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Performance Analysis of the Cooperative ZP-OFDM: Diversity, Capacity and Complexity

Abstract

In this paper, we investigate the diversity, capacity and complexity issues of cooperative Zero-Padding (ZP)-Orthogonal Frequency Division Multiplexing (OFDM) communication. We consider cooperative ZP-OFDM communication over a multipath Rayleigh channel and with multiple Carrier Frequency Offsets (CFOs) existing at different relays. We use a cooperative tall Toeplitz scheme to achieve full cooperative and multipath diversity, while simultaneously combat the CFOs. Importantly, this full diversity scheme only requires Linear Equalizers (LEs), such as Zero-Forcing (ZF) and Minimum Mean Square Error (MMSE) equalizers, an issue which reduces the system complexity when compared to a Maximum-Likelihood Equalizer (MLE) or other near-MLEs. Theoretical analysis of the proposed cooperative tall Toeplitz scheme is provided on the basis of the analytical upper bound of the channel orthogonality deficiency derived in this paper. Utilizing only low-complexity linear equalizers, theoretical analysis and simulation results show that the proposed Toeplitz scheme achieves the full cooperative, multipath and outage diversity.

Abbreviations

AF:

Amplify-and-Forward

AWGN:

Additive White Gaussian Noise

BER:

Bit Error Rate

CDFs:

Cumulative Density Functions

CF:

Compress-and-Forward

CFOs:

Carrier Frequency Offsets

CP:

Cyclic Prefix

CR:

Cognitive Radio

DF:

Decode-and-Forward

DPS:

Digital Phase Sweeping

ECMA:

European Computer Manufacturers Association

FFT:

Fast Fourier Transform

IFFT:

Inverse Fast Fourier Transform

ISI:

Inter-Symbol-Interference

LEs:

Linear Equalizers

MB:

Multi-Band

MLE:

Maximum-Likelihood Equalizer

MMSE:

Minimum Mean Square Error

od :

orthogonality deficiency

OFDM:

Orthogonal Frequency Division Multiplexing

OLA:

Overlap and Add

PSD:

Power Spectral Density

SD:

Sphere Decoding

SNR:

Signal-to-Noise Ratio

STC:

Space Time Coding

STFC:

Space-Time-Frequency Coding

SFC:

Space Frequency Coding

UWB:

Ultra Wide Band

ZF:

Zero-Forcing

ZP:

Zero-Padding

(·)T :

Transpose of (·)

(·)* :

Conjugate of (·)

(·)H :

Hermitian of (·)

(·)−1 :

Inverse of (·)

\({\left(\cdot \right)^{\dagger}}\) :

Pseudo inverse of (·)

\({\forall}\) :

For all

\({\left| \cdot \right|}\) :

Absolute value of a scalar or cardinality of a set

\({\left\| \cdot \right\|\quad}\) :

2-Norm of a vector/matrix

argue min (·):

Argument of minimum of (·)

diag (·):

Diagonal matrix with main diagonal (·)

det (·):

Determinant of (·)

lim (·):

Limit of (·)

log (·):

Logarithm with base 10

log 2 (·):

Logarithm with base 2

max (·):

Maximum of (·)

od (·):

Orthogonality deficiency of matrix (·)

O (·):

Landau notation

Prob (·):

Probability of (·)

References

  1. 1

    Liu K. J. R., Sadek A. K., Su W., Kwasinski A. (2009) Cooperative communications and networks. Cambridge University Press, Cambridge

  2. 2

    Nosratinia A., Hunter T. E., Hedayat A. (2004) Cooperative communciation in wireless networks. IEEE Communications Magazine 42: 74–80

  3. 3

    Cover T. M., El Gamal A. A. (1979) Capacity theorems for the relay channel. IEEE Transactions on Information Theory IT-25: 572–584

  4. 4

    Kramer G., Gastpar M., Gupta P. (2005) Cooperative strategies and capacity theorems for relay networks. IEEE Transactions on Information Theory 51: 3037–3063

  5. 5

    Laneman J. N., Tse D. N. C., Wornell G. W. (2004) Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Transactions on Information Theory 50(12): 3062–3080

  6. 6

    Sendonaris A., Erkip E., Aazhang B. (2003) User cooperation diversity—Part I: System description. IEEE Transactions Communications 51(11): 1927–1938

  7. 7

    Sendonaris A., Erkip E., Aazhang B. (2003) User cooperation diversity—Part II: Implementation aspects and performance analysis,”. IEEE Transactions Communications 51(11): 1939–1948

  8. 8

    Letaief K. B., Zhang W. (2009) Cooperative communications for cognitive radio networks. Proceedings of IEEE 97: 878–893

  9. 9

    Oyman O., Laneman J. N., Sandhu S. (2007) Multihop relaying for broadband wireless mesh networks: From theory to practice. IEEE Communications Magazine 45: 116–122

  10. 10

    Muquet B., Wang Z., Giannakis G. B., Courville M., Duhamel P. (2002) Cyclic prefixing or zero padding for wireless multicarrier transmissions. IEEE Transaction on Communications 50(12): 2136–2148

  11. 11

    Lu, H., Nikookar, H. & Chen H. (2009). On the potential of ZP-OFDM for cognitive radio. In Proc. WPMC’09, Sendai, Japan, 7–10 Sept 2009.

