Impact of Multi-Diagonal Code on High-Speed ...

Arab J Sci Eng DOI 10.1007/s13369-012-0517-6

RESEARCH ARTICLE - ELECTRICAL ENGINEERING

Impact of Multi-Diagonal Code on High-Speed Spectral Amplitude Coding Optical Code Division Multiple-Access Networks Thanaa Hussein Abd · S. A. Aljunid · Hilal Adnan Fadhil · M. N. Junita · M. Saad

Received: 24 August 2011 / Accepted: 24 November 2011 © King Fahd University of Petroleum and Minerals 2012

Abstract In this paper, we present the impact of the multi-diagonal (MD) code for high data rate in spectral amplitude coding optical code division multiple-access (SAC-OCDMA) networks. An MD code structure for SACOCDMA system is presented. The MD code can be flexibly generated for any weight and number of users by using a combination of diagonal matrices. Several advantages of the MD code family are: the cross-correlation value remains zero even in case of an increased number of users for any weight value, supporting large number of users, and easy code construction. Thus, these features show the MD code family be a solution to enhance the data rate of SAC-OCDMA networks. Based on the theoretical and simulation evaluation, MD code is shown to provide a much better performance compared to existing SAC-OCDMA codes. The ability of MD code to support simultaneous transmissions at different high data rate has been successfully established through the simulated results of the 0.24 Tbps (12 Gbps × 20 users) and 0.3 Tbps (15 Gbps × 20 users) data-rate transmission networks. Keywords Spectral amplitude coding optical code division multiple access (SAC-OCDMA) · Multi-diagonal (MD) code · Modified quadratic congruence (MQC) code

T. H. Abd (B) · S. A. Aljunid · H. A. Fadhil · M. N. Junita School of Computer and Communication Engineering, University Malaysia Perlis, Perlis, Malaysia e-mail: [email protected] S. A. Aljunid e-mail: [email protected] M. Saad Electrical and Electronic Engineering Department, University Teknologi Petronas, Perak, Malaysia

1 Introduction Due to the attractive features of optical code division multiple-access (OCDMA) technology considered as an attractive candidate for the multiple-access scheme in future high-speed fiber-optical networks [1]. OCDMA is a multiplexing technique that can multiplex a number of channels on a single wavelength and same time slot by assigning a different optical code to each channel [2,3]. In addition, OCDMA is a method of sharing the bandwidth of optical fiber among a number of users in the local-area network [4]. Recent demonstrations of several multiuser- and Mbps OCDMA systems showed the capability of OCDMA technique as a multiple-access scheme in several Gbps systems [5]. While existing multiplexing techniques such as wavelength division multiplexing (WDM) technique and optical time-division multiplexing (OTDM) are ideally suited

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to long-haul networks, their requirement of high-accuracy wavelength-stabilization techniques and strict synchronization control limits their overall flexibility and scalability [6]. In contrast, OCDMA has several advantages for the implementation of the local-area network, such as capability to support asynchronous access networks, dynamic bandwidth assignment, all optical processing, and the ability to support multimedia services [6–8]. In general, OCDMA system is limited by the multi access interference (MAI) that results from the other users transmitting at the same time and in the same common channel [9]. Furthermore, there are other noises arising from the physical effect of the system design itself, such as phase-induced intensity noise (PIIN), thermal noise, and shot noise [10], where the PIIN is related to the MAI due to the overlapping of the spectra from the different users [9]. The spectral amplitude coding OCDMA (SAC-OCDMA) technique offers the ability to completely eliminate the MAI by spectral coding [11]. Therefore, many codes have been proposed for SACOCDMA networks such as: Khazani-Syed (KS) code [12], enhanced double-weight (EDW) code [13], modified frequency hopping (MFH) [14], modified quadratic congruence (MQC) [15], random diagonal (RD) code [16], and modified double-weight (MDW) code [9]. These codes suffered from several limitations, such as the code length is very long, the code construction is limited by the code parameters, or the cross-correlation is increased with an increase in the weight number. Therefore, to improve the performance of SAC-OCDMA networks, we propose the multi-diagonal (MD) code. The MD code has zero cross-correlation properties and no overlapping of spectra for different users, which cancels the MAI. We also present the impact of MD code in high data-rate transmission networks. The MD code showed a better performance than the existing SAC-OCDMA codes, reduced the receiver complexity, and improved the SAC-OCDMA performance. This paper is organized as follows. In Sect. 2, the principle of MD code construction are explained. In Sect. 3, the performance analysis of the new proposed system is done. Section 4 is devoted to numerical and simulation evaluation, and finally, conclusions are given in Sect. 5.

