# Frequency domain soft-constraint multimodulus blind equalization for uplink SC-FDMA

- Kabiru Akande
^{1}, - Naveed Iqbal
^{1}, - Azzedine Zerguine
^{1}Email author, - Naofal Al-Dhahir
^{2}and - Abdelmalek Zidouri
^{1}

**2016**:23

https://doi.org/10.1186/s13634-016-0317-3

© Akande et al. 2016

**Received: **27 June 2015

**Accepted: **1 February 2016

**Published: **20 February 2016

## Abstract

Single-carrier frequency division multiple access (SC-FDMA) has been adopted and employed as the standard in the 3rd Generation Partnership Project (3GPP) Long-Term Evolution (LTE) uplink multiple-access scheme. It offers comparable performance and complexity to orthogonal frequency multiple access scheme (OFDMA) with a lower peak to average power ratio (PAPR) offering power-efficient transmission and longer battery life to mobile terminals. However, due to its single-carrier nature, SC-FDMA performance degrades in channels with long impulse responses and becomes prohibitive to equalize when implemented in time domain (TD). Furthermore, of the seven SC-FDMA symbols in the LTE uplink slot, one full symbol is used for channel estimation leading to about 14 *%* throughput degradation. In this work, a novel frequency domain soft-constraint satisfaction multimodulus blind algorithm (FDSCS-MMA) is developed and proposed. The frequency domain approach results in computational complexity reduction while blind implementation ensured improved spectral efficiency and throughput. The algorithm convergence is further improved by normalization of each of the frequency bin in the weight update. Simulation results show superior performance of the developed algorithm over other blind algorithms.

## Keywords

## 1 Introduction

The demand for high data transmission rates has been on the rise in recent years with organizations and individuals requiring ultra high-speed data transmission scheme. Broadband wireless transmission is employed in delivering this high speed data requirement to subscribers in a very hostile radio environment which offers multipath to transmitted signal. The multipath could be severe requiring sophisticated corrective measures at the receiver. Orthogonal frequency multiple access scheme (OFDMA) is a popular technique which uses a low symbol rate modulation specially designed to cope with severe channel conditions in multipath environment [1]. However, it has high peak to average power ratio (PAPR) which imposes high-power penalty on the mobile users [2].

Single-carrier frequency division multiple access (SC-FDMA) is a variant of OFDMA with an additional discrete Fourier transform (DFT) processing block hence is referred to as DFT-coded OFDMA [3, 4]. It has been adopted in 3rd Generation Partnership Project (3GPP) Long-Term Evolution (LTE) uplink scheme due to its lower PAPR while maintaining comparable performance and complexity to OFDMA [5, 6]. The lower PAPR feature makes it suitable for uplink communication benefiting mobile users in terms of low-cost and improved, power-efficient transmission [5]. However, SC-FDMA is a single-carrier modulation technique whose performance degrades in a multipath environment and this gets worse with the severity of the multipath. Furthermore, frequency domain decision feedback equalization (FD-DFE) was proposed for SC-FDMA in both [7] and [8]. However, the solution assumed time-invariant and ideal channel estimate with reduction in bandwidth efficiency as pilot sequences are required for channel estimation. Both [9] and [10] equalized SC-FDMA without reference symbols but the equalizer was implemented in time domain making it unsuitable for channels with long impulse responses due to prohibitive computational complexity. An adaptive frequency domain DFE was also proposed in [11] with added complexity of encoder and decoder. The cost function in [12] seeks to minimize the average error for a block of received symbols which does not necessarily force/restore each of the transmitted symbols to its correct point on the signal constellation while [13] is essentially a time domain implementation and hence has high complexity [12]. However, since SC-FDMA technique is set up in frequency domain, it is easier to implement its equalization in frequency domain as this avoids a lot of complications [10].

