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

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.


Introduction
Orthogonal frequency division multiple access (OFDMA) and its variant forms are widely used as multicarrier modulation techniques in current wireless standards such as worldwide interoperability for microwave access (Wi-MAX), Long Term Evolution-Advanced (LTE-Advanced) etc. OFDMA has certain key features like high spectral efficiency, robustness against multi-path fading etc., which renders it particularly appealing to the contemporary and futuristic applications in the challenging wireless communication environment [1]. In particular, a modified version of OFDMA known as single carrier-frequency division multiple access (SC-FDMA) has the ability to ensure maximum amplifier efficiency with minimal distortion and is thus used in LTE uplink [2]. This effectively deals with the peak to average power ratio (PAPR) problem of OFDMA systems [3]. In SC-FDMA systems, data symbols are pre-coded with discrete Fourier transform (DFT), thereby converting the system into a single carrier system while preserving the properties of orthogonal frequency division multiplexing (OFDM) [4][5][6][7].
SC-FDMA systems offer multiple advantages and overcome many of the limitations of conventional OFDMA which are of paramount importance in contemporary wireless communication needs and demands [8,9]. It alleviates the high PAPR due to envelope fluctuation of the signals due to symbol mapping [10]. There are many secondary advantages to this like longer battery life, smaller packaging for batteries which are of importance in the proliferation of mobile devices. SC-FDMA systems with appropriate equalization are less sensitive to CFOs compared to conventional OFDMA systems [11,12]. It has better resilience to spectral nulling also [13].
Yet, all multiple access systems based on both SC-FDMA and OFDMA suffer from two major issues. These are the carrier frequency offset (CFO) and radio in-phase/quadrature phase (I/Q) imbalances [14,15]. The CFO arises as a consequence of the mismatch between local oscillator (LO) frequencies at the transmitter (TX) and receiver (RX) sides. It could also be due to the mobility of users. The CFOs experienced by different users in multiuser uplink systems will be distinct from each other. The deleterious consequence of this is that the mutual orthogonality between subcarriers is destroyed thus leading to intercarrier interference (ICI) and inter-user interference (IUI) [16,17].
Wireless communication systems, in general employ two physically generated signals, viz., a sine wave and its 90-degree delayed version for both up conversion and down conversion. These are called as the I and Q signal components. In practice, however, the phase delay need not be exactly 90 degrees. Also, the amplitudes of these component signals are never perfectly matched for each of the signal paths. This imbalance in amplitude and phase between the I and Q branches is termed as I/Q imbalance [17]. The mismatches associated with the I and Q signal components create interference between various users. This is especially predominant between subcarriers located on either side of the central frequency. I/Q imbalance may cause ICI through own signal image or IUI through image signal of other users located on mirror sub bands. This interference is known as mirror frequency interference [18]. For multi-user systems like SCFDMA/OFDMA, bandwidths and power levels of different users are disparate. Because of this, users with weaker power levels are bound to experience severe interference from users with higher power levels. Also, one or more narrow band users may interfere with a wideband user. This leads to strong IUI and thus drastic degradation in bit error rate (BER) performance. Hence, effective equalization schemes have a pivotal role on increasing the system performance of SC-FDMA uplink systems.
Even though, CFO compensation issues in OFDM based multiple access systems are a thoroughly explored area of research, compensation of I/Q imbalance has not received the necessary attention until recently, especially in the last five to eight years only. For example, in [19], the BER performance of OFDM systems impaired by CFO is analysed. Yet it does not propose an efficient compensation method. I/Q imbalance issues are treated in various ways in in [18,20,21]; but a careful perusal shows that the proposed methods have various limitations. For example, the impact of joint frequency-selective TX and RX I/Q imbalances on error vector magnitude is investigated in [20]. However, the methods to compensate these adverse effects are not discussed. The compensation of TX I/Q in an OFDM system is dealt with in [22], but it is not appropriate for multi-user systems employing SC-FDMA or OFDMA. In a similar manner, estimation and equalization of TX I/Q imbalances in OFDMA systems are explored in [23], but with the proposed compensation method having low spectral efficiency. Data detection and channel estimation algorithms are proposed for OFDMA systems under I/Q imbalance in [23]. The applicability of these algorithms is limited to time invariant channels only, which severely curtails its practical significance. A fixed point compensator for I/Q imbalance is discussed in [24]. However, the method is not suitable for multi carrier systems. In [25,26] the impact of I/Q amplitude and phase imbalance on the performance of receive space modulation techniques and OFDM is analyzed, respectively. These works gives insight to the impairments caused at the receiver due to I/Q imbalances. However, multiuser multicarrier uplink systems are not discussed in these works.
Moreover, the studies dealing with I/Q imbalance compensation for SC-FDMA systems are very few in number. Zero-forcing equalization for compensation of TX I/Q imbalance is proposed in [18]. An important limitation of this work is that, only the TX I/Q imbalance is considered, without giving consideration to the imbalance in RX I/Q. Also, the range considered for the CFOs is only −200 Hz to 200Hz, which is insignificant when we consider real systems. Another, widely linear equalization technique to compensate I/Q imbalance is proposed in [21]. However, the high complexity of the technique curtails its utility in practical systems. A rigorous survey of the related literature shows that, existing I/Q imbalance compensation and CFO synchronization techniques for OFDMA have one or more of the following limitations or drawbacks: (1) They have a limited range of operation in terms of CFO and temporal behaviour (many techniques limited to only time invariant systems), (2) addresses either the imbalance at the TX or RX only and not both together, (3) high complexity in practical implementation. To make it clearer, most of the existing works addresses TX I/Q imbalance only and make the assumption that RX imbalance is compensated a-priori. However, this assumption is not tenable in a practical sense. There are very few works that consider the joint compensation of both I/Q imbalances and CFOs. Also, the equalization techniques for SC-FDMA uplink systems under both the TX and RX I/Q imbalances are not addressed adequately in the existing literature. Thus, there is a great need for highly effective techniques which can jointly handle both TX and RX radio imbalances and synchronization errors. Our contributions in this paper are summarized as follows: • We propose two novel synchronization and equalization techniques for SC-FDMA uplink system under TX and RX I/Q imbalances. Even though we consider SC-FDMA systems in our simulation studies, the proposed methods can be applied to OFDMA systems also with slight modifications. The proposed methods effectively perform the equalization for SC-FDMA uplink system under TX and RX I/Q imbalances, CFOs and channel distortions.

