Mitigating the Interference Caused by Pilot Contamination in Multi-cell Massive Multiple Input Multiple Output Systems Using Low Density Parity Check Codes in Uplink Scenario

The next generation 5G technology is at the doorsteps to serve the need of an increase in demand of high data rates for a capacious range of applications such as real-time monitoring, vehicle to vehicle communication which have a scope of very low latency. Another important necessity is the usage of spectral efficient systems to provide high speed connectivity so that the crucial resource can be used sagaciously to accommodate a considerable number of subscribers. The probable, reliable, and robust technology capable of serving the above mentioned requisitions is massive multiple input multiple output (m-MIMO) which has considerably large number of BS antennas that are used for accommodating a relatively lesser number of users [1]. It has been proposed that the optimal scheduling of user equipments can increase spectral efficiency (SE) [2]. Thus the technology is termed as multi-user massive MIMO (MU-m-MIMO). Zhang et al. [3] have proposed energy efficient multiple antenna technologies such as cell free massive MIMO, beam space MIMO, and intelligent reflecting surfaces which provide considerably higher gain at a decent cost for enabling the overall scalable connectivity. Murthy and Sankhe [4] have said that spectral efficiency is increased with a confined number of radio frequency chains through a combination of spatial modulation and SM. However, increase in the subscriber base results in limited availability of resources, the same frequency has to be provided in multiple cells resulting in frequency reuse. So, the challenge is to provide connectivity to subscribers present in different cells using the same frequency.

In a typical wireless propagation scenario due to the Rayleigh fading, the signal received is the sum of a large number of multi-path components that differ in phase, time, and amplitude as given in Eq. (1).

$y(t)=s(t) \sum_{i=0}^{Z-1} a_{i} e^{-j 2 \pi f_{c} \tau_{i}}$  (1)

where, Z represents the number of multiple paths, a_i and τ_i are the attenuation and delay of the i^th path. s(t) is the transmitted signal. Jerez et al. [5] have proposed a channel model scheme in which dominant multipath components are considered which have a significant impact on the performance of m-MIMO. In m-MIMO, the number of links between a subscriber and a BS is equal to the number of BS antennas. It has been shown that the estimated signal can be obtained from the combined signal version by using efficient MIMO receivers such as matched filter (MF) and zero forcing (ZF) which require the estimation of the channel are given respectively by Eq. (2) [6].

$\tilde{\mathbf{x}}=\mathbf{H}^{H} \mathbf{y}$ and $\tilde{\mathbf{x}}=\left(\mathbf{H}^{H} \mathbf{H}\right)^{-1} \mathbf{H}^{H} \mathbf{y}$  (2)

where, $\tilde{\mathbf{X}}$ is the estimated signal, H and y are the channel matrix and received signal respectively.

It is clear from (2) that CE is utmost crucial in finding $\tilde{\chi}$. If the number of users and pilot symbol vectors for a single cell are K and P respectively, then P≥K so that the pilot matrix becomes orthogonal signifying that the interference between transmitted pilot symbols from K users is zero. However, the orthogonal pilot symbols are optimal when the time slots for pilot training are large enough to support the orthogonality, and if not then the minimum value of error variance shifts from zero to a finite value irrespective of the type of pilot symbols used [7].

In a scenario of L number of multi-cells having a K number of users in each cell as illustrated in Figure 1, if the number of users and pilot vectors required for CE are KL and P respectively, then to maintain orthogonality, the condition P≥KL should be satisfied. This leads to a huge overhead for estimating the channel in MC-MU-m-MIMO as the number of pilot transmission increases by a factor L. If the overhead is avoided by compromising with the orthogonality between pilot symbols, it leads to PC which is due to interference in the pilot from other cells [8]. However, if the ratio of the coherence time of the channel to the number of subscribers in each cell is large, the optimality in the number of pilots also grows resulting in less aggressive pilot reuse [9]. Al-Hubaishi et al. [10] have proposed an effective scheme of assigning the pilot symbols to counter PC when the minimum uplink rate has been maximized and hence inter-cell interference is minimized by considering large-scale characteristics of fading channel. A Hungarian method based optimum pilot allocation scheme has been proposed for maximizing the uplink sum- rate of the system [11].