This paper presents a novel frequency domain implementation of soft-constraint satisfaction multimodulus algorithm (FDSCS-MMA) for equalization of SC-FDMA. The proposed frequency domain implementation is based on SCS-MMA [14] which was derived by applying the principle of soft-constraint satisfaction to relax the constraints in Lin’s cost function [15]. This implementation avoids the use of reference symbols in order to improve the spectral efficiency and throughput. This is highly desired due to the fact that in the LTE uplink, a frame has 20 slots and each slot contains 7 SCFDMA symbols. Of these seven, one full training SC-FDMA symbol (preamble) is used followed by six data symbols (which has no training) and the channel is estimated (with channel-estimate-based approach, e.g., least squares) using this single preamble [16]. Hence, one out of seven SC-FDMA symbols in the LTE uplink is already designated for channel estimation leading to approximately 14 *%* throughput degradation [3]. Therefore, blind algorithms provide attractive solution for SC-FDMA equalization. Also, the frequency domain (FD) implementation greatly reduces the computational complexity [17] that is associated with time domain implementation in channels with long impulse responses and has many other advantages [18]. Therefore, the frequency domain approach results in computational complexity reduction, while blind implementation ensured improved spectral efficiency and throughput [19–22]. Furthermore, FDSCS-MMA achieve lower mean square error (MSE) than both the normalized FD-modified constant modulus algorithm (NFDMMA) [23] and the popular constant modulus algorithm (CMA). Finally, FDSCS-MMA convergence is greatly improved by normalization of each of the frequency bin in the weight update. We have used the square root of the spectral power of the equalizer input for our normalization rather than the spectral power considered in [14] as we found that this gives better performance. Specific contributions as presented in this paper include: (1) frequency domain implementation of SCS-MMA, (2) convergence improvement of FDSCS-MMA to realize normalized FDSCS-MMA, (3) adaptation and implementation of NFDSCS-MMA for the equalization of SC-FDMA, (4) reduced overhead and improved bandwidth efficiency compared to channel estimation algorithms, (5) superior phase recovery and intersymbol interference (ISI) optimization capability compared to other popular blind algorithms such as CMA and MMA.

This paper is organized as follows: Section 2 details the mathematical description of SC-FDMA system. Section 3 provides the time domain (TD) implementation of the blind algorithms. Section 4 describes the FD implementation of the proposed algorithms. Section 5 shows simulation results of the performance for the proposed equalizers. Section 6 concludes the paper.

## 2 Description of SC-FDMA

*a*

_{ n }} are first modulated into symbols using any of the modulation methods (BPSK, QPSK or M-QAM). For the

*q*th user, where

*Q*represents the total number of users in the system, data block

**x**consisting of

*N*symbols, is generated from the resulting modulation scheme as

**x**is taken as

**X**=

**F**

_{ N }

**x**to yield frequency coefficients which are then assigned orthogonal subcarriers for transmission over the channel. From the DFT operation,

**x**represents DFT outputs for

*q*

^{ t h }user given as

**F**

_{ N }is an

*N*×

*N*DFT matrix defined as

**D**given in [8]. After allocating the subcarriers, M-point (

*M*>

*N*) inverse DFT (IDFT) is taken to convert the signal to time domain. The resulting signal is given as \(\textbf {S}=\boldsymbol {{F_{M}^{H}}}\textbf {X}\) where

**S**is the

*k*th SCFDMA symbol consisting of all the users’ signal

*M*×

*M*IDFT matrix and H is an Hermitian operator. The total number of users in the SC-FDMA system equals bandwidth expansion factor

*Q*=

*M*/

*N*where

*M*is the total number of subcarriers. In order to complete an SC-FDMA block, the time domain signal is converted from parallel to serial arrangement and is cyclically extended by addition of cyclic prefix. A cyclic prefix (CP), which is typically removed at the receiving section before any major processing, is obtained by prefixing a symbol with its tail end to achieve mainly two purposes. If the CP length is the same or longer than the length of multipath channel delay spread, it helps prevent interblock interference (IBI) and also enable convolution between the channel impulse response and transmitted signal to be modeled as circular as opposed to normal linear convolution. This makes frequency domain equalization easy at the receiver. It is this second purpose that we have taken advantage of in adapting the FD blind algorithms to equalizing SC-FDMA symbols. The transmitted SC-FDMA block is

*P*is the length of the appended CP. In matrix format, both the transmitted and received signals can be written, respectively, as

_{ P×M }is a matrix used in copying the last

*P*row of I

_{ M }, O

_{ M×P }is an

*M*×

*P*zero matrix and I

_{ M }is an

*M*×

*M*identity matrix. H is (

*P*+

*M*)×(

*P*+

*M*) channel matrix and V is (

*P*+

*M*)×1 noise vector. The received signal undergoes the reverse of what it has undergone during the transmitting phase as shown in Fig. 1, hence the input to the equalizer is

^{ ′ }is an

*N*×

*N*diagonal matrix containing the channel frequency response for the

*q*th user and V

^{ ′ }is the effective 1×

*N*noise vector. They are given as

Equation (10) results from the fact that addition and removal of CP turns channel matrix into a circulant matrix, and the resulting circulant matrix is diagonalized by DFT processing [24].

