Bi, J.; Rohling, H.: Complementary Binary Code Design based on MismatchedFilter. IEEE Transaction on Aerospace and Electronic Systems, Vol.48, No.2,January 2012.

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Complementary Binary Code Design based onMismatched Filter

Complementary code (CC) pairs have the interesting

characteristic that the superposition of the two related

autocorrelation functions (ACFs) has zero sidelobes in range

direction. This is an important property for radar applications

to avoid range sidelobes. However, CC pairs exist only for

limited codeword lengths. Therefore, the general idea of CC

design is extended here by applying a mismatched filter (MMF)

procedure to the binary phase codeword pairs. There are two

objectives considered. The first objective is to design MMF

impulse responses to fulfill the zero sidelobe property. The second

objective is to find binary phase codeword pairs in cooperation

with the MMFs which have a high gain in signal-to-noise ratio

(SNR). It is shown that for all codeword lengths (even where the

classic CC pair does not exist) there exist binary phase codeword

pairs and the related MMF impulse response coefficients which

also have the zero sidelobe property. Furthermore, these codeword

pairs have high gains in SNR which are nearly the same as for the

classic matched filter technique.

I. INTRODUCTION

The mismatched filter (MMF) design procedurecan be considered as an extension of the classicmatched filter (MF) technique which has been appliedsuccessfully to periodic [1] and to aperiodic binarycodes [2, 3].A pair of aperiodic binary phase coded signals s1

and s2 is called a complementary code (CC) pair if thesuperimposed autocorrelation function (ACF) of thetwo codes has zero sidelobes [4] in range direction.Considering the ACF of an aperiodic signal meansimplicitly that the MF procedure is assumed andapplied in the receiver. The important zero sidelobeproperty of CC pairs can be described analytically inthe following equation.

A1(i) +A2(i) =

1Xn=¡1

s1(n) ¢ s1(n+ i)+1X

n=¡1s2(n) ¢ s2(n+ i)

=

½2N, i = 0

0, i 6= 0(1)

where A1,2(i) is the ACF of the signal s1,2 = fs1,2(n) j n= 0, : : : ,N ¡1g.

Manuscript received November 26, 2010; revised March 14, 2011;

released for publication July 5, 2011.

IEEE Log No. T-AES/48/2/943857.

Refereeing of this contribution was handled by C. Baker.

0018-9251/12/$26.00 c° 2012 IEEE

Equation (1) shows that any sidelobe interferences

can be avoided in stationary radar measurements and

radar applications where complementary binary phase

coded transmit signals are considered and the MF

procedure is applied in the receiver for each codeword

separately. The considered situation is demonstrated in

Fig. 1 for a CC pair example with a codeword length

of N = 10.

However the CC property of (1) is only fulfilled

for a very limited number of codeword lengths N.

Complementary binary code pairs are known for

codeword lengths N of the form:

N = 2a10b26c

where a, b, and c are nonnegative integer values [5].

In fact, for codeword lengths N · 100, only those CCpairs of length N = 1, 2, 4, 8, 10, 16, 20, 26, 32, 40,

52, 64, 80, 100 have been found [1].

To extend the theory of CCs in order to cover all

codeword lengths N, an alternative receiver structure

is considered in this paper. This means that the

application of the MF in the receiver is replaced by

an MMF design procedure for all codeword lengths N,

even where no CC pairs exist.

For an arbitrary binary phase codeword pair, two

digital MMF impulse responses, denoted by h1 and

h2, are calculated to have the same zero sidelobe

property as for classic CC pairs. This means that the

resulting cross-correlation functions (CCFs) will be

superimposed and should show zero sidelobes as

required for classic CC theory. These requirements

are described analytically by (2).

C1(i) +C2(i) =

1Xn=¡1

s1(n) ¢ h1(n+ i)+1X

n=¡1s2(n) ¢ h2(n+ i)

=

½2N, i= 0

0, i 6= 0(2)

where C1,2(i) is the CCF of the code s1,2 and the

related MMF impulse response h1,2 = fh1,2(n) j n=0, : : : ,N ¡ 1g. The MMF impulse response is requiredto have the same length N as the binary phase

code.

