Skip to main content
Log in

Symbol Error Rate Analysis of OFDM System with CFO Over TWDP Fading Channel

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

In this manuscript, an exact symbol error rate analysis of orthogonal frequency division multiplexing system is presented in the presence of carrier frequency offset over two wave with diffuse power (TWDP) fading channel. Both Binary Phase Shift Keying and Quadrature Phase Shift Keying modulation techniques are considered in this study over TWDP channel which contains Rayleigh, Rician and two ray fading as its special cases. The results are validated by means of Monte Carlo simulations and also verified from the benchmark results available in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Zhang, C. J., et al. (2016). New waveforms for 5G networks. IEEE Communications Magazine, 54(11), 64–65.

    Article  Google Scholar 

  2. Keller, T., & Hanzo, L. (2000). Adaptive multicarrier modulation: A convenient framework for time–frequency processing in wireless communications. Proceedings of the IEEE, 88(5), 611–640.

    Article  Google Scholar 

  3. Rugini, L., & Banelli, P. (2005). BER of OFDM systems impaired by carrier frequency offset in multipath fading channels. IEEE Transactions on Wireless Communications, 4(5), 2279–2288.

    Article  Google Scholar 

  4. Zhou, P., Jiang, M., Zhao, C., & Xu, W. (2007). Error probability of OFDM systems impaired by carrier frequency offset in frequency selective Rayleigh fading channels. In IEEE ICC (pp. 1065–1070).

  5. Sathananthan, K., & Tellambura, C. (2001). Probability of error calculation of OFDM systems with frequency offset. IEEE Transactions on Wireless Communications, 49(11), 1884–1888.

    Article  Google Scholar 

  6. Dharmawansa, P., Rajatheva, N., & Minn, H. (2009). An exact error probability analysis of OFDM systems with frequency offset. IEEE Transactions on Communications, 57(1), 26–31.

    Article  Google Scholar 

  7. Hamza, A. M., & Mark, J. W. (2014). Closed form SER expressions for QPSK OFDM systems with frequency offset in Rayleigh fading channels. IEEE Communications Letters, 18(10), 1687–1690.

    Article  Google Scholar 

  8. Hamza, A. M., & Mark, J. W. (2015). Closed-form expressions for the BER/SER of OFDM systems with an integer time offset. IEEE Transactions on Communications, 63(11), 2279–2288.

    Article  Google Scholar 

  9. Saberali, S. A., & Beaulieu, N. C. (2013). New expressions for TWDP fading statistics. IEEE Wireless Communications Letters, 2(6), 643–646.

    Article  Google Scholar 

  10. Dixit, D., & Sahu, P. (2013). Performance of QAM signaling over TWDP fading channels. IEEE Transactions on Wireless Communications, 12(4), 1794–1799.

    Article  Google Scholar 

  11. Durgin, G. D., et al. (2002). New analytical models and probability density functions for fading in wireless communications. IEEE Transactions on Communications, 50(6), 1005–1015.

    Article  Google Scholar 

  12. Jeffrey, A., & Zwillinger, D. (2007). Table of integrals, series, and products. Cambridge: Academic Press.

    Google Scholar 

  13. Simon, M. K., & Alouini, M.-S. (2005). Digital communication over fading channels (2nd ed.). New York: Wiley.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daljeet Singh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1: Analytical Expression for BPSK

The integral (5) can be simplified by using the definition of Q function given in [13] and expressed as a sum of four similar terms-

$$\begin{aligned} P_s(\xi )=\frac{1}{2^{N-1}}\sum _{k=1}^{2^{N-2}} \left[ I_1 + I_2 + I_3 + I_4\right] , \end{aligned}$$
(9)

Here, the computation of \(I_1\) is considered as an example where, \(I_1\) is defined as-

$$\begin{aligned} I_1&=\frac{1}{\pi }\int _{0}^{\pi /2}\sum _{i=1}^{L} C_i \frac{\exp (-K)}{C_{\tau _1\tau _1}} \frac{\exp (\beta _iK)}{2} \int _{0}^{\infty } 2\tau \exp \left( \frac{-\tau ^2}{C_{\tau _1\tau _1}}\right) \\&\quad \exp \left( \frac{-\tau ^2[\mathfrak {R}(S_1)+a_k]^2}{(C_{\tau _1\tau _1}+b_k)\sin ^2\psi }\right) I_0\left( \tau \sqrt{\frac{4K(1-\beta _i)}{C_{\tau _1\tau _1}}} \right) d\tau d\psi . \end{aligned}$$
(10)

After some mathematical adjustments and using the identity [12, (6.61)], the above expression can be rewritten as

$$\begin{aligned} I_{1}&= \frac{1}{\pi }\int _{0}^{\pi /2} \sum _{i=1}^{L} \frac{C_i}{2} \exp (-K+\beta _iK) \\&\quad \bigg [\exp \left( \frac{K(1-\beta _i)\sin ^2\psi (C_{\tau _1\tau _1}+b_k)}{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2 + \sin ^2\psi (C_{\tau _1\tau _1}+b_k)}\right) \bigg ] \\&\quad \left( \frac{\sin ^2\psi (C_{\tau _1\tau _1}+b_k)}{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2+\sin ^2\psi (C_{\tau _1\tau _1}+b_k)}\right) . \end{aligned}$$
(11)

