A Study of Different Angles for the New Spread Matrix for BSOFDM in UWBChannels

Ibrahim S. Raad, Xiaojing Huang and Darryn LoweSchool of Electrical, Computer and Telecommunications Engineering

University of Wollongong, N.S.W Australiaibrahim@uow.edu.au

Abstract

This paper presents a study into different angles forthe New spread matrix developed for BSOFDM. It variesthe angles for the matrix to develop different constella-tion schemes which are useful in overcoming the frequencyselective channels which are encountered in mobile com-munication systems. Previously it has been discussed thatthis new matrix (the rotation matrix) has some advantagesover Hadamard and the rotated Hadamard matrix in cer-tain channels. This paper presents a study of varies an-gles with this new matrix over the UWB channels CM1 toCM4.1

Key Words-OFDM, Spreading Matrices, Block Spread-OFDM, UWB

1 Introduction

Many solutions have been presented to allow a com-munications system to improve its spectral efficiency ofthe modulation schemes by applying different schemessuch as adaptive modulation based on the Bit Error Rate(BER) or the signal to noise ratio. But these systems suf-fer from complexity issues and the fact many still, aftervaries algorithms to improve the spectral efficiency, endup using BPSK or QPSK. This paper continues the studyof a new method to increase the correlation between thesymbols through the use of a rotation of the modulatedsymbols, and depending on the rotation angle, α, a newand higher order modulation is used in the transmission ofthe system to increase the correlation between the trans-mitted symbols to improve the BER performance. Thisnew spreading matrix [1], as with other existing matriceslike the Hadamard, is used in Block Spreading OFDM andis discussed in detail in [2], [3], [4], [5] and [6]. At the re-ceiver, the decoder used is the Maximum Likelihood de-coder (ML). This is not the same as adaptive modulationas this does not retransmit the data when the new modula-tion scheme is created but rather has the same modulationscheme at the transmission. This paper has the followingsections. Section 2 provides a description of the system

1This research is sponsored by ARC DP 0558405

used to test the New matrix and discusses the advantagesand disadvantages of this New spread matrix over otherexisting spreading matrix. Section 3 discusses the newspreading matrix used in this study. A brief description isgiven of the four UWB channel models in Section 4. Sec-tion 5 gives the results achieved with the new spreadingmatrix and the comparisons between it and the existingspreading matrices over UWB channel models. And fi-nally in Section 6 provides a conclusion.

2 System Description

Primarily this new spreading matrix is used in whathas been described as Block Spread OFDM (BSOFDM),which is when the full set of subcarriers are divided intosmaller blocks and using spreading matrices to spread thedata across these blocks so to achieve multipath diversityacross each block at the receiver [2], [3], [4] and [5]. TheBSOFDM channel model is shown in Figure 1.

y = Cq + n (1)

The output of the receiver’s FFT processor is shownin Equation 1, where y is the FFT output, q ∈ AN isthe vector of transmitted symbols, each drawn from analphabet A, C is a diagonal matrix of complex normalfading coefficients, and n is a zero mean complex nor-mal random vector. Equalization of the received data isdone through multiplication by C−1 and then “quantizedindependently on each subcarrier to form the soft or harddecision q̂ which may be further processed if the data bitsare coded” [4]. There is no loss in performance when thedetection is performed independently on each carrier dueto the noise being independent and identically distributedwith fading been diagonal [4].

The block spreading matrices are used to introduce de-pendence among the subcarriers. N subcarriers are splitinto N

M of blocks of size M, where M = 2 for this ex-ample. Then each of the blocks are multiplied by a 2 × 2unitary matrix U2. The length two output vectors are in-terleaved using general block interleaving to ensure thesymbols are statistically independent so as to encounter

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Figure 1. Block diagram representation ofthe BSOFDM channel for a block length oftwo [4].

independent fading channels. This will ensure in a dis-persive frequency selective channel the data is statisticallyless likely to become corrupted and studies and simula-tions have shown this to be correct.