## 3 Blind algorithms

### 3.1 CMA

*z*(

*n*) is the output of the equalizer, E[·] denotes statistical expectation operator, and

*R*is a constant defined as

**y**(

*n*)= [

*y*(

*n*),

*y*(

*n*−1),…,

*y*(

*n*−

*N*+1)]

^{ T }and equalizer weight vector as

**w**(

*n*)=[

*w*

_{0}(

*n*),

*w*

_{1}(

*n*),…,

*w*

_{ N−1}(

*n*)]

^{ T }for an equalizer of length

*N*, the equalizer output is expressed as

*e*(

*n*) is the error factor and is given as

### 3.2 MMA

*R*and

*I*denote real and imaginary parts, respectively. However, including both real and imaginary parts of the equalizer output in the cost function and equalizing them separately sometimes results in diagonal solutions [29]. The error sample for MMA can be derived from (18) and is given as

### 3.3 SCS-MMA

SCS-MMA achieves equalization by forcing the real and imaginary parts of equalizer output onto a four-point contour with distance *R*
_{2} from the origin.

TD blind algorithms operate on a symbol-by-symbol basis processing a sample at a time. However, in order to take advantage of DFT processing, we need to formulate a block-by-block processing algorithm which will operate on a block of symbols at a time. This greatly improves computational cost and efficiency and is the most appropriate mode of processing for SC-FDMA FD equalization.

In the next section, we have taken advantage of CP embedded in the SC-FDMA block formation in adapting FD blind algorithms to its equalization. It should be noted that the frequency domain processing proposed in this work does not require the use of overlap-save and overlap-add signal processing techniques because these techniques are needed and employed in order to segment long streams of data for block processing and can be avoided with the inclusion of CP [30]. Additionally, since multiplication in frequency domain for discrete data is essentially circular convolution in time domain, overlap-save and overlap-add techniques helps in implementing linear convolution in frequency domain for cases where transmitted symbol is much longer than the channel impulse response. However, in SC-FDMA case, the received data are in blocks and these blocks of data, kept from IBI due to the appended CP, are fed into the equalizer for FD equalization.

## 4 Frequency domain blind algorithms

It is essential to point out the fundamental difference between the frequency domain equalization considered in this work and the frequency domain equalization (FDE) which is common in the literature. The FDE considered in works such as [23, 31] and [32] are linear convolution implemented through the use of overlap save method. In this work, cyclic-prefixed single-carrier system (CP-SCS) results in periodic transmitted symbols which trick the channel to perform circular convolution rather than linear convolution. The periodicity is then removed at the receiver before carrying out frequency domain equalization. This sort of transmission format eliminates the need for overlap save method. Therefore, we simply feed the received symbol represented by (9) into the equalizer.

*k*th received block will, respectively, yield

*k*th block of the equalizer output can be implemented with IDFT as

**D**, a 2

*N*×2

*N*matrix, is defined as

Both (28) and (34) completely describe the equalizer operation in frequency domain.

Blind algorithm equations

Algorithm | Cost function | Estimation error ( |
---|---|---|

CMA |
E{(| | 4 |

MMA |
E{[| | \(2\left [ z_{R}(n)\left ({z_{R}^{2}}(n)-R_{1,R}\right)+{jz}_{I}(n)\left ({z_{I}^{2}}(n)-R_{1,I}\right)\right ]\) |

SCS-MMA | \(\boldsymbol {E}\left \{ \frac {|z_{R}(n)|^{3}}{{3R_{2,R}}} - \frac {{z^{2}_{R}}(n)}{2} + \frac {{R_{2,R}}^{2}}{6} + \frac {|z_{I}(n)|^{3}}{{3R_{2,I}}} - \frac {{z^{2}_{I}}(n)}{2} + \frac {{R_{2,I}}^{2}}{6} \right \}\) | \(z_{R}(n)\left (1-\frac {|z_{R}(n)|}{{R_{2,R}}}\right)+ {jz}_{I}(n)\left (1-\frac {|z_{I}(n)|}{{R_{2,I}}}\right)\) |

*λ*is a forgetting factor and ⊘ is an element-wise division operator. A careful re-ordering of the normalized weight update equation reveals another insightful observation into its effectiveness in improving the equalizer convergence. It is seen that the normalization is tantamount to using variable step size in each of the frequency bin which amounts to power control on each bin, and such technique is especially useful in applications where the input level is uncertain or vary widely across the band as noted in [18]. The procedure outlined in this section is repeated to realize normalized FDMMA and normalized FDCMA (NFDMMA) from the equations given in Table 1, and the details of the algorithm are given in Fig. 3.

