 Research
 Open Access
Linear IC detectors for low to medium SNR illconditioned communication systems with unknown noise variance
 Abdelouahab Bentrcia^{1}Email author and
 Saleh A Alshebeili^{1, 2}
https://doi.org/10.1186/168761802013180
© Bentrcia and Alshebeili; licensee Springer. 2013
 Received: 9 March 2013
 Accepted: 19 November 2013
 Published: 5 December 2013
Abstract
In this paper, we introduce two new linear parallel interference cancellation (LPIC) detectors that are suitable for low to medium signaltonoise ratio illconditioned communication systems and do not require knowledge of the noise variance but perform close to the linear minimum mean square error detector, which needs such information. Particularly, we focus in this work on fast linear parallel interference cancellation detectors that are asymptotically equivalent to the steepest descent and conjugate gradient algorithms, respectively, and show that they exhibit a spectral filtering property and semiconvergence behavior. Consequently, a deterministic stopping rule to stop the LPIC iterations that is independent of the noise level (known as the Lcurve method) is investigated and tested. Simulation results are presented to support our theoretical findings.
Keywords
 Linear
 PIC
 Regularization
 Semiconvergence
 Linear MMSE
 Lcurve
1. Introduction
The capacity of the thirdgeneration cellular systems and optical networks using optical CDMA (OCDMA) technology is mainly limited by the multiaccess interference (MAI) [1]. Other systems suffer from other types of interference such as the intercarrier interference (ICI) in orthogonal frequency division multiple access (OFDMA) and interantenna interference (IAI) in multiinput multioutput (MIMO) systems, just to name a few [1].
In 4G and beyond wireless communication systems, the problem of interference is becoming of increasing importance because cells are getting smaller and condensed (i.e., femtocells) and new technologies that cause additional interference are introduced. For example, relay nodes are proposed to increase coverage and allow cooperative communication; however, they also bring additional interference into the network [2].
To combat these different types of interferences, various interference cancellation and multiuser detection algorithms are proposed. Multiuser detectors (MUDs) are mainly introduced to reduce the effect of interference in wireless/wired systems and consequently to boost the system capacity and throughput. Many multiuser detectors were developed in the literature and have found applications in various wireless/wired systems such as OCDMA, MIMOOFDM, and MIMOUWB, just to name a few [1–5]; however, due to the fact that the capacity of CDMA systems is essentially limited by MAI, a large part of the literature of MUDs focused on systems based on the CDMA technology.
The decorrelating detector is an effective multiuser detector to eliminate interference. It is also an important building block for nonlinear multiuser detectors. It enjoys several desirable features such as: (1) complete removal of interference and (2) independence of noise level information. The latter is very important in situations where the estimation of the noise variance is not possible or not accurate. This is the case where the noise term includes in addition to thermal noise, other types of background noise such as cochannel interference which may change significantly with time/frequency (particularly in frequencyhopping systems) and its variance can be considered as unknown to the receiver [6, 7].
However, the decorrelating detector suffers from two drawbacks: (1) Its relatively high computational complexity which is of the order of O(N ^{3}) [1], where N in the dimension of the system's crosscorrelation matrix and (2) noise enhancement effect. Therefore, the challenge is to maintain the advantages of the decorrelating detector and overcome its deficiencies, that is, to lower its computational complexity and combat the noise enhancement effect but without requiring the knowledge of the noise variance, like the linear minimum mean square error (LMMSE) detector.
Recent results reported in [8] proved the semiconvergence behavior of the conventional linear parallel interference cancellation (LPIC) detector and showed that early stopping rules can be used to combat the noise enhancement effect. They used the Morozov discrepancy rule to stop the LPIC iterations prior to final convergence in order to avoid noise magnification. However, the conventional LPIC detector is very slow and may require a very large number of stages to converge especially for illconditioned communication systems. Moreover, the Morozov discrepancy rule requires the noise level information and therefore cannot be used in many practical settings.