1.jpg

Figure 1. A typical scenario of MC-MU-m-MIMO system

CE in practical systems is imperfect resulting in CE error having finite variance. This adds to the PC problem and thus the output of the receiver in a multi-cell scenario has some proportion of user information present in L-1 cells along with the desired information resulting in erroneous output at the ZF or MF receiver due to the effect of PC. The signal to interference and noise ratio (SINR) reduces due to additional interference caused by channel vectors of the L-1 cells. For large number of BS antennas, the line of sight (LOS) components is PC free which can be found by the angle of arrivals of signals at users [12]. Akbar et al. [13] have proposed a generalized welch-bound equality (GWBE) base design of pilot sequence for mitigating PC in MC-MU-m-MIMO networks which fulfill the requirements of SINR for all subscribers in the network. Liu et al. [14] have proposed a graph coloring based pilot assignment scheme by utilizing the large-scale coefficients between the access points and user terminals which is capable of reducing the effect of pilot contamination. A method for optimization of coverage, positions, and energy consumption of m-MIMO for low power requirements has been proposed [15]. Al-hubaishi et al. [16] have proposed a partial pilot allocation scheme capable of increasing the uplink rate of users in a multi-cell scenario by exploiting the large- scale characteristics of fading channel so that users having poor channel conditions are saved from harsh interference. Also, with an increase in the number of BS antennas, sum-rate capacity increases as pairwise channel vectors become less correlated [17]. It is very tough to curtail the bit error caused by PC since the separation of desired information from a mess of different signals is challenging. An efficient approach for designing the pilot sequence has been proposed which is capable of achieving capacity and SINR fulfillments for users in presence of PC [18]. It has been proposed by Bhardwaj et al. [19] that the performance degradation in MU-m-MIMO systems with increase in the user base can be overcome by incorporating LDPC codes. Yin et al. [20] have proposed interference free CE performance through covariance aided framework using Bayesian estimator. Some authors have proposed a scheme in which grouping of users can be done in all cells depending on the similarity of the angle of arrivals and by implementing group matching and graph coloring based pilot allocation algorithms, the effect of PC can be minimized [21]. Benzida and Kadochb [22] have suggested the use of raptor codes with minimum mean square estimation (MMSE) to eliminate the use of pilot signals to find the channel state information resulting in reduced BER and power consumption at the user terminal. The problem due to PC can be reduced to some extent through error correction codes (EEC) which are capable of bringing down the BER if the block codes of large length are used. The class of EEC used in this article is LDPC codes which can approach Shannon’s capacity which iterates that reliable communication is possible over an unreliable channel if code rates exceed a specified limit [23]. The iterative procedure helps in obtaining the desired codeword using belief propagation (BP) or sum-product (SP) decoding which uses log-likelihood ratios (LLRs) corresponding to incoming channel information bits. The data bits are encoded by transferring the extrinsic information based on the values of code chart. Another widely used encoding algorithm is dense encoding (DE) technique. Hwang et al. [24] have proposed a lesser complex scheme for improving the m-MIMO performance by jointly detecting and decoding the information that relies on code chart technique. It has been proposed that the error correction coding in MU-m-MIMO systems with phase shift keying outperforms the conventional MIMO technology [25]. Incorporation of orthogonal frequency division multiplexing (OFDM) and LDPC codes with m-MIMO using non-regenerative relays improves the information quality at BS by implementing an MMSE detector with successive interference cancellation (SIC) [26].

In MC-MU-m-MIMO, users are present at different locations in all L cells. The time interval during which the channel impulse response of all the multipath components reaching the BS from a single user is essentially assumed to be constant. Ashikhmin et al. [27] have proposed a large-scale fading precoding concept which can reduce the inter-cell interference significantly by combining the messages aimed at subscribers from various cells that are reusing the training sequence.

3.1 Single cell MU-m-MIMO

Let’s consider an uplink scenario where a single cell is considered to have a M number of BS antennas and a K number of users. The channel vector M×1 between BS and user k is given as: $\boldsymbol{w}_{k}=\left[\begin{array}{l}w_{1 k} \\ w_{2 k} \\·\\ w_{m k}\\·\\ w_{M k}\end{array}\right]$, here the coefficient of channel between the k_th user and m_th antenna of the BS is w_mk.

w_mk ~ CN(0, b_k), E{w_mk}=0 and $E\left\{\left|w_{m k}\right|^{2}\right\}=b_{k}$, where variance b_k  models the shadow fading and geometric attenuation.

The considerably large physical distance among users in MU-m-MIMO leads to an existence of time interval during which the channel impulse response between M antennas and a user is fixed.

The M×1 received vector r at the BS is given below in Eq. (3):

$\mathbf{r}=\sqrt{s_{u}}\left[\begin{array}{llll}w_{1} & w_{2} & \ldots & w_{K}\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\ \cdot \\ \cdot \\ x_{K}\end{array}\right]+\left[\begin{array}{l}n_{1} \\ n_{2} \\ \cdot \\ \cdot\\ n_{M}\end{array}\right]=\sqrt{s_{u}} \mathbf{W} \mathbf{x}+\mathbf{n}$  (3)

here, W=[w₁w₂ … w_k] is the matrix having channel coefficients between K number of subscribers and M BS antennas. Further, w_mk =[W]_mk such that entries in W represents the effects of multi-path propagation, log-normal shadowing, and geometric attenuation, and hence it comprises of fast fading due to h_mk and large scale fading due to b_k. The noise vector n is AWGN noise such that n ~ CN(0,1). A k_th user has a symbol x_k and the average transmitted power associated with this signal is given by s_u such that

$\sqrt{s_{u}}\left[\begin{array}{l}x_{1} \\ x_{2} \\ \cdot \\ \cdot \\ x_{K}\end{array}\right]=\sqrt{s_{u}} \mathbf{x}$

where, $\sqrt{s_{u}}$ represents the average strength and $\sqrt{s_{u}} x$ is a column vector simultaneously transmitted by K users. As the BS antennas increases, the mean transmitted power s_u reduces. For ICSI present at the BS, $s_{u}=1 / \sqrt{M}$, while for accurate channel information, s_u=1/M [28].

The channel coefficient w_mk can thus be represented as $w_{m k}=h_{m k} \sqrt{b_{k}}$ and hence the channel matrix of the form W=HA^1/2 is given by Eq. (4) [28].

$\left[\begin{array}{llll}w_{1} & w_{2} & \ldots & w_{K}\end{array}\right]=\left[\begin{array}{llll}\mathbf{h}_{1} & \mathbf{h}_{2} & \ldots & \mathbf{h}_{K}\end{array}\right] \times$

$\left[\begin{array}{ccccc}\sqrt{b_{1}} & 0 & \cdot & \cdot & 0 \\ 0 & \sqrt{b_{2}} & 0 & \cdot & 0 \\ \cdot & 0 & \sqrt{b_{3}} & \cdot & 0 \\ \cdot & 0 & \cdot & \sqrt{b_{k}} & 0 \\ 0 & 0 & . & . & \sqrt{b_{K}}\end{array}\right] \leftarrow \mathbf{A}^{1 / 2}$  (4)

here matrix A is a square matrix that corresponds to geometric attenuation and shadow fading.