2 Multi-Diagonal Code Design The MD code is characterized by the following parameters (N , W, λc ) where N is the code length (number of total chips), W is the code weight (chips that have a value of 1), and λc is the in-phase cross-correlation. The crosscorrelation theorem could be defined as present in Appendix. So, if xi j is an entry from X and yi j is an entry from

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Y , then an entry from the product C = X Y is given by: Ci j =

N 

xi K y K j .

(1)

K =1

For the code sequences X = (x1 , x2 , x3 , . . . , x N ) and Y = (y1 , y2 , y3 , . . . , y N ), the cross-correlation function can be N xi yi . When λc = 0, it is conrepresented by: λc = i=1 sidered that the code possesses zero cross-correlation properties. The matrix of the MD code consists of a K × N matrix functionally depending on the value of the number of users (K ), and the code weight (W ). For MD code, the choice of the weight value is free, but should be more than 1 (W > 1). The following steps explain how the MD code is constructed. Step 1 Firstly, construct a sequence of diagonal matrices using the value of the weight (W ) and the number of subscribers (K ). According to these values, the number of rows (i) in each matrix and the number of diagonal matrices ( jW ) will be set. When K and W are positive integer numbers, (i = 1, 2, 3, 4, . . . , i n = K ) and ( jW = 1, 2, 3, 4, . . . , W ), where i n represent the maximum row number of each diagonal matrix. Step 2 Based on the next equations, the MD sequences will be computed for each diagonal matrix.  Si, jW = ⎡

Si,1

1 ⎢2 ⎢ ⎢ = ⎢3 ⎢. ⎣ .. K

(i n + 1 − i), For jW = even number i, For jW = odd number ⎤ ⎥ ⎥ ⎥ ⎥ , Si,2 ⎥ ⎦

⎡

K ⎢ .. ⎢. ⎢ = ⎢3 ⎢ ⎣2 1

⎤ ⎥ ⎥ ⎥ ⎥ , Si,3 ⎥ ⎦

⎡

1 ⎢2 ⎢ ⎢ = ⎢3 ⎢. ⎣ .. K

⎤ ⎥ ⎥ ⎥ ⎥ , . . . , Si,W ⎥ ⎦

⎡

1 ⎢2 ⎢ ⎢ = ⎢3 ⎢. ⎣ .. K

(2) ⎤ ⎥ ⎥ ⎥ ⎥ (3) ⎥ ⎦

Any elements of the Si,W matrices represent the position of one in Ti,W matrices with K × K dimensions; where Ti,1 = [Si,1 ] K ×K , Ti,2 = [Si,2 ] K ×K and Ti,W = [Si,W ] K ×K . Therefore, ⎡ 1 0 ··· ⎢0 1 ··· ⎢ Ti,1 = ⎢ . . . ⎣ .. .. . .

⎤ 0 0⎥ ⎥ .. ⎥ .⎦

⎡

⎤ 0 ··· 0 1 ⎢0 ··· 1 0⎥ ⎢ ⎥ , Ti,2 = ⎢ . . . . ⎥ , ⎣ .. . . .. .. ⎦ 0 0 · · · 1 K ×K 1 · · · 0 0 K ×K ⎡ ⎤ 1 0 ··· 0 ⎢0 1 ··· 0⎥ ⎢ ⎥ . . . , Ti,W = ⎢ . . . . ⎥ ⎣ .. .. . . .. ⎦ 0 0 · · · 1 K ×K

(4)

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Step 3 The total combination of diagonal matrices (3) represents the MD code as a matrix of power K × N .  .. .. .. (5) M D = Ti,1 .Ti,2 . · · · · · · .Ti,W K ×N

⎡

a1,1 a1,2 ⎢ a2,1 a2,2 ⎢ ⎢ M D = ⎢ a3,1 a3,2 ⎢ . .. ⎣ .. . ain ,1 ain ,2

· · · a1,N · · · a2,N · · · a3,N . · · · ..