  12. 12

    Batra A. (2004) Design of a multiband OFDM system for realistic UWB channel environments. IEEE Transactions on Microwave Theory and Techniques 52: 2123–2138

  13. 13

    Batra, A. (2004). Multi-band OFDM physical layer proposal for IEEE 802.15Task Group 3a IEEE P802.15-04/0493r1.

  14. 14

    Standard ECMA-368 (2008). High Rate Ultra Wideband PHY and MAC Standard (3rd ed.).

  15. 15

    Hassibi B., Vikalo H. (2005) On the sphere-decoding algorithm I. Expected complexity. IEEE Transactions on Signal Processing 53(8): 2806–2818

  16. 16

    Giannakis G. B., Liu Z., Ma X., Zhou S. (2007) Space-time coding for broadband wireless communications. Wiley, New York

  17. 17

    Su W., Safar Z., Liu K. J. R. (2005) Towards maximum achieveable diversity in space, time, and frequency: peraformance analysis and code design. IEEE Transactions on Wireless Communications 4(4): 1847–1857

  18. 18

    Su W., Safar Z., Liu K. J. R. (2005) Full-rate full-diversity space-frequency codes with optimum coding advantage. IEEE Transactions on Information Theory 51(1): 229–249

  19. 19

    Zhang W., Letaief K. B. (2007) Space-time/frequency coding for MIMO-OFDM in next generation broadband wireless systems. IEEE Wireless Communications Magazine 14(3): 32–43

  20. 20

    Ma X., Giannakis G. B. (2005) Space-time-multipath coding using digital phase sweeping or block circular delay diversity. IEEE Transactions on Signal Processing 53(3): 1121–1131

  21. 21

    Fang K., Leus G. (2010) Space-time block coding for doubly-selective channels. IEEE Transaction on Signal Processing 58(3): 1934–1940

  22. 22

    Ma X., Zhang W. (2008) Fundamental limits of linear equalizers: Diversity, capacity, and complexity. IEEE Transactions on Information Theory 54(8): 3442–3456

  23. 23

    Zhang W., Ma X., Gestner B., Anderson D. V. (2009) Designing low-complexity equalizers for wireless systems. IEEE Communications Magazine 47(1): 56–64

  24. 24

    Su B., Vaidyanathan P. P. (2008) New blind block synchronization for transceivers using redundant precoders. IEEE Transaction on Signal Processing 56(12): 5987–6002

  25. 25

    Manton J. H., Neumann W. D. (2003) Totally blind channel identification by exploiting guard intervals. Systems & Control Letters 48(2): 113–119

  26. 26

    Lu, H., Xu, T. & Nikookar, H. (2010). Performance analysis of the STFC for cooperative ZP-OFDM diversity, capacity and complexity. In Proc. WPMC’10, Recife, Brazil, 11–14 Oct 2010.

  27. 27

    Lu, H., Xu, T., & Nikookar, H. (2011). A cooperative scheme for ZP-OFDM with multiple carrier frequency offsets over multipath channel. In Proc. IEEE Vehicular Technology Conference (pp. 1–5). Budapest, Hungary.

  28. 28

    Zhang, J. K., Liu, J., & Wong, K. M. (2005). Linear Toeplitz space time block codes. In Proc. IEEE ISIT’05, Adelaide, Australia.

  29. 29

    Shang Y., Xia X. G. (2008) On space-time block codes achieving full diversity with linear receivers. IEEE Transactions on Information Theory 54: 4528–4547

  30. 30

    Wang H., Xia X. G., Yin Q. (2009) Distributed space-frequency codes for cooperative communication system with multiple carrier frequency offsets. IEEE Transactions on Wireless Communication 8: 1045–1055

  31. 31

    Avestimehr A. S., Tse D. N. C. (2007) Outage capacity of the fading relay channel in the low-SNR regime. IEEE Transactions on Information Theory 53(4): 1401–1415

  32. 32

    Telatar I. E. (1999) Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications 10(6): 585–595

  33. 33

    Pammer, V., Delignon, Y., Sawaya, W., & Boulinguez, D. (2003). A low complexity suboptimal MIMO receiver: The combined ZF-MLD algorithm. In Proc. Personal, Indoor and Mobile Radio Communications (Vol. 3, pp. 2271–2275). Beijing, China.

  34. 34

    Hassibi B., Vikalo H. (2005) On the sphere-decoding algorithm I. Expected complexity. IEEE Transactions on Signal Processing 53(8): 2806–2818

  35. 35

    Windpassinger C., Lampe L., Fischer R. F. H., Hehn T. (2006) A performance study of MIMO detectors. IEEE Transactions on Wireless Communications 5(8): 2004–2008

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Correspondence to Hao Lu.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Lu, H., Xu, T., Nikookar, H. et al. Performance Analysis of the Cooperative ZP-OFDM: Diversity, Capacity and Complexity. Wireless Pers Commun 68, 587–608 (2013). https://doi.org/10.1007/s11277-011-0470-9

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Keywords

  • Cooperative communication
  • OFDM
  • Zero-padding
  • Diversity
  • Capacity
  • Complexity
  • Carrier frequency offsets
  • Tall Toeplitz
  • Linear equalizers
  • Orthogonality deficiency