3
• We derive both the time and frequency domain analytical expressions for received signal in SC-FDMA systems under TX and RX I/Q imbalances, CFOs and channel distortions. This mathematical analysis provides insight into the unfavourable effects caused by radio imbalances and CFOs. • Based on the mathematical modelling of the received signal, we propose a joint equalization technique to compensate TX and RX imbalances, CFOs and channel, by applying the concepts of oblique projection (OP) technique which is exploited for separating the desired signal components. • Additionally, we propose a low complexity technique for the joint equalization of radio imbalances in TX and RX sections, CFOs and channel impairments. • Results of the simulation studies illustrate that the proposed equalization techniques offer satisfactory BER performance.
The remainder of this paper is organized as follows. Section 2 describes the transmitted and received signal models under radio imbalances and CFOs. An equalization technique based on OP and an alternative technique for the joint equalization of radio impairments, CFOs and channel are proposed in Sects. 3 and 4, respectively. The detailed study of the simulation results and performance evaluation is done in Sects 5. 6 concludes the paper. Notations Used: We use lower-case, boldface symbols like a to represent a vector and upper-case, boldface notations like A to represent a matrix. The notation (⋅) H is used for representing the Hermitian transpose of a vector or matrix and (⋅) T for the transpose of a vector or matrix. Convolution operation is denoted by ⋆ and (⋅) * represents complex conjugation operation. The (⋅) † is pseudo-inverse operation and (⋅) M denotes modulo-Moperation.
where R is a rotational matrix whose (i, m) th element is given by