3.2 MC-MU-m-MIMO model

Consider a scenario of L multiple cells sharing the same frequency band in an uplink case with M antennas present at the BS and K single antenna users transmitting in every cell. Referring to (3) of section 3.1, the model for a multi-cell scenario is given in Eq. (5).

$\mathbf{r}_{l}=\sqrt{s_{u}} \sum_{j=1}^{L} \mathbf{W}_{l j} \mathbf{x}_{j}+\mathbf{n}_{l}$   (5)

where, r_l is the M×1 vector received at the l^th BS and W_lj is the channel matrix between l^th BS and K users in the j^th cell. $\sqrt{S_{u}} \boldsymbol{x}_{j}$ is the K×1 transmitted vector of K users in the j^th cell and n_l is an AWGN vector having complex normal Gaussian random elements of average power unity at l^th BS.

The channel vector at l^th BS for i^th user present in the j^th cell is given by $w_{l j i}=\sqrt{b_{l j i}} h_{l j i}$ where b_lji represents geometric attenuation and shadow fading between l^th BS and the i^th user in the j^th cell.

The matrix W_lj is therefore given by Eq. (6).

$\mathbf{W}_{l j}=\mathbf{H}_{l j} \sqrt{\mathbf{A}_{l j}}$  (6)

$\mathbf{W}_{l j}=\left[\begin{array}{lllll}\boldsymbol{w}_{l j 1} & \boldsymbol{w}_{l j 2} & \ldots & \boldsymbol{w}_{l j K}\end{array}\right]=\left[\begin{array}{llll}\mathbf{h}_{l j 1} & \mathbf{h}_{l j 2} & \ldots & \mathbf{h}_{l j K}\end{array}\right]\times$

$=\left[\begin{array}{ccccc}\sqrt{b_{l j 1}} & 0 & . & . & 0 \\ 0 & \sqrt{b_{l j 2}} & 0 & . & 0 \\ . & 0 & \sqrt{b_{l j 3}} & . & 0 \\ . & 0 & . & \sqrt{b_{l j k}} & 0 \\ 0 & 0 & . & . & \sqrt{b_{l j K}}\end{array}\right]$

where, A_lj is a K×K diagonal matrix such that [A_lj]_ii= b_lji.

3.3  Channel estimation in multi cell m-MIMO

Considering a single-cell scenario having K users for which the model for CE at the BS is given in Eq. (7).

$\mathbf{Z}_{M \times K}=\sqrt{K s_{u}}\left[\begin{array}{llll}\boldsymbol{w}_{1} & \boldsymbol{w}_{2} & \ldots & \boldsymbol{w}_{K}\end{array}\right]\left[\begin{array}{llll}\phi_{1} & \phi_{2} & \ldots & \phi_{K}\end{array}\right]+$$\left[\begin{array}{llll}\mathbf{n}_{1} & \mathbf{n}_{2} & \ldots & \mathbf{n}_{K}\end{array}\right]$   (7)

Eq. (7) can be expressed in dimensional form by Eq. (8) [29].

$\mathbf{Z}_{M \times K}=\sqrt{t_{p p}} \mathbf{W}_{M \times K} \mathbf{\phi}_{K \times K}+\mathbf{N}_{M \times K}$  (8)

here Z_M×K is the matrix of data aided symbols at the BS. t_pp is the power of pilot matrix such that t_pp = Ks_u. $\phi_{K \times K}$ is matrix of the pilot symbols and $\phi_{k}$ in $\left[\begin{array}{llll}\phi_{1} & \phi_{2} & \ldots & \phi_{K}\end{array}\right]$ is k^th vector represented by the K pilot symbols. N_M×K represents the noise matrix whose columns represent the noise elements present at M antennas.

The pilot matrix $\phi_{K \times K}$ is chosen in such a way that $\phi \phi^{H}=I$ where I is a K×K identity matrix. The result of this equality leads to orthogonality in the pilot matrix which consequently ends up with minimum or very little interference among the transmitted pilot symbols from different subscribers.

In MC-MU-m-MIMO, the total number of users in L cells is K×L. The CE model for the multi-cell scenario is given in Eq. (9) considering Eq. (8) as the reference equation.

$\mathbf{Z}_{l}=\sqrt{t_{p p}} \sum_{i=1}^{L} \mathbf{W}_{l i} \phi_{i}+\mathbf{N}_{l}$  (9)

here, the number of pilot vectors is P_S. Z_l and N_l are received and noise matrices having dimensions of M×P_S at the l^th cell BS. As the number users and BS antennas are assumed to be the same in all cells, the dimensions of matrices W_li and $\phi_{i}$ are M×K and K×P_S respectively for all values of i.

Eq. (9) can be put in the form given in Eq. (10).

$\mathbf{Z}_{l}=\sqrt{t_{p p}}\left[\begin{array}{llll}\mathbf{W}_{l 1} & \mathbf{W}_{l 2} & \ldots & \mathbf{W}_{l L}\end{array}\right]\left[\begin{array}{l}\phi_{1} \\ \phi_{1} \\ \vdots \\ \phi_{L}\end{array}\right]+\mathbf{N}_{l}$  (10)

where, [W_l1W_l2 … W_lL] is the M×KL channel matrix and $\left[\begin{array}{l}\phi_{1} \\ \phi_{1} \\ \vdots \\ \phi_{L}\end{array}\right]$  is a KL×P_S pilot matrix corresponding to L cells. The received matrix can be expressed in the composite form given in Eq. (11).