In contrast to the MF procedure and the resulting

ACF, the filter coefficients in an MMF procedure are

not necessarily binary and the CCF is not necessarily

symmetric. The MMF procedure offers more freedom

in the receive filter design and makes it possible to

design binary phase codeword pairs to have the same

zero sidelobe property as CC pairs (see (2)) for any

arbitrary length N.

II. MMF DESIGN PROCEDURE AND ZERO SIDELOBEPROPERTY

In this section a pair of aperiodic binary phase

coded signals of length N is considered where a

CORRESPONDENCE 1793

Fig. 1. ACF for each code of the CC pair and superposition of both for codeword length of N = 10.

classic CC pair does not exist. For an arbitrary pair of

binary phase codewords s1 and s2, the correspondingMMF impulse responses h1 and h2 are calculated

to fulfill the zero sidelobe condition described in

(2). It is required that all range sidelobes of the

superimposed CCF should be zero.

The binary phase codewords s1 and s2 as well as

the MMF impulse responses h1 and h2 are representedin a vector form of length N:

s1 = [s1(0),s1(1),s1(2), : : : ,s1(N ¡ 1)]s2 = [s2(0),s2(1),s2(2), : : : ,s2(N ¡ 1)]h1 = [h1(0),h1(1),h1(2), : : : ,h1(N ¡ 1)]h2 = [h2(0),h2(1),h2(2), : : : ,h2(N ¡ 1)]:

Equation (2) can be formulated as a system with

linear equations of dimension (2N ¡ 1)£ (2N):S ¢h= d (3)

where the vector h= [h1,h2]T contains the coefficients

of the two MMF impulse responses h1 and h2 in acascaded way. The vector

d= [0, : : : ,0,2N,0, : : : ,0]T (4)

denotes the desired filter output which describes the

zero sidelobe property.

For these linear equations the signal matrix S hasthe form of two adjoined Toeplitz matrices, as shown

in (5).

S= [S1,S2]

with S1,2 =

266666666666666666664

s1,2(N ¡ 1) 0 0 ¢ ¢ ¢ 0

s1,2(N ¡ 2) s1,2(N ¡ 1) 0 ¢ ¢ ¢ 0

s1,2(N ¡ 3) s1,2(N ¡ 2) s1,2(N ¡ 1) ¢ ¢ ¢ 0

.... . .

s1,2(0) s1,2(1) s1,2(2) ¢ ¢ ¢ s1,2(N ¡ 1)0 s1,2(0) s1,2(1) ¢ ¢ ¢ s1,2(N ¡ 2)0 0 s1,2(0) ¢ ¢ ¢ s1,2(N ¡ 3)...

. . .

0 0 0 ¢ ¢ ¢ s1,2(0)

377777777777777777775

:(5)

The maximum rank of this signal matrix S is2N ¡ 1

rankS· 2N ¡ 1: (6)

It has been validated by computer search that thereare many codeword pairs s1 and s2 for any arbitrarycodeword length N which have a signal matrix S ofmaximum rank 2N ¡ 1. For these codeword pairs,the signal matrix S has 2N ¡1 linear independentcolumns and there are 2N MMF impulse responsecoefficients. Therefore the MMF impulse responsecoefficients are linear dependent on a single parameter® 2R and are solved as follows:

h= ® ¢ a+b: (7)

The vector a is a single solution of the homogeneoussystem of linear equations (8) and b is a singlesolution of the system of linear equations (3).