By using Maclaurin series for \(\exp (x) = \sum _{n=0}^{\infty } \frac{x^n}{n!}\) and by using [12], the integral in (11) can be solved as

$$\begin{aligned} I_{1}&= \sum _{i=1}^{L} \frac{C_i}{2} \Bigg [\exp (-K+\beta _iK) \sum _{l1=0}^{\infty } \frac{[K(1-\beta _i)]^{l1}}{l1!}\Bigg ] \\&\quad \left[ \frac{1}{2}\left( 1- \sqrt{\frac{\frac{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2}{C_{\tau _1\tau _1}+b_k}}{1+\frac{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2}{C_{\tau _1\tau _1}+b_k}}}\right) \right] ^{l1+1} \\&\quad \sum _{l2=0}^{l1}\begin{pmatrix} l1+l2 \\ l2\end{pmatrix} \left[ 1- \frac{1}{2}\left( 1- \sqrt{\frac{ \frac{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2}{C_{\tau _1\tau _1}+b_k}}{1+\frac{C_{\tau _1\tau _1}[\mathfrak {R}(S_1)+a_k]^2}{C_{\tau _1\tau _1}+b_k}}} \right) \right] ^{l2}. \end{aligned}$$
(12)

Similarly, the integrals \(I_2\), \(I_3\) and \(I_4\) can be solved and combined to form the complete SER expression given in (6).

Appendix 2: Analytical Expression for QPSK

The unconditional SER for QPSK is determined as

$$\begin{aligned} P_s(\xi ) = \int _{0}^{\infty } 1 - \frac{1}{2^{2N-2}} \sum _{k=1}^{2^{N-2}} \sum _{n=1}^{2^{N-2}} \sum _{m=1}^{4} Q\left( -\tau A_1\right) Q\left( -\tau A_2\right) f_\tau (\tau ) d\tau , \end{aligned}$$
(13)

where \(A_1=\frac{(S_A +\mu _{k,n}[1,m])}{\sqrt{\gamma C_{\tau _1\tau _1}/2} \sqrt{1+\frac{V_{k,n}[m]}{\gamma C_{\tau _1\tau _1}}}}, A_2=\frac{(S_B +\mu _{k,n}[2,m])}{\sqrt{\gamma C_{\tau _1\tau _1}/2} \sqrt{1+\frac{V_{k,n}[m]}{\gamma C_{\tau _1\tau _1}}}}\). By using the identity of product of two Q functions [7], and substituting the value of Q function from [13], \(f_\tau (\tau )\) from (3); the above expression (13) can be simplified as sum of four different integral terms-

$$\begin{aligned} P_s(\xi )=1-\frac{1}{2^{N-1}}\sum _{k=1}^{2^{2N-2}} \sum _{n=1}^{2^{N-2}} \sum _{m=1}^{4} \left[ I_1 + I_2 + I_3 + I_4\right] \end{aligned}$$
(14)

where \(I_A\) is defined as

$$\begin{aligned} I_1&=\frac{1}{2} \int _{0}^{\infty } \int _{0}^{\frac{\pi }{2}-\arctan (\frac{A_2}{A_1})} \exp \left( \frac{-\tau ^2A_1^2}{2 \sin ^2\theta } \right) \frac{2\tau }{C_{\tau _1\tau _1}} \exp \left( -K - \frac{\tau ^2}{C_{\tau _1\tau _1}} \right) \\&\quad \sum _{i=1}^{L} \frac{C_i}{2} \exp (\beta _i K) I_0\left( \frac{\tau }{\sqrt{C_{\tau _1\tau _1}/2}}\sqrt{2K(1-\beta _i)} \right) d \theta d\tau \end{aligned}$$
(15)

The outer integral in (15) can be solved by using [13] can be written as

$$\begin{aligned} I_{1}&= \frac{1}{2}\sum _{i=1}^{L} C_i \Bigg [ \exp (-K+\beta _i K) \sum _{l_1=0}^{\infty } \frac{1}{l_1!}\left( K(1-\beta _i)\right) ^{l_1} \Bigg ] \\&\quad \int _{0}^{\frac{\pi }{2}-\arctan (\frac{A_2}{A_1})} \left( \frac{ \sin ^2 \theta }{\frac{A_1^2 C_{\tau _1\tau _1}}{2} + \sin ^2 \theta } \right) ^{l_1+1} d\theta . \end{aligned}$$
(16)

Finally, using the identity from [13, App. A.5], the integral in (16) can be solved. Similarly, the integrals \(I_2\), \(I_3\) and \(I_4\) can be solved and combined to form the complete SER expression is given in (8).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, D., Kumar, A., Joshi, H.D. et al. Symbol Error Rate Analysis of OFDM System with CFO Over TWDP Fading Channel. Wireless Pers Commun 109, 2187–2198 (2019). https://doi.org/10.1007/s11277-019-06674-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-019-06674-7

Keywords

Navigation