$\mathbf{Z}_{l}=\sqrt{t_{p p}} \tilde{\mathbf{W}} \tilde{\boldsymbol{\phi}}+\mathbf{N}_{l}$   (11)

In the wireless propagation scenario of L cells, signals from all KL users are being received by the l^th BS under consideration. These signals are propagating through different paths which are associated with corresponding channel coefficients. The knowledge of channel coefficients is crucial at the receiver so that signals are separated effectively using the MIMO receiver. The key point is that whether the behavior of the channel is appropriate or abrupt, the estimated coefficients should approach the actual coefficients to get considerable SNR for every received symbol.

3.3.1 Least square estimate

The CE model as given in (11) can be used and Least Square Cost (LSC) function is utilized to get the channel estimate. To obtain the estimate of the channel matrix, the minimum of $\left\|Z_{l}-\tilde{\phi} \widetilde{W}\right\|^{2}$ is evaluated at the receiver for obtaining the output at the MIMO receiver [30].

Estimated channel coefficients after implementation of the LSC function is given in Eq. (12) [31].

$\tilde{\mathbf{W}}=\frac{1}{\sqrt{t_{p p}}} \mathbf{Z}_{l} \tilde{\boldsymbol{\phi}}^{H}\left(\tilde{\phi} \tilde{\phi}^{H}\right)^{-1}$  (12)

It is worth noting from (12) that the orthogonality feature of the pilot matrix is necessary to eliminate the interference among the pilot symbols of KL users. To achieve the orthogonality it is required that $\widetilde{\boldsymbol{\phi}}{\widetilde{\boldsymbol{\phi}}}^{H}=\boldsymbol{I}$. Thus to accomplish the above-mentioned criteria for negligible interference in pilot symbols, the condition in Eq. (13) has to be satisfied.

$P_{s} \geq K L$   (13)

The distinct number of pilot symbols needed for transmission for CE is KL which is same as the total number of users present in a multi-cell scenario. This results in huge overhead in the MC-MU-m-MIMO systems. It is practically difficult to provide such a significant overhead as it may increase the latency caused by the delay in allocation of the orthogonal symbols to the huge user base of L cells. Also for pilot transmission, both allocation of time-frequency resources and coherence time of channel are limited and hence the number of orthogonal pilot symbols is also limited [32].

3.4 Pilot contamination

Avoiding overhead leads to interference in pilot symbols from other cells resulting in PC. The impact of PC on the received signal at the BS of l^th cell under consideration can have undesirable implications as the received M×1 vector is the sum of signals from users of L cells and separation process of the information symbols of the desired l^th cell users results in the presence of signals from the corresponding users of remaining L-1 cells at the output. This is due to the fact that the channel information present at the l^th BS also contains channel fading coefficients of remaining L-1 cells due to PC. Further, as the number of BS antennas increases, the problem of PC may be persistent and the usage of orthogonal symbols might lead to improved performance if the effect of PC is significant [33].

3.4.1 Channel estimation with PC

The number of pilot vectors required for no pilot interference in a multi-cell scenario is KL and one possibility of reducing the overhead is to reuse the pilot matrix in all L cells so that the distinct number of orthogonal pilot symbols required for CE reduces to K as compared to the situation when PC is not considered. Pilot reuse in all cells is given below: $\boldsymbol{\phi}_{j}=\boldsymbol{\phi}_{K \times P_{S}}$ for all values of j.

The CE model with pilot reuse is given in Eq. (14).

$\mathbf{Z}_{l}=\sqrt{t_{p p}} \sum_{j=1}^{L} \mathbf{W}_{l j} \phi+\mathbf{N}_{l}$  (14)

From (14), it is clear that the number of minimal pilot vectors needed is equal to the distinct number of pilot symbols used in the system which is equal to K. So, the overhead is reduced by a factor L that comes at the cost of increased interference since $\boldsymbol{\phi}$ is now reused in all cells. The channel estimated at the l^th cell BS using the LSC function is given in Eq. (15).

$\tilde{\mathbf{w}}_{l l}=\frac{1}{\sqrt{t_{p p}}} \mathbf{Z}_{l} \phi^{H}$  (15)

$=\frac{1}{\sqrt{t_{p p}}}\left(\sqrt{t_{p p}} \sum_{j=1}^{L} \mathbf{W}_{l j} \boldsymbol{\phi}+\mathbf{N}_{l}\right) \phi^{H}$  (16)

$=\sum_{j=1}^{L} \mathbf{W}_{l j}+E_{r}$  (17)

where, $E_{r}=\frac{1}{\sqrt{t_{p p}}} \boldsymbol{N}_{l} \phi^{H}$ is the CE error.

The fading coefficients between the K users and the BS of the l^th cell include the coefficients from all L cells as mentioned in Eq. (18).

$\tilde{\mathbf{W}}_{l l}=\mathbf{W}_{l l}+\sum_{j=1 \atop j \neq l}^{L} \mathbf{W}_{l j}+E_{r}$  (18)

Here, W_ll is the matrix of channel coefficients between K users and M BS antennas of the l^th cell. If the i^th user in l^th cell is the desired user, then the estimated channel vector corresponding to this user is given in Eq. (19).