Building up on these results, a twofold approach is proposed in this work to overcome the drawbacks of the decorrelator detector cited above. First, we employ fast linear interference cancellation detectors to reduce its computational complexity, and then, we make use of an early stopping technique (known as the Lcurve method) that does not require noise level information to reduce the noise enhancement effect [9]. Up to our knowledge, this is the first work that investigates the possible use of these early stopping rules that do not require noise level information in the communication field. Preliminary results are promising and suggest that more improvements and extensions can be made.
The organization of this paper is as follows: in Section 2, a simplified system model of the OFDMA uplink that will be used throughout this work is briefly described. In Section 3, the decorrelator detector's solution is analyzed and some of the regularization techniques that are used to overcome its drawbacks are detailed. Section 4 briefly describes the fast LPIC detectors and shows their spectral filtering property. Section 5 analyzes the semiconvergence behavior of the LPIC detector and then investigates the Lcurve stopping rule and shows how it can be applied to the LPIC detector. Section 6 supports the theoretical findings by a number of simulations. Finally, Section 7 concludes the paper with some results and recommendations.
2. System model
In this work, we consider the interleaved subcarrier allocation scheme because it is well known that this scheme suffers the most from ICI compared to other subcarrier allocation schemes [11].
An OFDM symbol consisting of N _{ u } samples with sampling time T _{ u } where N _{ u } is the total number of data samples is transmitted using N orthogonal subcarriers. Without loss of generality, we assume that the total number of subcarriers of the IFFT matrix Ψ with elements ${\mathrm{\Psi}}_{{n}_{u},n}=\frac{1}{\sqrt{{N}_{u}}}{e}^{\frac{j2\pi {n}_{u}n}{{N}_{u}}}$, 1 ≤ n _{ u } ≤ N _{ u }, and 1 ≤ n ≤ N is divided equally among all users; therefore, the total number of subcarriers per user is N _{ k } = N/K.
where

$\tilde{\mathbf{\Psi}}=\mathbf{\Psi}\circ \left({\mathbf{1}}_{1,{N}_{k}}\otimes \mathbf{{\rm E}}\right)$ is a combination of the IFFT matrix Ψ and the normalized carrier frequency offset (NCFO) matrix E. Here, ^{○} and ⊗ denote the Schur and Kronecker products, respectively, and ${\mathbf{1}}_{1,{N}_{k}}$ denotes a 1byN _{ k } vector of ones. The NCFO matrix can be partitioned as $\mathbf{{\rm E}}=\left[\begin{array}{cccc}\hfill {\mathbf{\u03f5}}_{1}\hfill & \hfill {\mathbf{\u03f5}}_{2}\hfill & \hfill \cdots \hfill & \hfill \begin{array}{cc}\hfill {\mathbf{\u03f5}}_{k}\hfill & \hfill \begin{array}{cc}\hfill \cdots \hfill & \hfill {\mathbf{\u03f5}}_{K}\hfill \end{array}\hfill \end{array}\hfill \end{array}\right]$, where the vector ϵ _{ k } is given by ${\mathbf{\u03f5}}_{k}={\left[\begin{array}{cccc}\hfill {e}^{\frac{j2\pi {\u03f5}_{k}}{{N}_{u}}}\hfill & \hfill {e}^{\frac{j2\pi 2{\u03f5}_{k}}{{N}_{u}}}\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {e}^{\frac{j2\pi {n}_{u}{\u03f5}_{k}}{{N}_{u}}}\hfill & \hfill \cdots \hfill & \hfill {e}^{\frac{j2\pi {N}_{u}{\u03f5}_{k}}{{N}_{u}}}\hfill \end{array}\hfill \end{array}\right]}^{\mathit{T}}$, where ϵ _{ k } = Δf _{ k }/Δf is the NCFO of the k th user and Δf is the subcarrier spacing.

H(m) is the matrix of Rayleigh fading coefficients and it is given by: $\mathbf{H}\left(m\right)=\mathit{diag}\left(\begin{array}{cccc}\hfill {\mathbf{H}}_{1}\left(m\right)\hfill & \hfill {\mathbf{H}}_{2}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {\mathbf{H}}_{{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill {\mathbf{H}}_{{N}_{k}}\left(m\right)\hfill \end{array}\hfill \end{array}\right)$, where ${\mathbf{H}}_{{n}_{k}}\left(m\right)=\mathit{diag}\left(\begin{array}{cccc}\hfill {h}_{1,{n}_{k}}\left(m\right)\hfill & \hfill {h}_{2,{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {h}_{k,{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill {h}_{K,{n}_{k}}\left(m\right)\hfill \end{array}\hfill \end{array}\right)$.