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6)

· · · ain ,N

From the above basic matrix (5), the rows determine the number of users (K ). Notice that the association between code weight (W ), code length (N ) and number of subscribers (K )can be expressed as: N = K × W.

(7)

For example, to generate a MD code family according to the previous steps, let us say K = 4 and W = 3. Therefore, i = 1, 2, 3, 4, and jW = 1, 2, 3. The diagonal matrices can be expressed as: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 4 1 ⎢2⎥ ⎢3⎥ ⎢2⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Si,1 = ⎢ (8) ⎣ 3 ⎦ , Si,2 = ⎣ 2 ⎦ , Si,3 = ⎣ 3 ⎦ . 4 1 4 The MD code sequence for each diagonal matrix is shown as: ⎡ ⎡ ⎤ ⎤ 1000 0001 ⎢0 1 0 0⎥ ⎢0 0 1 0⎥ ⎢ ⎥ ⎥ T = Ti,1 = ⎢ i,2 ⎣0 0 1 0⎦ ⎣0 1 0 0⎦ 0 0 0 1 4×4 1 0 0 0 4×4 ⎡ ⎤ 1000 ⎢0 1 0 0⎥ ⎥ Ti,3 = ⎢ (9) ⎣0 0 1 0⎦ 0 0 0 1 4×4 The total MD code sequence will be: ⎡ ⎤ 100000011000 ⎢0 1 0 0 0 0 1 0 0 1 0 0⎥ ⎥ MD = ⎢ ⎣0 0 1 0 0 1 0 0 0 0 1 0⎦ 0 0 0 1 1 0 0 0 0 0 0 1 4×12

Table 1 System parameters Symbol

Parameter

e

Electron charge

I

Average photocurrent

B

Electrical bandwidth

Kb

Boltzmann constant

Tn

Receiver noise temperature

RL

Receiver load resistor

The MD code design shows that changing matrices’ elements in the same diagonal part will result in a constant property of zero cross-correlation, and it is constructed with zero crosscorrelation properties, which cancels the MAI. The MD code presents more flexibility in choosing the W, K parameters and with a simple design to supply a large number of users compared to existing SAC-OCDMA codes. Furthermore, there are no overlapping chips for different users.

3 System-Performance Analysis 3.1 Gaussian Approximation To analyze our system, Gaussian approximation is used for the calculation of BER [15]. We have considered the effect of thermal noise (σth ) and shot noise (σsh ) in the photo-detector. The SNR of an electrical signal is defined as the average signal power (I 2 ) to noise power SNR = (I 2 /σ 2 ). Due to the zero cross-correlation property of MD code, there is no overlapping in spectra of different users. For that reason, the effect of incoherent intensity noise has been ignored. The variation of the photo-detector is a result of the detection of an ideally unpolarized thermal light, which is generated by spontaneous emission. The variation can be expressed as: σ 2 = σsh + σth , σ 2 = 2eB I +

(10)

where K = 4, N = 12. So, the codeword for each user according to the above example would be: ⎧ user1 ⇒ λ1 , λ8 , λ9 ⎪ ⎪ ⎨ user2 ⇒ λ2 , λ7 , λ10 codeword = user3 ⇒ λ3 , λ6 , λ11 ⎪ ⎪ ⎩ user4 ⇒ λ4 , λ5 , λ12

4K b Tn B RL .

(11)

where the parameters that have been used in Eq. (11) are represented in Table 1. Let C K (i) denote the ith element of the Kth MD code sequences, and according to the properties of MD code, the direct detection technique can be written as:  N  W, For K = l C K (i)Cl (i) = (12) 0, Else i=1

The following assumptions are made [14,16]. 1. Each light source is ideally unpolarized and its spectrum is flat over the bandwidth [vo − v/2, vo + v/2] where

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∞ 

∞ G(v)dv = 0

 N K  Psr  dK C K (i)Cl (i)rect(i) dv. v K =1

0

i=1

(16) As a result, ∞ G(v)dV =

Psr W N

(17)

0

Fig. 1 The PSD of the received signal r (v)