$\tilde{\boldsymbol{w}}_{l l i}=\boldsymbol{w}_{l l i}+\sum_{j=1 \atop j \neq l}^{L} \boldsymbol{w}_{l j i}+\tilde{\mathbf{n}}_{l}$  (19)

where, the desired channel vector for the i^th user of the l^th cell is w_lli whereas the factor $\sum_{j=1 \atop j \neq l}^{L} \boldsymbol{w}_{l j i}$ representing the effect of PC constitutes corresponding channel vectors of the i^th user of L-1 cells. $\tilde{\boldsymbol{n}}_{l}$ is an M×1 noise vector corresponding to E_r. In terms of error and the desired channel vector, (19) can be re-written and is given by Eq. (20).

$\tilde{\boldsymbol{w}}_{l l i}=\boldsymbol{w}_{l l i}+\sum_{j=1 \atop j \neq l}^{L} \boldsymbol{w}_{l j i}+\tilde{\mathbf{n}}_{l}=\boldsymbol{w}_{l l i}+\mathbf{e}_{l l i}$  (20)

where, e_lli is the sum of interfering channel vectors and noise vector $\tilde{\boldsymbol{n}}_{l}$. The desired channel vector w_lli of the i^th user and $\tilde{\boldsymbol{n}}_{l}$ have variances b_lli and $\frac{1}{K s_{u}}$ respectively. So, it can be concluded that each element of e_lli has variance given in Eq. (21).

$\sigma_{e}^{2}=\sum_{j=1 \atop j \neq l}^{L} b_{l j i}+\frac{1}{K s_{u}}$  (21)

The variance in (21) has significance in interference caused at the output of the MIMO receiver. The variance due to multi-cell interference gets added with the variance of CE error to yield high BER when compared to the case of a single cell MU-m-MIMO system. The challenge lies ahead to overcome the effect of PC.

5.1 Analysis of error probability of MC-MU-m-MIMO using LDPC codes

A multi-cell scenario with K users in each cell is considered and a message block B_l for the users present in the l^th cell is formed. B_l is a K×f information matrix as given in Eq. (29) with f as the message bits of each user.

$\mathbf{B}_{l}=\left[\begin{array}{ccccc}I_{11} & I_{12} & I_{1 k} & . & I_{1 f} \\ I_{21} & I_{22} & I_{2 k} & . & I_{2 f} \\ . & . & . & . & . \\ I_{K 1} & I_{K 2} & I_{K k} & . & I_{K f}\end{array}\right]$   (29)

here I_Kk is k^th message bit of the user K. DE with coding rate 1/2 is implemented on each row resulting in a codeword with length equal to 2f. The K×n matrix E_l thus obtained having K codewords is given in Eq. (30).

$\mathbf{E}_{l}=\left[\begin{array}{ccccc}c_{11} & c_{12} & c_{1 k} & \cdot & c_{1 n} \\ c_{21} & c_{22} & c_{2 k} & \cdot & c_{2 n} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ c_{K 1} & c_{K 2} & c_{K k} & \cdot & c_{K n}\end{array}\right]$  (30)

where, [c₁₁ c₁₂ c_1k … c_1n] is the codeword obtained for user 1. In a scenario of L cells under consideration, the number of users in all cells is the same and hence the encoded matrices E₁, E₂, … E_L corresponding to L cells have the same dimensions K×n. The modulation of all the codewors of E_l is done employing QAM and 16-QAM techniques. The K×(n/log₂U) modulated matrix Q_l thus obtained has modulated symbols and the column vectors are sequentially transmitted by K users. log₂U is an integer value (2 or 4) which corresponds to symbol bits and depends on the order of modulation U. The model equation of MC-MU-m-MIMO is thus given in Eq. (31).

$\mathbf{R}_{l}=\sqrt{s_{u}} \sum_{j=1}^{L} W_{l j} \mathbf{Q}_{j}+\mathbf{N}_{l}$    (31)

here, R_l and N_l are M×(n/log₂U) received and AWGN noise matrices respectively at the l^th cell BS.

Each element of R_l is the summation of symbols obtained from K×L users. Signals separation of the l^th cell users is done through ZF receiver giving the output as a K×(n/log₂U) block matrix $\hat{\boldsymbol{X}}_{l}$ and is mentioned in Eq. (32), the rows of which represent the estimated symbols of l^th cell users in modulated form.

$\hat{\mathbf{X}}_{l}=\left[\begin{array}{ccc}D_{M 1}(1) & D_{M 1}(2) \cdots & D_{M 1}\left(n / \log _{2} U\right) \\ D_{M 2}(1) & D_{M 2}(2) \cdots & D_{M 2}\left(n / \log _{2} U\right) \\ D_{M k}(1) & D_{M k}(2) \cdots & D_{M k}\left(n / \log _{2} U\right) \\ \vdots & \vdots & \vdots \\ D_{M K}(1) & D_{M K}(2) & D_{M K}\left(n / \log _{2} U\right)\end{array}\right]$   (32)

here, D_Mk(2) is the modulated symbol of second transmission which is expected to be of the k^th user of the l^th cell at the ZF output. The symbol D_Mk(2) is given as: D_Mk(2)= D_Mk1(2)+ D_Mk2(2)+…D_Mkl(2) …+ D_MkL(2) where D_Mk1(2), D_Mk2(2) and D_MkL(2) represent the symbols of k^th users of 1, 2, and L^th cells. It clearly represents that D_Mk(2) also contains the information of the k^th user of remaining L-1 cells resulting from PC.