System Model
We consider an SC-FDMA uplink system with K active users sharing M subcarriers and the i th user is allocated with N i subcarriers for transmission with In SC-FMDA as well in OFDMA systems, it is a well-known fact that the signal undergoes frequency selective fading. Due to the mismatch in phase between the transmitter and the receiver, the faded signal at the receiver will get corrupted. The CFOs will lead to inter carrier interference (ICI) and multi-user interference. Apart from the effect of CFOs, the mismatch in I and Q branches of the transmitter and receiver processing units will lead to mirroring effects. In our proposed work, we consider a scenario where all the above-mentioned effects come in to picture. The block diagram representation of SC-FDMA system under I/Q imbalances and CFOs and its corresponding equalization scheme is shown in Fig. 1. We consider generalized carrier assignment scheme (GCAS) with all K active users communicating with the base station simultaneously. In GCAS, each user is assigned with a set of mutually exclusive subcarriers with respect to other users in the uplink. Firstly, the data bits related to each user is converted into parallel where s i (m) ∈ M is the data symbol of the i th user which belongs to the given constellation and S i (k) is the precoded data symbol. The precoded data symbol S i (k), is then assigned to a particular subcarrier based on the GCAS. Note that the subcarrier indices are generated in pseudorandom manner in GCAS. Hence, the proposed technique is applicable under any subcarrier assignment scheme. The data on the m th subcarrier is denoted by represents the indices of subcarriers allocated to i th user, which has N i elements. After performing the subcarrier mapping, the data symbols are converted in to time domain, complex baseband signal by taking the M-point inverse-fast Fourier transform (IFFT) operation. To avoid inter symbol interference (ISI), a cyclic prefix (CP) of length N cp is then added in such a way that, the length of CP is greater than the maximum length of channel taps. The time domain signal obtained after performing the IFFT operation and appending the CP is expressed as As mentioned earlier, the baseband signal which passes through the modulator for up-conversion will get distorted due to TX I/Q imbalance. This impairment will lead to mirroring effect. The baseband equivalent of the transmit radio frequency signal impaired by TX I/Q imbalance is given by [18] where we define the terms in (4) as z i and i T being the amplitude and phase imbalances between the I and Q branches of the modulator, respectively [18]. Note that, h i I (t) and h i Q (t) are the impulse responses of I and Q branches of the modulator. In an ideal situation without any I/Q imbalance we have g i However, the effect of I/Q imbalance will lead to the addition of mirrored signal with the original signal. The adverse effects of I/Q imbalance is the interference from mirror subcarriers as illustrated in Fig. 2. The frequency domain impaired transmitted signal corresponding to i th user can be expressed in vector form as  Fig. 2 Mirroring effects due to I/Q imbalance where H i is the M × M frequency domain matrix that contains the i th user channel frequency response on its main diagonal and H i is the diagonal matrix with frequency response of channel corresponding to i th user. In this paper, we consider a realistic scenario with both transmitter and receiver I/Q imbalances. In such a situation the LOs of TX and RX will not be perfectly synchronized and it will lead to CFO. By including the CFO effect to (6), the signal received at the front end of uplink is a superposition of signals from all the users. It can be expressed as . Note that, is the normalized CFO and i is the phase shift introduced due to the CFO of the i th user. As mentioned earlier, when the signal gets processed through the demodulator section at the receiver by down converting, it will get impaired due to RX I/Q imbalance. This is due to the mismatch between the amplitude and phase of the parallel signal processing sections of demodulator. Hence, the overall received signal of SC-FDMA uplink system under TX, RX imbalances and CFO impairments in time domain complex baseband representation is given as is the sum of individual received signals in time domain and is impaired by CFO. Herer(t) = IFFT(R) represents the baseband time domain symbol at the receiver front end. The noise term under RX I/Q imbalance is given as ñ(t) = z 1R (t) ⋆ n(t) + z 2R (t) ⋆ n(t) * . Here z 1R (t) and z 2R (t) are RX compleximbalance filters given by z 1R (t) ≅ h I (t) + g R exp −j R h Q (t) ∕2 and z 2R (t) ≅ h I (t) − g R exp j R h Q (t) ∕2 , respectively. Note that g R and R represent the gain imbalance and phase imbalance of the receiver, respectively, at the demodulator section. Also, it is a well-known fact that the convolution operation in (8) will change into multiplication operation in frequency domain. As a result of it, the spectra of different users may overlap each other which may result in interference between the active users in the uplink. Hence, the mirroring effect will lead to inter user interference also. Therefore, the overall received symbol vector in frequency domain can be represented as where Z 1R and Z 2R are M × M frequency domain diagonal matrices whose diagonal elements are M -point DFTs of RX I/Q imbalance filters z 1R and z 2R , respectively. The conjugated and mirrored version of R is denoted as R . Hence it is evident that the received signal is interfered with the conjugated mirrored version of the signal received at the base station. In this paper, we propose efficient equalization techniques at the uplink to compensate the adverse effects due to channels, CFOs and I/Q imbalance due to transmitter as well as the receiver. The upcoming sections elaborates the equalization schemes.