The photocurrent I can be found as: vo is the central optical frequency and v is the optical source bandwidth expressed in Hz. 2. Each power spectral component has an identical spectral width. 3. Each user has an equal power at the transmitter. 4. Each bit stream from each user is synchronized. The above assumptions are important for mathematical straightforwardness. Devoid of these assumptions, it is difficult to analyze the system; for example, if the power for each spectral component is not identical and each user has a different power at the receiver. The power spectral density (PSD) of the received optical signals can be written as [16]: r (v) =

Psr v

K 

dK

K =1

N 

I =

c K (i)rect(i)

(13)

i=1

where u(v) is the unit step function expressed as:  1, v ≥ 0 u(v) = 0, v < 0

(15)

To compute the integral of G(v), let us first consider an  example of the PSD (denoted by G (v) of the received superimposed signal), which is shown in Fig. 1, where A(i) is the amplitude of the signal of the ith spectral slot with a width of v N [14]. From Eq. (13) the integration of the power spectral density at the photo-detector of the lth receiver during one period can be written as:

G(v)dv

(18)

0

where  is the responsivity of the photo-detectors given by ηe [13]. Here, η is the quantum efficiency, h is Planck’s  = hv c constant, and vc is the central frequency of the original broadband optical pulse. Then, Eq. (18) can be expressed as: ∞ I =

G(v)dv =

Psr W N

(19)

0

Substituting Eq. (19) in Eq. (11), we obtain: σ2 =

where Psr is the effective power of a broadband source at the receiver, K is the active users, and N is the MD code length, and d K is the data bit of the Kth user, which is either “1” or “0”. The rect(i) function in Eq. (13) is given by  v (−N + 2i − 2) rect(i) = u v − v0 − 2N   v v (−N + 2i) = u −u v − v0 − (14) 2N N

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∞

2eBPsr W 4K b Tn B . + N RL

(20)

Note the probability of sending bit “1” at anytime for each user is 1/2, thus Eq. (20) becomes σ2 =

4K b Tn B eBPsr W + N RL

(21)

Lastly from Eqs. (19) and (21) we can calculate the average SNR as: ⎤ ⎡ 2  Psr W

N ⎢ SNR = ⎣ eBP W sr + N

4K b Tn B RL

⎥ ⎦

(22)

Using Gaussian approximation, the bit error rate (BER) can be expressed as [16]:   SNR 1 BER = Pe = erfc . (23) 2 8 3.2 MD Code Transceiver Description The transceiver design of the MD code is constructed based on direct detection technique. Figure 2 illustrates the schematic diagram of a MD code transceiver. As observed in the transmitter part, an optical external modulator (OEM) is used to modulate the modulated electric data with the distinct code sequence. On the other hand, each codeword of the

Arab J Sci Eng Fig. 2 Illustration of the block diagram of an MD code system with direct detection technique

SAC-OCDMA based on MD code consists of different wavelengths. An optical combiner is used to combine all modulated code sequences sent out through the optical fiber. On the receiver side, a de-multiplexer is used to separate the different modulated codewords. The received code sequence will be filtered out using the fiber Bragg grating (FBG) filter, and then decoded by the photo detector. However, based on the property of zero cross-correlation of the MD code, the receiver was designed based on the direct-detection technique. In the direct detection technique, there is only one single decoder and a single detector required for each user. Therefore, no subtraction or balance is required. This is achievable for a simple reason; the information can be adequately recovered from any of the chips that do not overlap with any other chips from other code sequences. Thus, the decoder will only need to filter the clean chips and be detected by the photodiode.

4 Numerical and Simulation Evaluation 4.1 Numerical Results Using Eqs. (22–23), the BER of the MD code is tested numerically with other existing SAC-OCDMA codes. The parameters of Eqs. (22–23) used in our numerical calculation are listed in Table 2. These parameters have been chosen based on the published results for these practical codes [12–16]. Figure 3 shows the BER as a function of active users, for MD, KS, RD, MFH, and MQC codes. The calculated results for MD code were achieved under 622 Mbps, the power spectral density of light source (Psr = −10 dBm), code weight (W = 4), and the number of active users, K , up to 160.