The estimated modulated symbols of $\hat{\boldsymbol{X}}_{l}$ are demodulated to obtain a K×n matrix $\hat{\boldsymbol{B}}_{l}$. The rows of $\hat{\boldsymbol{B}}_{l}$ are the estimated vectors which correspond to codewords of K users of the l^th cell. The decoding of every row vector of $\hat{\boldsymbol{B}}_{l}$ is performed through the algorithm as discussed in section 4.2 which gives the output as a K×f block matrix $\widetilde{\boldsymbol{D}}_{l}$. All the row vectors of $\widetilde{\boldsymbol{D}}_{l}$ are the estimated information bitstream of K users of the l^th cell under consideration. The comparison of $\widetilde{\boldsymbol{D}}_{l}$ and $\mathbf{B}_{l}$ in terms of bit errors provides the performance analysis of pilot Contaminated MC-MU-m-MIMO system.

Algorithm 1. Analysis of error probability

Initialize SNR and BER

1. n_Block: data blocks
2. K: users in each cell
3. L: multiple cells with same frequency
4. L_block: block length
5. N_R: receiving antennas at BS
6. N_T: transmitting antennas in a cell
7. N_SYM: symbols transmitted for CE
8. for $j \in[1: L]$
9. rawbitsj=randombits([0,1],[N_T,L_block]): Information matrix of K users of j^th cell
10. f_j=[ ]: Encoded matrix initialization for j^th cell users
11. for i=1:K
12. f_j=[f_j encode(rawbitsj)’(:,i)]: Encoded matrix of K users of j^th cell
13. end
14. s_j=[ ]: Modulated matrix initialization for j^th cell users
15. for i=1:K
16. s_j=[s_j modulate f_j(:i)]: Matrix of the modulated symbols of K users of j^th cell
17. end
18. bits_j=conj(s_j’): Transverse of s_j
19. end
20. for i=1:n_Block
21. subscript 1j: between BS antennas of cell 1 and K users of the j^th cell where cell 1 is the desired cell
22. H_1j=randn(N_R1,N_Tj)+j×randn(N_R1,N_Tj); $j \in[1: L]$: complex channel matrix
23. A_1j; $j \in[1: L]$: Matrix having parameters signifying geometric attenuation and log-normal fading.
24. $W_{1 j}=H_{1 j} \sqrt{A_{1 j}}$; $j \in[1: L]$: channel coefficients matrix corresponding to fast and slow fading.
25. $S_{u}=E_{u s e r} / \sqrt{N_{R}}$: average user power
26. P_pilot=N_SYMs_u: Total power of pilot matrix
27. M_P: matrix of pilot symbols
28. $N_{1}=\sqrt{1 / N_{R}}\left(\right.$ randomnumber $\left(N_{R}, N_{S Y M}\right)+$$\operatorname{jrandn}\left(N_{R}, N_{S Y M}\right)$: Noise matrix at BS antennas of cell 1.
29. $Z_{1}=\sqrt{P_{\text {pilot }}} W_{11} M_{P}+\sqrt{P_{\text {pilot }}} W_{12} M_{P}+\cdots+$$\sqrt{P_{\text {pilot }}} W_{1 L} M_{P}+N_{1}$ : Received pilot for CE at cell 1.
30. $\hat{W}_{1}=\sqrt{\frac{1}{P_{p i l o t}}} \times Z_{1} \times \operatorname{conj}\left(M_{P'}\right)$: Matrix having estimates of channel coefficients at cell 1due to PC.
31.  for K = 1: (length of SNR)
32. N_1c=randn(N_R,L_block/y)+jrandn(N_R,L_block/y): BS noise at cell 1 when transmission of actual data occurs, y has value 4 and 2 for modulation orders 16 and 4 respectively.
33. SNR(K)=(SNRindB(K)/10)×10: conversion of dB to the standard value of SNR
34. $b=S N R(K) / \sqrt{N_{R}}$: scaling of SNR
35. $T_{j}=\sqrt{b} \times$ bits $_{j}$; $j \in[1: L]$: Transmitted symbol matrix of j^th cell
36. R₁=W₁₁T₁+W₁₂T₂+…+W_1LT_L+N_1c: Received signal at cell 1
37. $M F_{1}=\operatorname{conj}\left(\hat{W}_{1'}\right) \times R_{1}$: Output of MIMO receiver at cell 1.
38. r₁=[ ]: Demodulator matrix initialization
39. for i=1:K
40. r₁=[r₁de mod ulate(MF₁’(:,i))]: A demodulated matrix for cell 1.
41. end
42. x₁=[ ]: initialize decoder (BP decoding)
43. for i=1:K
44. x₁=[x₁ decode(r₁(:,i))]: LDPC decoded matrix for cell 1
45. end
46. BER(K)=sum(sum(x₁~=rawbits1))+BER(K):BER value
47. end: for K
48. end: for n_Block
49. BER=BER/(n_Block×L_block×N_T)

The algorithm executes for L=2,4,6&8

5.2 Sum rate capacity

In this section, the analysis of sum-rate capacity has been done for l^th cell users. The expressions of signal to interference and noise ratio (SINR) and capacity are derived for the first user of the l^th cell in a scenario of L cells sharing the same frequency band. Thereafter, generalized sum expression for rate will be given for all the K users present in the l^th cell.

The modified model equation of the MC-MU-m-MIMO system given in equation 5 can be expressed by Eq. (33).

$\mathbf{r}_{l}=\sqrt{s_{u}}\left(\mathbf{W}_{l 1} \mathbf{x}_{1}+\mathbf{W}_{l 2} \mathbf{x}_{2}+\ldots \mathbf{W}_{l l} \mathbf{x}_{l}+\ldots+\mathbf{W}_{l L} \mathbf{x}_{L}\right)+\mathbf{n}_{l}$   (33)

where, W_ll is the channel matrix between M BS antennas and K users of the l^th cell. Considering a general case where l=1 in which user 1 is the desired user, so the received vector is given in Eq. (34).