A is the matrix of amplitudes and it is given by: $\mathbf{A}=\mathit{diag}\left(\begin{array}{cccc}\hfill {\mathbf{A}}_{1}\hfill & \hfill {\mathbf{A}}_{2}\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {\mathbf{A}}_{{n}_{k}}\hfill & \hfill \cdots \hfill & \hfill {\mathbf{A}}_{{N}_{k}}\hfill \end{array}\hfill \end{array}\right)$, where ${\mathbf{A}}_{{n}_{k}}=\mathit{diag}\left(\begin{array}{cccc}\hfill {a}_{1,{n}_{k}}\hfill & \hfill {a}_{2,{n}_{k}}\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {a}_{k,{n}_{k}}\hfill & \hfill \cdots \hfill & \hfill {a}_{K,{n}_{k}}\hfill \end{array}\hfill \end{array}\right)$. It is used to weight the signals of different users with different powers to simulate nearfar scenarios.

b(m) is the vector of transmitted data symbols and it can be partitioned as: $\mathbf{b}\left(m\right)={\left[\begin{array}{cccc}\hfill {\mathbf{b}}_{1}\left(m\right)\hfill & \hfill {\mathbf{b}}_{2}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {\mathbf{b}}_{{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill {\mathbf{b}}_{{N}_{k}}\left(m\right)\hfill \end{array}\hfill \end{array}\right]}^{T}$, where ${\mathbf{b}}_{k}\left(m\right)=\left[\begin{array}{cccc}\hfill {b}_{1,{n}_{k}}\left(m\right)\hfill & \hfill {b}_{2,{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill \begin{array}{ccc}\hfill {b}_{k,{n}_{k}}\left(m\right)\hfill & \hfill \cdots \hfill & \hfill {b}_{K,{n}_{k}}\left(m\right)\hfill \end{array}\hfill \end{array}\right]$.

n(m) is an Nlength vector of independently and identically distributed additive white Gaussian noise samples with zero mean and variance ρ ^{2}.
And finally, the combination $\tilde{\tilde{\mathbf{\Psi}}}\left(m\right)=\tilde{\mathbf{\Psi}}\mathbf{H}\left(m\right)\mathbf{A}$ is the system matrix and it results from the multiplication of the IFFT, the NCFO, the channel gain, and power weighting matrices, respectively. To simplify the notations, we drop in all subsequent equations the OFDM symbol index m from all matrices and vectors used in Equation (1).
3. Regularization techniques for combating the noise enhancement effect
where ${\tilde{\tilde{\mathbf{\Psi}}}}^{\u2020}$ is the pseudoinverse of the system matrix $\tilde{\tilde{\mathbf{\Psi}}}$ and ${\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}\tilde{\tilde{\mathbf{\Psi}}}$ is the ICI matrix.
It is clear from Equation (6) that as long as $\left{\mathbf{u}}_{n}^{H}\mathbf{r}\right<{\sigma}_{n}$ for large n, the norm of y ^{DEC} will not go unbounded. This is what is known as the Picard condition [12]; however, due to the contamination of the received signal with noise, the Fourier coefficients $\left{\mathbf{u}}_{n}^{H}\mathbf{r}\right$ do not decay monotonically to zero, instead they settle at certain level ρ while the singular values decay to zero. Therefore, all singular values below ρ contribute to the noise enhancement effect.
Consequently, these factors filter out solution components pertaining to small singular values.
This regularization technique is known also as spectral cutting technique and relies on cutting off all solution components under a certain threshold determined by the discrete regularization parameter n’.
Another category of regularization methods is regularization by early stopping [13], that is, apply an iterative method to the least square problem of Equation (2) and stop the iterations prior to convergence. This is motivated by the fact that for illconditioned systems, linear iterative methods exhibit a semiconvergence property and tend to generate good solutions at early iterations but after a certain number of iterations, the noise starts dominating the solution and the performance of the iterative method worsens.