Table 2 Typical parameter used in the numerical analysis Symbol

Parameter

Value

η

Photodetector quantum efficiency

0.6

B

Electrical bandwidth

311 MHz

Rb

Data bit rate

622 Mbps

Tn

Receiver noise temperature

300 K

RL

Receiver load resistor

1030 

e

Electron charge

1.6 × 10−19 C

h

Planck’s constant

6.66 × 10−34 Js

Kb

Boltzmann’s constant

1.38 × 10−23 J/K

While for KS, MFH, RD and MQC codes, the weight code was 4, 12, 5, and 14, respectively. The figure shows that the performance of MD code is better than other SAC-OCDMA codes; even when compared to high weight value like MQC code or with the same code weight, such as KS code. Furthermore, the maximum acceptable BER of 10−9 was achieved by multi-diagonal code with 90 active users, while achieving 26, 43, 23 and 59 active users by MFH code, MQC code, KS code, and RD code, respectively. This better performance is due to the zero cross-correlation property of MD code, which reduced the effect of MAI between the spectral light of other users. Figure 4 shows the variation traces of BER versus the data rate for different effective light power for 90 active users. However, the effective light power of RD, and MQC code was fixed at 0 dBm, while for MD code, it is −7 and −5 dBm. Moreover, the weight value of RD, and MQC code were 5 and 12, respectively. The mathematical results showed that the MD code can support 90 active users, each carrying a bit

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Arab J Sci Eng Fig. 3 BER against the number of active users for various codes employing the SAC-OCDMA technique

1.00E-01 1.00E-04 1.00E-07

BER

KS code W=4 1.00E-10

MD code W=4

1.00E-13

MFH code W=12

1.00E-16

MQC code W=14

1.00E-19

RD code W=5

1.00E-22 1.00E-25 0

20

40

60

80

100

120

140

160

Active Users

BER

Fig. 4 Variation of BER as a function of data rate for various codes employing the SAC-OCDMA technique

1.00E+00 1.00E-03 1.00E-06 1.00E-09 1.00E-12 1.00E-15 1.00E-18 1.00E-21 1.00E-24 1.00E-27 1.00E-30 1.00E-33 1.00E-36

RD code W=5, Psr= 0 dBm MQC code W=12, Psr= 0 dBm MD code W=4, Psr=-7dBm MD code W=4, Psr=-5dBm 0

2

4

6

8

10

Data rate (Gb/s)

rate of 10 Gbps with −5 dBm. Nevertheless, in the case of decreasing the effective light power the MD code system can still support the same number of users but with a lower rate of 2.5 Gbps. However, Fig. 4 clearly shows that the performance of MD code is better than that of the previous SAC-OCDMA codes even when these codes have the higher effective light power or higher weight value. It is obviously evident that the MD codes can outperform other previous SAC-OCDMA codes in terms of the number of users with a higher data rate. 4.2 Simulation The performance analysis of MD code was simulated by using the simulation software, Optisystem Version 9.0. The tests were carried out at a data rate of 0.24 Tbps (20 users × 12 Gbps) and 0.3 Tbps (20 users × 15 Gbps) with different fiber lengths according to the ITU-T G.652 standard single-mode optical fiber (SMF). Tests were also carried out under different power levels of the light source such as, −5 dBm and −10 dBm with 0.4 nm spectral width for each chip. All the attenuation α (i.e., 0.25 dB/km), dispersion (i.e., 18 ps/nm km) and non-linear parameters like self-phase modulation (SPM), cross-phase modulation (XPM) and fourwave mixing (FWM) have been activated in the simulation

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model and specified according to the typical industry values to simulate the real environment as closely as possible. The performance of the system was characterized by referring to the BER and eye pattern. Figure 5 shows the BER versus the transmission distance at different data bit rates and light strengths. The data rate carried by the whole network is 0.24 Tbps (12 Gbps × 20 users) and 0.3 Tbps (15 Gbps × 20 users). However, the longer fiber length will present a larger dispersion and attenuation, as well as the increasing of BER. To achieve the optimum optical system design, the maximum fiber length should be as short as possible, to carry a higher data rate and to achieve the desired system performance. For an SAC-OCDMA network employing MD code, the systems will significantly compensate for the dispersion effect, and therefore, the performances are limited by the fiber losses. However, the BER increased exponentially with an increase in fiber length when the data rate is increased to 15 Gbps and with the light source strength decreased to −10 dBm. In contrast, the MD code network with 0.24 Tbps shows a better performance for different fiber lengths with the light source strength higher than −10 dBm. The performance of the MD code was tested and compared with other SAC-OCDMA codes such as RD and MQC