$\mathbf{r}_{1}=\sqrt{s_{u}} \mathbf{W}_{11} \mathbf{x}_{1}+\sqrt{s_{u}}\left(\sum_{j=2}^{L} \mathbf{W}_{1 j} \mathbf{x}_{j}\right)+\mathbf{n}_{1}$   (34)

r₁ is M×1 vector received at BS of cell 1.

In terms of channel vector form, the above equation can be rewritten as given in Eq. (35).

$\mathbf{r}_{1}=\sqrt{s_{u}} \mathbf{w}_{111} x_{11}+\sqrt{s_{u}} \sum_{i=2}^{K} \mathbf{w}_{11 i} x_{1 i}+\sqrt{s_{u}} \sum_{j=2}^{L} \sum_{i=1}^{K} \mathbf{w}_{1 j i} x_{j i}+\mathbf{n}_{1}$  (35)

here x₁₁ and x_1i are the information symbols of user 1 and i^th user of cell 1. x_ji is the information symbol of the i^th user present in the j^th cell. w₁₁₁ is the M×1 vector having fading coefficients between user 1 and M BS antennas of cell 1. w_11i is the M×1 channel vector between M BS antennas and i^th user of cell 1. w_1ji is the M×1 channel vector between M BS antennas of cell 1 and i^th user of the j^th cell. n₁ is the M×1 noise vector at M BS antennas of cell 1.

The received vector r₁ is processed at the BS of cell 1 using an MF receiver. The separated signals thus obtained at the output of MF receiver are considered as the symbols of K users of cell 1 which also contain the interfering symbols from remaining L-1 cells. The optimal beamformer for Maximal Ratio Combiner (MRC) for the desired user is $\widetilde{\boldsymbol{W}}_{111}$. Hence the output of spatial MF receiver is thus given by Eq. (36).

Output $_{M F}=\tilde{\mathbf{w}}_{111}^{H} \mathbf{r}_{1}=\tilde{\mathbf{w}}_{111}^{H}\left(\sqrt{s_{u}} \mathbf{w}_{111} x_{11}+\sqrt{s_{u}} \sum_{i=2}^{K} \mathbf{w}_{11 i} x_{1 i}+\right.$$\left.\sqrt{S_{u}} \sum_{j=2}^{L} \sum_{i=1}^{K} \mathbf{w}_{1 j i} x_{j i}+\mathbf{n}_{1}\right)$ (36)

$=\sqrt{s_{u}} \tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{111} x_{11}+\sqrt{s_{u}} \sum_{i=2}^{K} \tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i} x_{1 i}+\tilde{\mathbf{w}}_{111}^{H} \mathbf{n}_{1}+$$\sqrt{s_{u}} \sum_{j=2}^{L} \sum_{i=1}^{K} \tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i} x_{j i}$   (37)

From (20), the estimated channel vector of user 1 can be written as $\tilde{\boldsymbol{W}}_{111}=\boldsymbol{w}_{111}+\boldsymbol{e}_{111}$, where e₁₁₁ is the error caused due to the sum of interfering channel vectors of corresponding users of L-1 cells and vector $\tilde{\boldsymbol{n}}_{1}=\left(1 / \sqrt{t_{p p}}\right) \boldsymbol{n}_{1} \phi^{H}$.

The desired signal part in Eq. (37) is $\sqrt{S_{u}} \widetilde{\boldsymbol{W}}_{111}{ }^{H} \boldsymbol{w}_{111} x_{11}$ which can be written as:

$\sqrt{s_{u}} \tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{111} x_{11}=\sqrt{s_{u}}\left(\mathbf{w}_{111}+\mathbf{e}_{111}\right)^{H} \mathbf{w}_{111} x_{11}$

$=\sqrt{s_{u}}\left\|\mathbf{w}_{111}\right\|^{2} x_{11}+\sqrt{s_{u}} \mathbf{e}_{111}^{H} \mathbf{w}_{111} x_{11}$

The SINR for user 1 present in cell 1is given in Eq. (38).

$\operatorname{SINR}_{11}=\frac{s_{u}\left\|\mathbf{w}_{111}\right\|^{4}}{s_{u} E\left\{\left|\mathbf{e}_{111}^{H} \mathbf{w}_{111}\right|^{2}\right\}+s_{u} \sum_{i=2}^{K} E\left\{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2}\right\}+}$$S_{u} \sum_{i=1}^{K} \sum_{j=2}^{L} E\left\{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i}\right|^{2}\right\}+E\left\{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{n}_{1}\right|^{2}\right.$   (38)

$\operatorname{SINR}_{11}=\frac{s_{u}\left\|\mathbf{w}_{111}\right\|^{2}}{s_{u} E\left\{\frac{\left|\mathbf{e}_{111}^{H} \mathbf{w}_{111}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}+s_{u} \sum_{i=2}^{K} E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}+}$$s_{u} \sum_{i=1}^{K} \sum_{j=2}^{L} E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}+\frac{\left\|\tilde{\mathbf{w}}_{111}\right\|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}$      (39)

where, $\widetilde{\boldsymbol{w}}_{111}^{H} \boldsymbol{w}_{11 i}$ is the correlation of the estimated vector of the desired user with the channel vector of the i^th user of cell 1. Also

$E\left\{\frac{\left|\mathbf{e}_{111}^{H} \mathbf{w}_{111}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}=\frac{b_{111}\left\|\mathbf{e}_{111}\right\|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}=\frac{\left\|\mathbf{e}_{111}\right\|^{2}}{M}$  (40)

where, $\frac{\left\|w_{111}\right\|^{2}}{M}=b_{111}$ is the variance of the channel vector of the desired user as obtained by implementing the law of large numbers.

