay of the bounced return than the direct one. By searching in a small time window preceding the bounced return, the much smaller
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direct return could be identified. For this approach to work, zero sidelobes are required to avoid confusion between the sidelobes of the bounced return and the mainlobe of the direct one. Also, resistance to fading is required to avoid any significant re-appearance of the sidelobes. Both these requirements are satisfied by the proposed class of codes. The complete characterization of these codes is a topic for further research. Acknowledgments The authors would like to thank the anonymous reviewers for their helpful comments and pointing out some important references especially in the area of ternary complementary codes. References [1] M.J.E. Golay, Multislit spectrometry, J. Opt. Soc. Amer. 39 (1949) 437. [2] M.J.E. Golay, Complementary series, IRE Trans. Inform. Theory IT-7 (April 1961) 82–87. [3] C.-C. Tseng, C.L. Liu, Complementary sets of sequences, IEEE Trans. Inform. Theory IT-8 (September 1972) 644–652. [4] R. Sivaswamy, Multiphase complementary codes, IEEE Trans. Inform. Theory IT-24 (5) (March 1973) 214–218.
[5] R.L. Frank, Polyphase complementary codes, IEEE Trans. Inform. Theory IT-26 (6) (October 1980) 641–647. [6] T. Misaridis, J.A. Jensen, Use of modulated excitation signals in medical ultrasound. Part I: basic concepts and expected benefits, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 177–191. [7] T. Misaridis, J.A. Jensen, Use of modulated excitation signals in medical ultrasound. Part II: design and performance for medical imaging applications, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 192–207. [8] R.Y. Chiao, X. Hao, Coded excitation for diagnostic ultrasound: a system developer’s perspective, IEEE Trans. Ultrasonics Ferroelectrics Frequency Control 52 (2) (February 2005) 160–170. [9] J.O. Mutton, Advanced pulse compression techniques, in: Proceedings of the IEEE NAECON, May 1975, pp.141–148. [10] A. Gavish, A. Lempel, On ternary complementary sequences, IEEE Trans. Inform. Theory 40 (2) (March 1994) 522–526. [11] R. Craigen, C. Koukouvinos, A theory of ternary complementary pairs, J. Combin. Theory Ser. A 96 (2001) 358–375. [12] M. Gysin, J. Seberry, On ternary complementary pairs, Austral. J. Combin. 23 (2001) 153–170. [13] R. Craigen, Boolean and ternary complementary pairs, J. Combin.Theory Ser. A 104 (2003) 1–16. [14] R. Craigen, S. Georgiou, W. Gibson, C. Koukouvinos, Further explorations into ternary complementary pairs, J. Combin. Theory Ser. A 113 (2006) 952–965. [15] F.F. Kretschemer Jr., K. Gerlach, Low sidelobe radar waveforms derived from orthogonal matrices, IEEE Trans. Aerospace Electronics Systems 27 (1) (January 1991) 92–101.