Equalization of TX-RX I/Q Imbalances, Channels and CFOs Using Projection
Our proposed equalization scheme makes use of OP technique for separating the desired signal components from the received uplink signal. A brief review on OP is provided in Appendix-A. At first, we rewrite the equation of the received signal. After performing the conjugate mirroring operation on (9), we obtain the mirrored version of the overall received signal vector as By stacking together (9) and (10), the system of equations can be of the form as shown below The equation (11) can be simplified as Where Ÿ is the vector obtained by vertically concatenating the overall received vector Y and its conjugated mirror vector Y , A 1 is the 2M × M matrix that is obtained by vertically concatenating matrices Z 1R and Z 2R . Similarly, A 2 is the 2M × M matrix that is obtained by vertically concatenating matrices Z 2R and Z 1R , respectively. From (12), it is evident that Ÿ contains two distinct components, Y 1 = A 1 R and Y 2 = A 2 R . Among the two components in (12), the first component is the desired part, and the second component is the component that arises due I/Q imbalance. It is evident that Ÿ is a superposition of signals Y 1 and Y 2 . Hence, the function of the equalizer is to extract the desired component from the overall impaired receive signal. We apply the principle of oblique projection technique to efficiently separate the undesired signal components [27]. More details regarding the OP projection operator and its significance is given in Appendix-A. We define OP operator to separate the signal Y 1 , from Ÿ by projecting Ÿ obliquely on to the column space of Y 1 and null space of Y 2 . The OP operator for achieving this task is given by [27,28] where A H 1 is the Hermitian transpose of A 1 , and P ⟂ Using the OP operator, the estimate of Y 1 can be obtained as [28] Hence, the first component can be separated out using (14). By using the first useful component, we can estimate the sum of individual received signals R as, where R is the received signal, which is a superposition of signals from all active users, and it is also free from receiver I/Q imbalance. From (14) and (15), it is evident that the . By making use of (13), we obtain � . The next step is to find the estimate of the transmitted data symbols from the sum of individual received signals. We can also rewrite R as the superposition of individual user signal as given below For a multi-user scenario, by substituting the expression for impaired TX signal in frequency domain, this can be further reduced to where the matrices and are M × KM matrices that characterizes the effect of CFO, channel and TX I/Q imbalances. Note that the vector, X = X 1 , X 2 , … , X K T represents data symbols of all users and X = X 1 ,X 2 , … ,X

K T
represents the vector containing mirrored and conjugated data symbols corresponding to X . The conjugated mirror subcarrier of R is obtained as where � R is the conjugated mirrored version of R . Also, Ḡ 1T and Ḡ 2T are M × KM matrices. By combining (17) and (18), the system of equations can be rearranged in a matrix form as This can be simplified as � R is the vector obtained by vertically concatenating the overall received vector R and conjugate of its mirror vector � R , that is Using OP operator given in (21), the estimate of first component in � R is obtained as From the estimate of first component, the estimate of X can be given as Using (22) and (23), we have � After obtaining X , the individual user data is separated out by making use of the CAS and N-point inverse DFT (IDFT) is performed. Further, de-mapping is done to get the estimate of original data bits of each user.

Alternate Technique for Equalization of TX and RX I/Q Imbalances and CFOs
The mismatch in carrier frequency between the LOs used at TX and RX sections results in ICI as well as IUI. Similarly, radio imbalances cause interference among mirror users. Hence, recovering the individual user data from the overall received signal is a challenging task. The expression for overall received vector in frequency domain representation for SC-FDMA systems under TX I/Q imbalances, RX I/Q imbalances and CFOs is given by (9). The overall received signal contains both the desired component and the interference component. The conjugated mirror vector of the received vector is given by (10). In order to separate out the desired component from the overall received vector, (11) can be simplified as where A is the 2 M × 2M matrix obtained by horizontally concatenating the matrices A 1 and A 2 in (12) and r is the vector obtained by vertically concatenating r and r . Now (24) is of the form b = Ax which can be solved for x as x = A −1 b . Therefore, estimate of r is obtained as By solving (25), 2M values are obtained out of which the first M elements represent the estimate of the overall received signal without RX imbalance and the remaining M elements represent its mirrored and conjugated version. Now, proceeding in the same way, (20) can be simplified as where B is a 2 M × 2KM matrix obtained by horizontally concatenating the matrices B 1 and B 2 in (20), which represents the distortion effects caused by TX I/Q imbalance, CFOs and channel. This system of equations has greater number of unknowns than the number of variables. In order to solve this system of equations without any rank deficiency problem, the columns of B are chosen corresponding to the non-zero entries in ẍ . Hence, a new matrix B is obtained with dimension 2 M × 2M . So, by solving this, a set of non-zero values are obtained in which the first KN elements correspond to the original data of each user and the remaining KN elements correspond to its mirrored and conjugated versions.
By applying N-point IDFT on the first KN elements and subsequent demodulation gives the estimate of original data bits of each user.