Many linear interference cancellation schemes have been shown to be equivalent to certain iterative methods [1], and therefore, they tend to have the same semiconvergence behavior for illconditioned systems. Therefore, linear IC detectors equipped with efficient early stopping mechanisms that do not require the noise level information can be used to implement a lowcomplexity decorrelating detector that resists noise amplification without requiring knowledge of the noise level. In the following, we focus on the LPIC detector and use it as an iterative regularizing scheme to achieve two simultaneous objectives: lowcomplexity and resistance to noise enhancement.
4. Intrinsic regularization property of the LPIC detector
where ω _{ p } is the step size and d _{ p } is the search direction.
and ${T}_{1}^{\mathrm{SD}}\left(\theta \right)=0$.
where ${\mathbf{G}}_{p}^{H}={T}_{p}^{\mathit{SD}}\left({\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}\tilde{\tilde{\mathbf{\Psi}}}\right){\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}$ for the LPIC detector based on the RNSD and ${\mathbf{G}}_{p}^{H}={T}_{p}^{\mathit{CG}}\left({\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}\tilde{\tilde{\mathbf{\Psi}}}\right){\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}$ for the LPIC detector based on the CGLS and G _{ p } = [g _{ p,1} g _{ p,2} ⋯ ⋯ g _{ p,n } ⋯ g _{ p,N }].
5. Semiconvergence behavior of the LPIC detector and early stopping using the Lcurve method
It is clear that the two detectors, that is, the one equivalent to the RNSD method (RNSDLPIC) and the one equivalent to the CGLS method (CGLSLPIC), exhibit a semiconvergence behavior where they reach their minimum average BER at 10 and 45 stages, respectively, and converge to the decorrelator detector's solution.
The RNSDLPIC detector exhibits a slower convergence behavior and wide flat minimum while the CGLSLPIC detector exhibits faster convergence and a narrow minimum.
It can be seen that the MSE can be decomposed into two components: the bias (also known as the data error) and the variance (also known as the noise error). The data error is caused by using a modified inverse of the ICI matrix ${\tilde{\tilde{\mathbf{\Psi}}}}^{\mathit{H}}\tilde{\tilde{\mathbf{\Psi}}}$ instead of the true inverse whereas the noise error is caused by the noise enhancement effect. It is evident from Equation (28) that if F ⋍ I, the data error is small but the noise error is large due to the noise enhancement effect; however, if F ⋍ 0, the noise error is small but the data error is large, and as a result, the solution is heavily damped and a large part of it is lost.
Therefore, a proper choice of the filtering matrix F should balance between the data and noise errors. Because the amount of filtering introduced by the filtering matrix is proportional to the stage index p, a proper stopping rule needs to be devised. Many stopping rules have been developed in the literature but roughly they can be classified into three broad techniques [9]:

Methods requiring the knowledge of the exact noise level.

Methods requiring the knowledge of the approximate noise level.

Methods not requiring the knowledge of the noise level.
Due to the fact that we are assuming the absence of the noise level information, we neglect the first two categories and focus on the last one. Under this category, the most known stopping rule technique is the Lcurve method [15]. This method has been used successfully in many areas such as spectroscopy, seismography, and medical imaging. The Lcurve method exhibits some desirable features such as:

Does not need the noise level information. This is important for communication systems where this information is not always available or it is not accurate.

Works well under colored Gaussian noise. This is important when the additive white Gaussian noise assumption in communication systems is violated, or it becomes colored because of some signal processing operations such as matched filtering.
respectively. It is clear that if f _{ p,n } ≈ 1 (at early stages), we are in the upper part of the curve. A small change in f _{ p,n } leads to a larger change in ${f}_{p,n}^{2}$ than in (1 − f _{ p,n })^{2}; hence, we expect ∥y _{ p }∥_{2} to change the most in this region, i.e., we expect the curve to be steep. On the other hand, if f _{ p,n } ≈ 0 (at late stages), we are in the lower part of the curve. A small change in f _{ p,n } leads to a larger change in (1 − f _{ p,n })^{2} than in ${f}_{p,n}^{2}$; hence, we expect ${\u2225\mathbf{r}\tilde{\tilde{\mathbf{\Psi}}}{\mathbf{y}}_{p}\u2225}_{2}$ to change the most in this region, i.e., we expect the curve to be flat here.