The transmitter’s IFFT has the interleaved data passedthrough it and this data is sent across the frequency selec-tive channel. The data is passed through an FFT processorat the receiver and deinterleaved before using block byblock processing.

The spreading matrices are generally used to increasethe correlation between the transmitted symbols after thetransmission has occurred. Unlike adaptive modulationschemes where depending on the system, a higher or-der modulation scheme is used to retransmit the data de-pending on the conditions presented, this scheme utilizesspreading matrices to increase the correlation between thesymbols, rather than retransmitting. This is depicted inFigure 2. So say at the transmission the system modulatesthe data using QPSK modulation, with spreading matri-ces a higher order modulation is used to increase corre-lation and therefore overall system performance. Thereare a number of matrices available and well studied, thispaper continues the study of the new matrix presented in[1] concentrating on the various angles which can be usedwith this new matrix. Below the description of the newmatrix is presented and discussed in terms of the differentpossible combinations achievable.

3 New Spreading Matrix

The following is a description of the new spread ma-trix for BSOFDM, where for example to make QPSK into16QAM the choice of tan(α) = 0.5. This achieves differ-ent modulation schemes to that of Hadamard and RotatedHadamard.

A =[

1 tan(α)tan(α) −1

]

QPSK 16QAM QPSK

Transmission Rotation Receiver

Figure 2. The process through which thetransmitted modulation is converted into ahigher order modulation and then returnedat the receiver.

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Qua

drat

ure

In−Phase

QPSK modulation

Figure 3. The QPSK constellation points.

Depending on the choice of α, different modulationschemes are possible. Naturally, not all angles can bechosen since this would not yield a better result than theHadamard matrix. For example an angle of α = π

4would result in one, which would mean that the matrixis a Hadamard matrix. Other angles which cannot be usedwhen using QPSK are α = π and π

2 since the rotationof QPSK would rotate back onto it self and the new rota-tion would be the same as the rotated, that is QPSK. Otherangles then can be used and are discussed below.

The following figures depict the varying constellationpoints which are achievable using the new spreading ma-trix. Figure 3 depicts the constellation scatter plot ofQPSK which will be used for this study. Figures 4, 5, 6,7, 8 and 9 depict the different constellation points whichare achievable using the new matrix. As can be seen fromFigure 9, since tan(π

4 ) = 1, the end result is the same asthe Hadamard matrix.

4 UWB channels

Ultra-wideband (UWB) systems occupy - by definition- a signal spectrum of more than 500 MHz or more than20% with regards to their centre frequency. The appli-cation of such large bandwidths enables communicationsystems with unique novel properties, like high-precision

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−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Qua

drat

ure

In−Phase

pi/3 new matrix

Figure 4. The new matrix for block spreadOFDM with rotation π

3 .

−1 −0.5 0 0.5 1

x 1016

−1

−0.5

0

0.5

1

x 1016

Qua

drat

ure

In−Phase

pi/2 −new matrix

Figure 5. The new matrix for block spreadOFDM with rotation π

2 .

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Qua

drat

ure

In−Phase

pi − new matrix

Figure 6. The new matrix for block spreadOFDM with rotation π.

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Qua

drat

ure

In−Phase

pi/6 new matrix

Figure 7. The new matrix for block spreadOFDM with rotation π

6 .

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Qua

drat

ure

In−Phase

Scatter plot

Figure 8. The new matrix for block spreadOFDM with rotation π

9 .

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Qua

drat

ure

In−Phase

Scatter plot

Figure 9. The new matrix for block spreadOFDM with rotation π

4 .

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CM1 CM2 CM3 CM4Channel Model LOS NLOS NLOS NLOS

0 − 4m 0 − 4m 4 − 10m 10mMean Ex.delay(ns) 5 9.9 15.9 30.1

RMS Delay(ns) 5 8.0 15.0 25.0No.of multi-arrival 12.5 15.3 24.9 41.2

within 10dB of peak

Table 1. UWB channels defined by IEEE [7]

indoor positioning.Based on the Saleh-Valenzuela model for indoor

multipath radio propagation channels, a set of statisti-cally UWB channel models has been produced by theIEEE.802.15.3a task group [7]. Line of Sight (LOS)and Non-Line of Sight (NLOS) cases can be modelledusing these standard channel models. The four differentscenarios are summarized in table 1. Each are identifiedfrom CM1 to CM4. These channels are used in our studyof the various angles with the new spreading matrix forBSOFDM.