Computational Complexity
The computational complexity of the proposed methods is evaluated based on the number of multiplications and addition involved in the computations. The computational complexity of the proposed methods for K= 2 users and for a single iteration is shown in Table 1. It can be seen that, the computational complexity of the alternate equalization method is less than that of the equalization based on OP.

Results and Discussions
In this section, we present the performance results of the proposed equalization techniques.

Simulation Scenario and Associated Parameters
We carried out extensive simulation studies to quantify the performance of the proposed equalization techniques. We considered a multicarrier system with a total of 1024 subcarriers.
Alternate equalization technique independently generated for each user and their effects are introduced to the uplink signals. The received signal is thus a sum of impaired transmitted signals which is further affected by CFO and RX I/Q imbalance. The proposed equalization techniques extract the data symbols of each individual user after compensating TX and RX I/Q imbalances as well as CFOs.

Performance Evaluation
The BER performance of the proposed equalization techniques as a function of signalto-noise ratio (SNR) is shown in Fig. 3. It also depicts the variations in BER when the number of users has increased from K = 2 to K = 4 . The x-axis in the figure represents the SNR expressed in dB and y-axis represents the corresponding BER values. The normalized CFOs are assumed to be uniformly distributed random variables in the range of (−0.5, 0.5) which corresponds to CFOs in the range of (−7.5, 7.5) kHz. It is evident from Fig. 3 that both the methods offer comparable performance when K = 2 and there is not so much performance gap between the two proposed methods. When K = 4 , OP-based equalization technique outperforms the alternate equalization technique. However, the difference in BER is not so high. Even though radio imbalances cause severe performance degradation, the proposed methods can equalize the impaired overall received data and thereby greatly reduce the BER. The performance of alternate technique is affected with the number of users. The performance difference is more evident at higher SNRs. For example, in a 4-user scenario, the alternate equalization scheme exhibits a BER of 2 × 10 −2 at a SNR of 30 dB , whereas the same BER can be achieved using OP based technique with a SNR of 20 dB. The BER performance of proposed methods at different normalized CFOs at an SNR of 35 dB are shown in Fig. 4. It is evident from the simulation studies that both the methods can offer BER of 10 −3 for normalized CFOs in the range of −0.4 to 0.4. Between the two proposed techniques, the OP based method gives improved performance as compared to alternate equalization technique at higher CFO values.
The computational complexity of the proposed techniques in terms of number of multiplications and additions involved in the equalization process is illustrated in Fig. 5. In this example we have considered M = 2048 . The number of users, K is varies from 2 to 64, so that the value of N = M∕K . It is found that the computational complexity of the OP based technique is nearly five times that of alternate equalization technique. However, the superior performance of OP technique is clearly visible in Fig. 3 as the number of active users in the uplink increases.

Conclusion
In this paper, we presented two novel equalization techniques for the joint compensation of TX and RX I/Q imbalances in the SC-FDMA uplink systems. We introduced an OP-based technique which offers satisfactory performance by incorporating both TX and RX I/Q imbalances under the presence of CFOs and channel impairments. It also ensures good performance even at higher values of normalized CFOs. Additionally, we proposed an alternative equalization technique which can compensate the TX and RX I/Q imbalances with reduced computational complexity. The alternative method is also able to compensate the large range of normalized CFOs and the adverse effects caused by the channel, making it very much suitable for the practical scenario. The results of simulation studies illustrate that the proposed equalization techniques effectively compensate the impairments and offer good performance making them an extremely attractive choices for implementation in upcoming wireless systems with low-cost modulators. Even though the focus is on SC-FDMA system, the proposed methods are equally applicable to OFDMA systems. The proposed methods can also be extended to generalized subcarrier scheme and to their corresponding multiple input-multiple output (MIMO) versions by employing multiple number of antennas at TX and the RX sides. Let col{M} and col{N} denote the column space of matrices M m1×m2 and N m1×m3 , respectively, that intersect trivially. Then the OP operator onto col{M} along col{N} can be defined as [27] where P ⟂ N = I m1 − P N , with P N as the orthogonal projection matrix whose range is col{N} and is defined as For an OP operator with range M and null space N , E M|N M = M and E M|N N = 0 m1×m3 . Hence, the col{N} form the null space of the OP operator E M|N [28].

Author Contributions
The authors contributed equally to this research. All authors read and approved the final manuscript.

Availability of Data and Material
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.

Declarations
Conflicts of interest I/we certify that there is no actual or potential conflict of interest in relation to this article.