## 5 Results and discussion

The algorithms proposed above were investigated by means of computer simulations in MATLAB environment. Specifically, we have evaluated the performance of both frequency domain soft-constraint multimodulus algorithm (FDSCS-MMA) and improved FDSCS-MMA and compared their performance with the well-known constant modulus algorithm CMA and its modified version MMA. In order to simulate multi-user environment, we use transmitter FFT size of 256 equivalent to the total available subcarriers in the system (*M*), input FFT size for a user is 64 same as the number of subcarriers available for each user (*N*), and length of CP is 20 samples (*P*). This makes a total number of four users whose data were transmitted simultaneously.

The MSE convergence curve in decibels was obtained as ensemble average and is plotted as a function of the number of iterations where each iteration represent an SC-FDMA symbol consisting of all the users’ signal for that transmission time. The filter taps are of the order of *N* with center spike initialization. The modulation scheme employed for SCFDMA transmisson is 4 QAM. The localized carrier transmission mode is used in LTE uplink since it offers much better performance with the arrangement of pulse-shaping filter. Simulation results are averaged over 100 Monte Carlo iterations and are done for LFDMA since DFDMA is no longer supported in three GPP LTE standards though a scenario is shown for comparison of both allocation schemes [8, 33]. The values of *R*
_{2,R
}, *R*
_{2,I
}, and *λ* are 1, 1, and 0.55, respectively. The step size for the equalizers are 4×10^{−3}, 3×10^{−3}, 3×10^{−4}, and 1×10^{−4} for NFDSCS-MMA, FDSCS-MMA, NFDMMA, and NFDCMA, respectively. The channel considered is frequency selective with six paths and each path fades independently, according to the Rayleigh distribution. A high speed of 360 km/h is used to account for time variation in the channel [34, 35]. The additive white Gaussian noise have been chosen such that the signal to noise ratio (SNR) at the input of the equalizer is 20 dB. SNR of 10 dB is also considered for comparison of low and high SNR performance. The simulation parameters described above are implemented except stated otherwise.

*%*improvement in symbols saving over the algorithm without normalization for the same residual MSE. It can be deducted from the curves in Fig. 5 that the effect of appropriate normalization is to provide better convergence seeing that both algorithms achieve the same residual MSE. Based on the preceding discussion, only normalized versions of the blind algorithms proposed in this work are considered in the remaining discussion.

*n*th iteration is given as

where *s*(*n*)=*h*(*n*)∗*w*
^{∗}(*n*), *s*(*n*) is the overall impulse response of the transmission channel, *h*(*n*), and equalizer, *w*(*n*). |*s*(*n*)|_{max} is the component with maximum absolute value among all the components of |*s*(*n*)| and [ ∗] denotes convolution.

The results show that all the algorithms are able to remove ISI but NFDSCS-MMA has better convergence performance than both NFDMMA and NFDCMA. The algorithms achieve same residual ISI but NFDSCS-MMA converges fastest for both low and high SNR scenarios and as a result, gives better performance.

*k*th received block is given as

Both \(\boldsymbol {\mathcal {H}}_{k}\) and \(\boldsymbol {\mathcal {N}}_{k}\) represent the channel response and noise component for the *k*th received block. We have assumed perfect knowledge of the channel in our simulation of the optimum equalizers. In order to assess the BER performance of NFDSCS-MMA, knowledge of the first two received symbols has also been assumed since SCS-MMA only minimizes the dispersion between real and imaginary parts of the received signal and four-point contours of distance *R*
_{2}. This assumption is required to correct the received signal phase [23] as “blind” in blind equalizers is with respect to the phase; hence, they are said to be blind to the “phase”. It is shown in Fig. 13 that both NFDSCS-MMA and FDSCS-MMA achieve similar BER performance which is slightly less than that of linear MMSE. In situations where blind equalizers are used to open the eye of the signal constellation, a probability of symbol error of 10^{−2} is considered acceptable [29]. From Fig. 13, it is seen that to achieve this acceptable performance, 8 dB is required for NFDSCS-MMA as compared to that of 7 dB for linear MMSE which is a small tradeoff compare to 14 *%* improvement in throughput.