The two regions, that is, the vertical and horizontal, are dominated by two types of errors: data error and noise error, respectively. The corner formed by the conjunction of these two regions balances between the data and noise errors, and therefore, the stage index corresponding to this corner is selected as the best regularization parameter that compromises between data and noise errors.
where η(p) = ∥y _{ p }∥_{2} and $\rho \left(p\right)={\u2225\mathbf{r}\tilde{\tilde{\mathbf{\Psi}}}{\mathbf{y}}_{p}\u2225}_{2}$ and denotes differentiation with respect to the regularization parameter p.
In order to implement the above stopping rule, we have to evaluate the norm of the residual error for each OFDM symbol a certain number of stages till the Lshaped curve is obtained and calculate the curvature information to ultimately determine the optimal stopping stage. This is too expensive in practice and consequently, another alternative should be sought. Fortunately, simulation results reveal that the optimal stage index p _{opt} is insensitive to variations in the condition number of system matrix and is almost constant from one OFDM symbol to another, and therefore, it is sufficient to obtain p _{opt} for a few OFDM symbols (practically in the range of 10 to 100 OFDM symbols) and then average it and use it for the rest of the OFDM symbols. This works well in practice and costs only a marginal addition to the overall LPIC computational complexity.
Another problem with the LPIC detector based on the CGLS is that the norm of the residual error ${\u2225\mathbf{r}\tilde{\tilde{\mathbf{\Psi}}}{\mathbf{y}}_{p}\u2225}_{2}$ behaves erratically for very illconditioned systems [14] where the norm of the residual error may increase occasionally and causes the maximum curvature detector to mistakenly detect the maximum curvature at the wrong stage index. To overcome this problem, we exploit the fact that the residual error decay exponentially and therefore can be fitted by a decaying exponential. The resulting exponentially fitted curve is used instead in computing the maximum curvature and hence the optimal stage index. This method proved to be efficient in combating the erratic behavior of the residual error norm of the LPIC detector based on the CGLS.
Lastly, the Lcurve method has obviously some limitations as discussed in [15]. The main one is that the regularized solution does not converge to the true solution as the noise variance vanishes to zero. Therefore, in our application, we expect that the LPIC detector equipped with the Lcurve stopping rule will work well for only a certain range of SNRs. Fortunately, this range covers low and medium SNRs where the noise enhancement effect is most prominent. More insight about this issue is given in the simulation results.
6. Computational complexity
The computational complexity of the basic LPIC detector with fixed step size used in [8] and the ones proposed here exhibit a computational complexity in the order of O(N ^{2}), namely the conjugate gradient based LPIC detector needs 4 N ^{2} +8 N1 cflops^{a} per iteration and the steepest descent based LPIC detector needs 4 N ^{2} +6 N1 cflops per iteration, and finally, the conventional LPIC detector requires 2 N ^{2} +2 N + 1 cflops per iteration. Even though the conjugate gradient based LPIC detector exhibits a slightly higher computational complexity per iteration compared to the other detectors, it however needs the least number of iterations; this is why it is the mostly used iterative method in practice.
Since early stopping methods, whether they require the noise variance information or not can be applied to any LPIC detector, we evaluate in the following the computational complexity of the Lcurve early stopping rule and compare it to that proposed in [8] (known as the Morozov early stopping rule).
 1.
Norm calculation of η(p) = ∥y _{ p }∥_{2}: 2 N cflops.
 2.
Norm calculation of $\rho \left(p\right)={\u2225\mathbf{r}\tilde{\tilde{\mathbf{\Psi}}}{\mathbf{y}}_{p}\u2225}_{2}$ 2 N cflops.
 3.
Curvature information of κ(p): 2p3 flops.
 4.