Arab J Sci Eng

BER

Fig. 5 BER versus the transmission distance at different data bit rates and light source power

1.00E-02 1.00E-03 1.00E-04 1.00E-05 1.00E-06 1.00E-07 1.00E-08 1.00E-09 1.00E-10 1.00E-11 1.00E-12 1.00E-13 1.00E-14

15 Gb/s, Psr= -5 dBm 15 Gb/s, Psr= -10 dBm 12 Gb/s,Psr= -5 dBm 12 Gb/s,Psr= -10 dBm

10

15

20

25

30

Fiber length (km)

Fig. 6 Eye diagram for MD code with 20 users and weight = 2 at 12 Gbps, 20 km

codes. The MD code test was carried out at 12 Gbps with 20 active users for 20 km, while the RD code was evaluated for four active users with 10 Gbps at 20 km distance and, the MQC code was tested for four active users with 10 Gbps at 10 km. The eye pattern shown in Figs. 6, 7 and 8 clearly shows that the MD code network performs better with a large eye opening, even though it was tested with a large number of users and higher data rate. Although the BER of MD code is lower than the RD code, it must be noted that it is still performing better than the other codes because it is performing for a larger number of users and at higher data rate. This is achieved because of the properties of MD code such as zero cross-correlation code with no overlapping between the light spectra. The results in Fig. 6 clearly show that an SAC-OCDMA network using MD code is suitable for the metropolitan area network (MAN) environment with a high data rate.

5 Conclusions This paper has presented the impact of MD code on high data rates in SAC-OCDMA networks. The results showed that the design of the network based on the MD code is simple due to the properties of MD code, such as zero crosscorrelation and no overlapping between the light spectra. The MD code network was also investigated with 0.3 Tbps (20 users ×15 Gbps) and the results showed that data can be transmitted efficiently without amplifiers or side pumps for 15 km. However, for a longer transmission of up to 30 km, the design of a 0.24 Tbps (20 users × 12 GBps) network with −5 dBm was observed to be the best choice. It is concluded from the numerical and simulation results shown in this research that MD code is a better solution in high data rate optical network design for use in the MAN environment.

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Arab J Sci Eng Fig. 7 Eye diagram for RD code with four users and weight = 3 at 10 Gbps, 20 km

Fig. 8 Eye diagram for MQC code with four users and weight = 3 at 10 Gbps, 10 km

Appendix The cross-correlation theorem could be defined as follows:

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In linear algebra, the identity matrix or unit matrix of size N is the N × N square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I N , or simply by I if

Arab J Sci Eng

the size is immaterial or can be trivially determined by the context. ⎡ ⎤  100 10 I1 = [1] , I2 = , I3 = ⎣ 0 1 0 ⎦ , 01 001 ⎡ ⎤ 10 0 0 ⎢0 1 0 0⎥ ⎢ ⎥ (A-1) . . . , IN = ⎢ ⎥ ⎣ 0 0 ... 0 ⎦ 0 0 ··· 1 Using the notation that is sometimes used to concisely describe diagonal matrices, we can write: I N = diag(1, 1, . . . , 1). An orthogonal matrix is a square matrix with real entries whose columns (and rows) are orthogonal unit vectors (i.e., orthogonal). Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse: A T A = A A T = I . Alternatively, A T = A−1 . A square matrix whose transpose is also its inverse is called an orthogonal matrix; that is, A is orthogonal if A T A = A A T = I N , the identity matrix, i.e., A T = A−1 . For example, A(N × N ) square matrix, A is said to be orthogonal if A A T = A T A = I N ×N . The cross-correlation theorem states that certain sets of complementary sequences have cross-correlation functions that sum to zero using all pairwise permutations. Here, all cross-correlation function permutations are required, such that their sum be identically equal to zero. For example, if the rows and columns of a (K × N ) matrix are orthogonal and all the columns except one sum to zero, then the sum of all cross-correlations between non-identical codewords is zero.

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