## 6 Conclusions

In this paper, we have implemented a novel frequency domain soft-constraint multimodulus algorithm for single carrier. It is shown that the proposed algorithm outperforms the popular blind algorithm, CMA and its modified version, MMA in both residual MSE and convergence rate. Phase recovery capability of the proposed algorithm is also demonstrated with acceptable BER performance. This suggests that SC-FDMA can be perfectly equalized in broadband systems using the proposed algorithm with the resultant lower MSE, faster convergence, and improved spectral efficiency.

## Declarations

### Acknowledgements

The authors like to thank the anonymous reviewers for their constructive suggestions which have helped improve the paper. The authors acknowledge the support provided by the Deanship of Scientific Research at KFUPM under Research Grant RG1312.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- JAC Bingham, Multicarrier modulation for data transmission: an idea whose time has come. IEEE Commun. Mag.
**28**(5), 5–14 (1990).View ArticleMathSciNetGoogle Scholar - K Paterson, Generalized Reed-Muller codes and power control in OFDM modulation. IEEE Transac. Inf. Theory.
**46**(1), 104–120 (2000).View ArticleMathSciNetMATHGoogle Scholar - HG Myung, DJ Goodman,
*Single Carrier FDMA: A New Air Interface for Long Term Evolution*(John Wiley & Sons Inc., New York, NY, 2008).View ArticleGoogle Scholar - FE Abd El-Samie, FS Al-kamali, AY Al-nahari, MI Dessouky,
*SC-FDMA for Mobile Communications*(CRC Press Inc, Boca Raton Raton, FL USA, 2013).View ArticleGoogle Scholar - H Myung, J Lim, D Goodman, Single carrier FDMA for uplink wireless transmission. IEEE Veh. Technol. Mag.
**1**(3), 30–38 (2006).View ArticleGoogle Scholar - N Benvenuto, R Dinis, D Falconer, S Tomasin, Single carrier modulation with nonlinear frequency domain equalization: an idea whose time has come again. Proc. IEEE.
**98**(1), 69–96 (2010).View ArticleGoogle Scholar - C Zhang, Z Wang, Z Yang, J Wang, J Song, Frequency domain decision feedback equalization for uplink SC-FDMA. IEEE Trans Broadcast.
**56**(2), 253–257 (2010).View ArticleGoogle Scholar - G Huang, A Nix, S Armour. Decision feedback equalization in SC-FDMA, IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications, (2008), pp. 1–5.Google Scholar
- S Yameogo, P Jacques, L Cariou. A semi-blind time domain equalization of SCFDMA signal, IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), (2009), pp. 360–365.Google Scholar
- S Yameogo, J Palicot, L Cariou. Blind time domain equalization of SCFDMA signal, IEEE Vehicular Technology Conference, VTC-Fall, (2009), pp. 1–4.Google Scholar
- S Yameogo, J Palicot, in International Conference on Consumer Electronics. Frequency domain equalization of SC-FDMA signal without any reference symbols, (2009), pp. 1–2.Google Scholar
- X Wang. Blind equalization for single carrier frequency domain equalization with low complexity, China-Japan Joint Microwave Conference, (2008), pp. 748–750.Google Scholar
- R Martin, Fast-converging blind adaptive channel-shortening and frequency-domain equalization. IEEE Transac Signal Process.
**55**(1), 102–110 (2007).View ArticleGoogle Scholar - S Abrar, A Zerguine, M Deriche, Soft constraint satisfaction multimodulus blind equalization algorithms. IEEE Signal Processing Letters.
**12**(9), 637–640 (2005).View ArticleGoogle Scholar - J-C Lin, Blind equalisation technique based on an improved constant modulus adaptive algorithm. IEE Proc.-Commun.
**149**(1), 45–50 (2002).View ArticleGoogle Scholar - HG Myung,
*3GPP Long Term Evolution: A Technical Overview*(John Wiley & Sons Inc., New York, NY, 2010).Google Scholar - K Akande, N Iqbal, A Zerguine, J Chambers, Normalised frequency-domain soft constraint satisfaction multimodulus blind algorithm. Electronics Letters (2015). [Online]. Available: http://digital-library.theiet.org/content/journals/10.1049/el.2015.2845.