5 Results

The results below depict the simulation results for dif-ferent angles used with the new spreading matrix. Thechannel used for this study is UWB described in Table1. In [1], a comparison of the new Spreading matrix wasdiscussed and the results of a similar study is shown inFigure 10 using the UWB CM1 model. As can be seenthe new matrix outperforms the Hadamard matrix signifi-cantly. The Rotated Hadamard has a very similar perfor-mance to the new matrix and to the Rotated new matrixas can be seen from Figure 10. So the advantage of thisnew matrix is its flexibility in achieving the same matricesas that of the Hadamard and Rotated Hadamard and othercombinations as well based on the rotation of the angle.This study sets out to study the angle which achieves thebest result in UWB channel models.

In Figure 11, using the UWB channel model CM1, itcan be seen that the angle π

3 achieves the best performancein terms of BER in a BSOFDM system.

In Figure 12, the packet error rate of the angle π3 can be

seen having the best performance for the BSOFDM sys-tem over UWB channel model CM1.

As can be seen in Figure 13, the performance of differ-ent angles with the new spreading matrix with the UWBchannel CM1, the angle π

3 outperforms the other angles.Angle π

6 has a similar performance. It is interesting to notethat the angle π

4 has a similar performance to that of theHadamard matrix since the tan(π

4 ) = 1 and substitutingthis value into the new matrix will result in a Hadamardmatrix. This can be seen from the constellation point dia-gram seen in Figure 9. As discussed earlier, the two anglesπ2 and π will yield the same result as the QPSK modula-tion, this can be seen in Figure 13. QPSK modulation ismodelled under the heading normal. The reason for thisis that the two mentioned angles simply rotate the QPSK

0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

EbN0

BE

R

BER of Different Spreading Options [CM1] − ALL ML

Normal

pi/6

pi/3

pi/2

Had

rotate Had

rotate−new (pi/3)

Figure 10. Comparing the new matrix withvarying angles versus the Hadamard andRotated Hadamard.

0 2 4 6 8 10 12 14 1610

−5

10−4

10−3

10−2

10−1

EbN0

BE

R

BER of Different Spreading Options [CM1]

Normalpi/6 (ML)pi/3 (ML) pi/7 (ML)

Figure 11. The new matrix with angles π6 , π

3and π

7 .

0 2 4 6 8 10 12 14 1610

−2

10−1

100

EbN0

SE

R

Symbol Error Rate (SER) of Different Spreading Options [CM1]

Normalpi/6 (ML)pi/3 (ML)pi/2 (ML)

Figure 12. The new matrix with angles π6 , π

3and π

7 .

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0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

EbN0

BE

R

BER of Different Spreading Options [CM1] − ALL ML

Normalpi/6pi/3 pi/2pi/7pi/9pi/4

Figure 13. The new matrix with anglesπ6 ,π

3 ,π2 , π

7 ,π9 and π

4 using CM1 channel.

0 5 10 15 20 2510

−5

10−4

10−3

10−2

10−1

100

EbN0

BE

R

BER of Different Spreading Options [CM2] − ALL ML

Normalpi/6pi/3 pi/2pi/7pi/9pi/4

Figure 14. The new matrix with anglesπ6 ,π

3 ,π2 , π

7 ,π9 and π

4 using CM2 channel.

modulation back onto it self. This can be seen from theconstellation points presented above in Figures 5 and 6.

For the UWB channel model CM2, it can be seen inFigure 14 that the angle π

7 out performs all the others,again we see that Hadamard is outperformed significantly.