S ¢ a= 0 (8)

S ¢b= d: (9)

Both vectors a and b are of dimension 2N and aredependent on the signal matrix S and the desiredMMF output d. The above equations have been solvedby using the Gauss-Jordan elimination technique.For a single codeword pair, all resulting MMF

impulse responses h of (7) have the zero sidelobeproperty for each value of parameter ®. This kind offreedom is used for an additional optimization stepwhere the receive signals are superimposed by someadditive noise. For a given codeword pair the

1794 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 2 APRIL 2012

Fig. 2. MMF coefficients for codeword length N = 14.

parameter ® is used to maximize the gain in

signal-to-noise ratio (SNR) for the MMF procedure,

denoted by SNRMMF in the following. It is calculated

in (10), for a binary phase coded input signal in the

presence of independent and identically distributed

(IID) white Gaussian noise.

SNRMMF =(2N)2PN¡1

n=0 h21(n) +

PN¡1n=0 h

22(n)

=(2N)2P2N¡1

j=0 (®a(j) + b(j))2

(10)

where a(j) and b(j) are the jth element of vectors aand b.Maximizing SNRMMF is equivalent to minimizing

the MMF energy E:

E =

N¡1Xn=0

h21(n) +

N¡1Xn=0

h22(n) =

2N¡1Xj=0

(®a(j)+ b(j))2: (11)

The optimal real valued ® is obtained, if the filter

energy E is minimized.

@E

@®= 0

! ®=¡P2N¡1j=0 a(j)b(j)P2N¡1j=0 a2(j)

:

(12)

The MMF impulse responses h1 and h2 for a

binary phase codeword pair of length N are finally

calculated, by inserting (12) into (7).

Example: As an example, an arbitrary binary

phase codeword pair of length N = 14 is considered

and the corresponding MMF impulse responses

have been calculated in accordance with the analysis

above (see Fig. 2). The superposition of the CCFs

fulfills the zero sidelobes property as shown in

Fig. 3.

CORRESPONDENCE 1795

TABLE I

CC Pairs for MMF Design with Maximum Normalized Gain G in SNR

Code Length G (%) Code Pair Example

4 100

5 73.64 [1 ¡1 1 1 1][1 1 ¡1 1 ¡1]

6 82.04 [1 ¡1 ¡1 1 1 1][1 1 1 ¡1 1 ¡1]

7 82.97 [1 ¡1 ¡1 ¡1 1 ¡1 1][1 ¡1 1 1 1 1 ¡1]

8 100

9 84.72 [1 ¡1 ¡1 1 ¡1 1 ¡1 ¡1 ¡1][1 1 1 ¡1 1 ¡1 ¡1 ¡1 1]

10 100

11 86.34 [1 ¡1 ¡1 1 1 1 ¡1 1 ¡1 ¡1 ¡1][1 1 1 ¡1 1 1 1 1 ¡1 ¡1 1]

12 95.15 [1 ¡1 1 1 ¡1 ¡1 ¡1 ¡1 1 ¡1 ¡1 ¡1][1 1 1 ¡1 1 ¡1 ¡1 ¡1 1 1 ¡1 1]

13 90.66 [1 ¡1 ¡1 ¡1 ¡1 1 1 ¡1 1 1 1 ¡1 1][1 ¡1 1 1 1 1 ¡1 1 1 1 1 1 ¡1]

14 92.10 [1 ¡1 ¡1 ¡1 1 ¡1 1 1 ¡1 ¡1 ¡1 ¡1 1 ¡1][1 ¡1 1 1 1 ¡1 ¡1 1 ¡1 ¡1 ¡1 ¡1 ¡1 1]

15 91.23 [1 ¡1 1 1 1 ¡1 1 ¡1 1 ¡1 ¡1 ¡1 ¡1 1 1][1 1 ¡1 ¡1 ¡1 ¡1 1 ¡1 ¡1 1 ¡1 ¡1 ¡11 ¡1]

16 100

17 91.21 [1 ¡1 1 1 ¡1 ¡1 ¡1 1 1 ¡1 1 1 1 1 1 ¡1 1][1 ¡1 1 1 1 1 ¡1 ¡1 ¡1 1 ¡1 1 1 ¡1 ¡1 1 ¡1]