- N Bershad, P Feintuch, A normalized frequency domain LMS adaptive algorithm. Acoustics, Speech and Signal Process. IEEE Transact. on.
**34**(3), 452–461 (1986).View ArticleGoogle Scholar - T-L Kung, K Parhi, English Semiblind frequency-domain timing synchronization and channel estimation for OFDM systems. EURASIP J. Adv. Signal Process.
**2013**(1) (2013). [Online]. Available: doi:http://dx.doi.org/10.1186/1687-6180-2013-1. - S Caekenberghe, A Bourdoux, L der Perre, J Louveaux, Preamble-based frequency-domain joint CFO and STO estimation for oqam-based filter bank multicarrier. EURASIP J. Adv. Signal Process.
**2014**(1), 118 (2014). [Online]. Available: http://asp.eurasipjournals.com/content/2014/1/118.View ArticleGoogle Scholar - S Minhas, P Gaydecki, A hybrid algorithm for blind source separation of a convolutive mixture of three speech sources. EURASIP J. Adv. Signal Process.
**2014**(1) (2014). [Online]. Available: doi:http://dx.doi.org/10.1186/1687-6180-2014-92. - Y Wang, L DeBrunner, V DeBrunner, D Zhou, Blind channel equalization with colored source based on constrained optimization methods. EURASIP J. Adv. Signal Process.
**2008**(1), 960295 (2008). [Online]. Available: http://asp.eurasipjournals.com/content/2008/1/960295.Google Scholar - HH Dam, S Nordholm, H Zepernick. Frequency domain constant modulus algorithm for broadband wireless systems, IEEE Global Telecommunications Conference, GLOBECOM, (2003), pp. 40–44.Google Scholar
- DH Carlson, Review: Philip j. davis, circulant matrices. Bulletin (New Series). Am. Math. Soc.
**7**(2), 421–422 (1982).View ArticleGoogle Scholar - J Treichler, C Johnson, MG Larimore,
*Theory and Design of Adaptive Filters*(Prentice-Hall, Englewood Cliffs, NJ, 2001).Google Scholar - D Godard, Self-recovering equalization and carrier tracking in two-dimensional data communication systems. Commun. IEEE Transac.
**28**(11), 1867–1875 (1980).View ArticleGoogle Scholar - J Treichler, B Agee, A new approach to multipath correction of constant modulus signals. Acoustics, Speech and Signal Process. IEEE Transac.
**31**(2), 459–472 (1983).Google Scholar - KN Oh, Y Chin. Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm, IEEE International Conference on Communications, (1995), pp. 498–502.Google Scholar
- J Yang, J-J Werner, G Dumont, The multimodulus blind equalization and its generalized algorithms. Sel. Areas Commun. IEEE J.
**20**(5), 997–1015 (2002).View ArticleGoogle Scholar - D Falconer, S Ariyavisitakul, A Benyamin-Seeyar, B Eidson, Frequency domain equalization for single-carrier broadband wireless systems. IEEE Commun. Magazine.
**40**(4), 58–66 (2002).View ArticleGoogle Scholar - C Chan, M Petraglia, J Shynk, 2. Frequency-domain implementations of the constant modulus algorithm, Twenty-Third Asilomar Conference on Signals, Systems and Computers, (1989), pp. 663–669.Google Scholar
- MM Usman Gul, SA Sheikh, Design and implementation of a blind adaptive equalizer using frequency domain square contour algorithm. Digit. Signal Process.
**20**(6), 1697–1710 (2010).View ArticleGoogle Scholar - I Ahmed, A Mohamed, I Shakeel. On the group proportional fairness of frequency domain resource allocation in L-SC-FDMA based LTE uplink, IEEE GLOBECOM Workshops (GC Wkshps), (2010), pp. 1312–1317.Google Scholar
- N Iqbal, A Zerguine, N Al-Dhahir, Adaptive equalisation using particle swarm optimisation for uplink SC-FDMA, Electronics Letters. 50(6), 469–471 (2014).Google Scholar
- N Iqbal, A Zerguine, N Al-Dhahir, Decision feedback equalization using particle swarm optimization. Signal Process.
**108**(C), 1–12 (2015). [Online]. Available: doi:http://dx.doi.org/10.1016/jsigpro.2014.07.030. - J Medbo, P Schramm, Channel models for HIPERLAN/2 in different indoor scenarios, ETSI/BRAN 3ERI085B (1998).Google Scholar