For both the UWB channel models CM3 and CM4 theangles π

3 and π7 again outperform the other angles shown,

this can be seen in Figures 15 and 16. Again it can beseen that the Hadamard matrix (also the angle π

4 ) are out-performed by a significant margin. It can be seen that theangle π

6 has a similar performance to that of the angles π3

and π7 . The angle π

9 does not perform as well, although itdoes out perform the Hadamard. This is due to the separa-tion of the constellation points for each scheme. The moreequal the distance apart is from each constellation point,the better the performance will be. Future work will con-centrate on finding the optimum angle for this matrix toachieve that goal.

0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

EbN0

BE

R

BER of Different Spreading Options [CM3] − ALL ML

Normalpi/6pi/3 pi/2pi/7pi/9pi/4

Figure 15. The new matrix with anglesπ6 ,π

3 ,π2 , π

7 ,π9 and π

4 using CM3 channel.

0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

EbN0

BE

R

BER of Different Spreading Options [CM4] − ALL ML

Normalpi/6pi/3 pi/2pi/7pi/9pi/4

Figure 16. The new matrix with anglesπ6 ,π

3 ,π2 , π

7 ,π9 and π

4 using CM4 channel.

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6 Conclusion

In conclusion this paper gave a study for varying angleswhich can be used for the new spreading matrix proposedin [1] for BSOFDM system over the four models pro-posed for the IEEE.15.3a task group. This paper stud-ies what the different constellation points are and what,if any, improvement take place for the common modula-tion schemes. It can be said that the limitations of such aspread matrix is the angles which reproduce the same con-stellation scheme as that of the modulation scheme used.In other words when using for example a QPSK modula-tion scheme, the use of the angle for example π

2 , wouldsimply rotate the modulation on it self as has been shownand end up using the same modulation scheme again.

The advantages can be noted is its flexibility in de-termining different structures of matrices and the sim-ple angle rotation allows an improvement to take placeover more traditional spreading matrices. At the sametime, this matrix can reproduce existing matrices depend-ing on the angle used, for example π

4 would result in theHadamard matrix.

It can be stated at the end of this simple study, that theangles which perform the best over UWB channel modelsare the angles π

3 , π6 and π

7 . Angles π4 , π

2 and π do notachieve the same performance as those listed above.

Future work on this matrix will include finding the the-oretical optimum angle to achieve the best result for thisnew matrix.

References

[1] Ibrahim Raad, Xiaojing Huang and Raad Raad, “ANew Spreading Matrix for Block Spread OFDM”10th IEEE International Conference on Communica-tion Systems 2006 (IEEE ICCS’06), Singapore, 31October - 3 November 2006.

[2] Michael L. McCloud, “Coded block OFDM for thefrequency selective fading channel”, Signals, Sys-tems and Computers, Conference Record of theThirty-Eighth Asilomar Conference on, volume 1,2004, 329-333, November.

[3] Michael L. McCloud, “Analysis and Design of ShortBlock OFDM Spreading Matrices for Use on Mul-tipath Fading Channels”, Communications, IEEETransactions on, volume 53, 4, 2005, 656-66, April.

[4] Michael L. McCloud, “Optimal Binary Spreadingfor Block OFDM on Multipath Fading Channels”,WCNC / IEEE Communications Society, volume 2,2004, 965-970, March.

[5] Ibrahim S. Raad, Xiaojing Huang, “Exploiting Timediversity to improve Block Spread OFDM”, FirstIEEE International Conference on Wireless Broad-band and Ultra Wideband Communications, AusWireless 2006, Sydney, 13 - 16 March.

[6] Ibrahim S. Raad and Xiaojing Huang , “Exploit-ing Time Diversity to Improve Block Spread OFDMin a Multipath Environment” The Second Interna-tional Conference on Information & CommunicationTechnologies - From Theory to Applications, IEEEICTTA’06, Damascus, Syria, 24 - 28 April 2006.

[7] Darryn W. Lowe , “Real - time FPGA Realizationof an UWB transceiver physical layer” Thesis forMasters of Engineering Research at the Universityof Wollongong, The School of Electrical, Computerand Telecommunications Engineering, 2005.

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