Obtaining the index of the maximum curvature: ${p}_{\mathit{opt}}=\underset{1\le p\le P}{argmax}\left\{\frac{\rho \left(p\right)\text{'}\text{'}}{{\left(\rho \left(p\right){\text{'}}^{2}+1\right)}^{\frac{3}{2}}}=\kappa \left(p\right)\right\}$: p flops.
Total complexity for M symbols (M in practice is between 10 to 100) is: 4MN cflops +3 M(p1) flops.
On the other hand, the Morozov stopping rule relies mainly on the estimation of the noise variance. A typical noise variance estimation algorithm used in wireless communication systems and specifically within the context of spectrum sensing [18], solves a yulewalker set of equations using LevinsonDurbin algorithm and needs a complexity of 2MN ^{2} cflops [19].
It is clear from the above expressions that the computational complexity of the Lcurve early stopping method is less than that of Morozov early stopping method by an order of magnitude. Therefore, in terms of computational complexity, using the Lcurve, early stopping method is much cheaper than using the Morozov early stopping method.
7. Simulation results
In the following, we evaluate the performance of the LPIC detector equipped with the Lcurve stopping rule and based on the residual norm steepest descent and conjugate gradient least squares, respectively (which are referred to by RNSDLPIC and CGLSLPIC detectors, for conciseness).
First, we show how the Lcurve method works. For this reason, we plot the norm of the residual error of both the RNSDLPIC and CGLSLPIC detectors for one OFDM symbol versus the stage index p, and we plot also its curvature using Equation (32). We set the carrier frequency to 3.5 GHz, SNR to 5 dB, N to 128, and K to 4 with frequency offsets (ε _{1} = −0.35, ε _{2} = 0.38, ε _{3} = 0.36, ϵ _{4} = −0.39) and speeds 80, 120, 90, and 100 km/h, respectively.
As depicted in Figure 7, the condition number of the system matrix varies from one OFDM symbol to another and it is dependent on the fading channels of the different users. It can be seen that the condition number of the system matrix varies largely in magnitude where it can go from 10^{3} to 10^{6}.
As depicted in Figures 8 and 9, it is clear that the stopping stage determined by the Lcurve method for both the RNSDLPIC and CGLSLPIC detectors is almost fixed with respect to the condition number variations of the system matrix over 5,000 OFDM symbol (see Figure 7) for each SNR. Consequently, it is sufficient to evaluate the stopping stage index for a few OFDM symbols (say 100 OFDM symbol) and then get the average stopping stage index and use it for the subsequent OFDM symbols. This is a very important result as it allows avoiding the calculation of the stopping stage for every OFDM symbol and thus renders this scheme very efficient in terms of computational complexity.
It also is clear that the stopping stage is almost constant on average with increasing SNR while in principle, it should increase. This is one of the deficiencies of the Lcurve method, and it is due to the fact that the Lcurve method is not a converging stopping rule, in the sense that the solution obtained using the Lcurve method does not converge to the true solution if the noise variance vanishes to zero.
This suggests that the stopping stage determined using the Lcurve method should be updated whenever the total number of subcarriers N in the system changes, for example, if one user needs a higher data rate and is assigned a higher number of subcarriers, or if one user enters or leaves the system, etc.
8. Conclusion
In this work, we introduced new linear interference cancellation detectors for low to medium SNR illconditioned communication systems that perform close to the LMMSE detector though they do not require the knowledge of the noise variance information. These linear IC detectors are based on early an stopping rule known as the Lcurve method. Simulation results indicate that these detectors are insensitive to both the condition number of the system matrix and SNR and can work well up to SNR of 16 dB.
Appendix
In the following, we develop the average BER of the RNSDLPIC and CGLSLPIC detectors.
The QPSK scheme can be seen as two superimposed BPSK channels, these channels are orthogonal to each other and do not mutually interfere; therefore, the BER of QPSK scheme is simply the same as that of the BPSK scheme, and it is enough to conduct the BER probability analysis using the real or imaginary part only. In this derivation, we consider the real part of the decision vector y _{ p }.
Endnote
^{a}cflop states for any complex addition, complex subtraction, complex multiplication, or complex division.
Declarations
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.