18 93.42 [1 ¡1 ¡1 1 ¡1 ¡1 ¡1 1 1 1 1 1 ¡1 1 ¡1 1 1 1][1 1 1 ¡1 1 ¡1 1 1 ¡1 ¡1 ¡1 1 1 1 ¡1 1 1 ¡1]

19 91.25 [1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1 1 ¡1 1 1 1 ¡1 1 ¡1 1 1 ¡1][1 ¡1 ¡1 1 ¡1 1 ¡1 ¡1 ¡1 1 1 1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1]

24 95.15 [1 1 ¡1 1 1 1 1 ¡1 ¡1 1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1 1 ¡1 1 ¡1 ¡1 ¡1 1][1 ¡1 ¡1 ¡1 1 ¡1 1 1 ¡1 ¡1 ¡1 1 ¡1 1 ¡1 1 1 ¡1 ¡1 ¡1 ¡1 1 ¡1 ¡1]

Fig. 3. CCF for each code of CC pair based on MMF and superposition of both for codeword length of N = 14.

III. RESULTS

There are two objectives considered in this

paper. The first objective is to design MMF impulse

responses for binary phase codeword pairs in order

to fulfill the zero sidelobe property in target range

direction for all codeword length N. In this case

a high flexibility is given in the waveform design

procedure by selecting codeword pairs of different

length N. The second objective is to find binary phase

codeword pairs in cooperation with the resulting MMF

impulse responses which have a high gain in SNR.

The gain in SNR for the MF technique is

calculated as follows:

SNRMF =(2N)2PN¡1

n=0 s21(n) +

PN¡1n=0 s

22(n)

=(2N)2

N +N

= 2N: (13)

1796 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 2 APRIL 2012

Fig. 4. Normalized gain in SNR over codeword length.

The gain in SNR for the MF procedure cannot

be outperformed by the MMF scheme. However,

an optimum codeword pair and the resulting MMF

coefficients show nearly the same gain as the MF

technique. The normalized gain in SNR between

MMF and MF is calculated as follows:

G =SNRMMFSNRMF

· 1: (14)

The normalized gain is used to rank all possible

codeword pairs for a given length N.

All codeword pairs s1 and s2 are considered in

this paper which have a signal matrix S of maximum

rank 2N ¡ 1. For these codeword pairs the MMFimpulse responses are calculated according to (7). It

is furthermore required that in combination with the

MMF impulse responses these codeword pairs have

high normalized gains G in SNR.

Therefore, a computer search has been carried out

by investigating all possible binary phase codes for

each codeword length N. Some codeword pairs are

given in Table I. The resulting normalized gain G in

SNR is approximately 90% (see Fig. 4).

IV. CONCLUSION

It has been shown in this paper and analyzed by

computer search that for each arbitrary codeword

length N, there exist binary phase codeword pairs and

the related MMF impulse responses which have the

zero sidelobe property in range direction. In addition

the resulting gain in SNR for the MMF procedure is

nearly the same as the MF technique.

JINGYING BI

HERMANN ROHLING

Department of Telecommunications

Hamburg University of Technology

Eissendorfer Strasse 40

21073 Hamburg

Germany

E-mail: (jingying.bi@tu-harburg.de)

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Mismatched-filter design for periodic binary phased

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IEEE Transactions on Aerospace and Electronic Systems,

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[2] Ackroyd, M. and Ghani, F.

Optimum mismatched filters for sidelobe suppression.

IEEE Transactions on Aerospace and Electronic Systems,

AES-9, 2 (1973), 214—218.

[3] Seidler, P.

Mismatched filtering for coded aperture imaging with

minimum sidelobes.

Electronics Letters, 17, 2 (1981), 96—97.

[4] Golay, M.

Complementary series.

IRE Transactions on Information Theory, IT-7, 2 (1961),

82—87.

[5] Turyn, R.

Hadamard matrices, Baumert-Hall units, four-symbol

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CORRESPONDENCE 1797