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Equalization Techniques for SC-FDMA Systems Under Radio Imbalances at Both Transmitter and Receiver

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Abstract

Orthogonal frequency division multiple access (OFDMA) is a multi-carrier, multiple access (MA) technique, which is widely adopted in contemporary wireless standards. Single carrier-frequency division multiple access (SC-FDMA) is a modified version of OFDMA which employs single carrier transmission by pre-coding the data symbols using discrete Fourier transform (DFT). However, these systems are highly vulnerable to the adverse effects arising in the channel, carrier frequency offset (CFO) and in-phase/quadrature phase (I/Q) imbalances. In the uplink communication scenario, the signal received at base station is the superposition of signals from all the active users. Even though the adverse effects caused by the communication channel and CFOs are addressed in the related literature extensively, the effect of I/Q-imbalances at the transmitter and the receiver is rarely considered. The effect of I/Q-imbalances will make the equalization process at the base station more complex. It is because the overall effective channel with radio impairments must be included in the system modelling. Hence the receiver processing should contain the equalization for effect of the channel, CFOs and I/Q imbalances. In this paper, we propose a novel technique based on the oblique projection (OP) technique for equalizing the channel, radio imbalances and synchronization errors caused by both transmitter (TX) and receiver (RX) for OFDMA/SC-FDMA uplink systems. We also propose an equalization technique with reduced computational complexity to overcome the adverse effects caused by I/Q imbalances and CFOs. The results of the simulation studies illustrate that the proposed techniques can compensate all the above-mentioned effects and they offer very good performance under both TX and RX I/Q imbalances.

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The datasets supporting the results of this article are included within the article.

Code Availability

The code generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number (GRP/131/43).

Author information

Authors and Affiliations

  1. Department of Electronics and Communication Engineering, MES College of Engineering, Kuttippuram, Kerala, India

    P. Sheeba

  2. Department of Electrical Engineering, College of Engineering, King Khalid University, Abha, Saudi Arabia

    P. Muneer, V. P. Thafasal Ijyas & Mohammed Usman

  3. Department of Electronics Engineering, AMU, Aligarh, India

    Mohd Wajid

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  1. P. Sheeba
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  2. P. Muneer
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  3. V. P. Thafasal Ijyas
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  4. Mohammed Usman
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The authors contributed equally to this research. All authors read and approved the final manuscript.

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Correspondence to P. Muneer.

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Appendix A Brief review on oblique projection

Appendix A Brief review on oblique projection

The OP operator can be applied to improve the signal component along the desired subspace while cancelling out the unwanted component of signal that belongs to an undesired subspace. The basic idea of OP is illustrated in Fig. 6. Let \(R\left( \varvec{M}\right)\) and \(R\left( \varvec{N}\right)\) represent two subspaces of \({\mathbb {C}}^{m1}\) that are intersecting trivially, then\(\ R\left( \varvec{M}\right) \cap R\left( \varvec{N}\right) \varvec{=}\varvec{0}\). Then the OP operator on \(R\left( \varvec{M}\right)\) along \(R\left( \varvec{N}\right)\) is the linear operator \({\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\) and it satisfies the following conditions [27]:

Fig. 6
figure 6

Graphical representation of OP operator [27]

Full size image

  • \(\forall {\varvec{m}}\in R\left( \varvec{M}\right) ,{\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{m}\varvec{=}\varvec{m}\)

  • \(\forall {\varvec{n}}\mathrm {\in }R\left( \varvec{N}\right) ,\varvec{\ }{\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{n}\varvec{=}\varvec{0}\)

  • \(\forall {\varvec{z}}\in {\mathbb {C}}^{m1},\varvec{\ }{\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{z}\varvec{=}{\varvec{z}}_{\varvec{M}}\in \varvec{\ }R\left( \varvec{M}\right)\)

Let \(col\left\{ \varvec{M}\right\}\) and \(col\left\{ \varvec{N}\right\}\) denote the column space of matrices \({\varvec{M}}_{m1\times m2}\) and \({\varvec{N}}_{m1\times m3}\), respectively, that intersect trivially. Then the OP operator onto \(col\left\{ \varvec{M}\right\}\) along \(col\left\{ \varvec{N}\right\}\) can be defined as [27]

$$\begin{aligned} {\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{=}\varvec{M}{\left( {\varvec{M}}^H{\varvec{P}}^{\perp }_{\varvec{N}}\varvec{M}\right) }^{-1}{\varvec{M}}^H{\varvec{P}}^{\perp }_{\varvec{N}} \end{aligned}$$
(27)

where \({\varvec{P}}^{\perp }_{\varvec{N}}={\varvec{I}}_{m1}-{\varvec{P}}_{\varvec{N}}\), with \({\varvec{P}}_{\varvec{N}}\) as the orthogonal projection matrix whose range is \(col\left\{ \varvec{N}\right\}\) and is defined as

$$\begin{aligned} {\varvec{P}}_{\varvec{N}}\varvec{=}\varvec{N}{\left( {\varvec{N}}^H\varvec{N}\right) }^{-1}{\varvec{N}}^H \end{aligned}$$
(28)

For an OP operator with range \(\varvec{M}\) and null space \(\varvec{N}\), \({\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{M}\varvec{=}\varvec{M}\) and \({\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\varvec{N}\varvec{=}{\varvec{0}}_{\varvec{m}\varvec{1}\varvec{\times }\varvec{m}\varvec{3}}\). Hence, the \(col\left\{ \varvec{N}\right\}\) form the null space of the OP operator \({\varvec{E}}_{\varvec{M}\left| \varvec{N}\right. }\) [28].

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Sheeba, P., Muneer, P., Thafasal Ijyas, V.P. et al. Equalization Techniques for SC-FDMA Systems Under Radio Imbalances at Both Transmitter and Receiver. Wireless Pers Commun 129, 2563–2581 (2023). https://doi.org/10.1007/s11277-023-10246-1

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