Further, from (20), $\boldsymbol{e}_{111}=\sum_{j=2}^{L} \boldsymbol{w}_{1 j 1}+\tilde{\boldsymbol{n}}_{1}$ and hence using the Law of large numbers, (40) is modified as follows:

$E\left\{\frac{\left|e_{111}^{H} w_{111}\right|^{2}}{\left\|w_{111}\right\|^{2}}\right\}=\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}$, here $\sum_{j=2}^{L} b_{1 j 1}$ and $\frac{1}{K s_{u}}$ represent the variances of inter-cell interference due to PC and CE error respectively. Eq. (41) gives the expression for SINR₁₁ by using the above relationship.

$S I N R_{11}=\frac{s_{u}\left\|\mathbf{w}_{111}\right\|^{2}}{s_{u} \sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K}+s_{u} \sum_{i=2}^{K} E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}+}$

$S_{u} \sum_{i=1}^{K} \sum_{j=2}^{L} E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}+\frac{\left\|\tilde{\mathbf{w}}_{111}\right\|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}$  (41)

$\frac{\left\|\tilde{\mathbf{w}}_{111}\right\|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}=\frac{\left\|\tilde{\mathbf{w}}_{111}\right\|^{2} / M}{\left\|\mathbf{w}_{111}\right\|^{2} / M}=\frac{b_{111}+\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}}{b_{111}}$  (42)

Also, $E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2}}{\left\|\mathbf{w}_{111}\right\|^{2}}\right\}=E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2} / M}{\left\|\mathbf{w}_{111}\right\|^{2} / M}\right\}$

$\Rightarrow E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{11 i}\right|^{2} / M}{\left\|\mathbf{w}_{111}\right\|^{2} / M}\right\}=\frac{\left(b_{111}+\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}\right) b_{11 i}}{b_{111}}$  (43)

Further, $\begin{aligned} E\{&\left\{\begin{array}{l}\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i}\right|^{2} \\ \hline\left\|\mathbf{w}_{111}\right\|^{2}\end{array}\right\}=E\left\{\frac{\left|\tilde{\mathbf{w}}_{111}^{H} \mathbf{w}_{1 j i}\right|^{2} / M}{\left\|\mathbf{w}_{111}\right\|^{2} / M}\right\} \\ &=\frac{\left(b_{111}+\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}\right) b_{1 j i}}{b_{111}} \end{aligned}$   (44)

Substituting Eqns. (42), (43), and (44) in Eq. (41), the expression for SINR₁₁ in terms of variances of channel vectors is given in Eq. (45).

$SIN{{R}_{11}}=\frac{{{s}_{u}}\parallel {{\mathbf{w}}_{111}}{{\parallel }^{2}}}{{{s}_{u}}\sum\limits_{j=2}^{L}{{{b}_{1j1}}}+\frac{1}{K}+{{s}_{u}}\sum\limits_{i=2}^{K}{\frac{\left( {{b}_{111}}+\sum\limits_{j=2}^{L}{{{b}_{1j1}}}+\frac{1}{K{{s}_{u}}} \right){{b}_{11i}}}{{{b}_{111}}}+}}$

$s_{u} \sum_{i=1}^{K} \sum_{j=2}^{L} \frac{\left(b_{111}+\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}\right) b_{1 j i}}{b_{111}}+\frac{b_{111}+\sum_{j=2}^{L} b_{1 j 1}+\frac{1}{K s_{u}}}{b_{111}}$  (45)

The expressions given in (43) and (44) are representing intra-cell and inter-cell interference respectively and hence the SINR₁₁ in (45) has been degraded due to the presence of inter-cell interference when compared to a single cell MU-m-MIMO where the factor representing inter-cell interference is nil. The average power of a user s_u can be scaled as $\frac{E_{u}}{\sqrt{M}}$ where E_u=s_u using which constant rate can be maintained. So, the expression of asymptotic SINR is given in Eq. (46).

$SIN{{R}_{11}}=\frac{Kb_{111}^{2}E_{u}^{2}}{KE_{u}^{2}\sum\limits_{j=2}^{L}{b_{1j1}^{2}+1}}$   (46)

The rate expression corresponding to SINR₁₁ is given in Eq. (47)

$R_{1}=\log _{2}\left(1+\operatorname{SINR}_{11}\right)$   (47)

The sum-rate expression for K users is given in Eq. (48).

$R=\sum_{i=1}^{K} \log _{2}\left(1+S I N R_{1 i}\right)$   (48)

where, SINR_1i represents the SINR of the i^th user of cell 1.

In the MC-MU-m-MIMO system, the location of the cell sites is different in a geographical area due to which the power received by the BS of cell 1 from users of L-1 cells is comparatively lesser than that received from users of cell 1. This fact gives an important conclusion on the variance of channel coefficient between M antennas of cell 1 and users of remaining L-1 cells. The shadowing effect leading to large-scale fading increases between two cells which results in a decrease in the power of channel coefficients and hence the variance b_1ji reduces due to co-channel cell separation where j and i vary from 2 to L and 1 to K respectively. Hence, the sum-rate capacity can be improved when the large-scale fading increases. Consequently, the separation between two cells where frequency reuse has been implemented has to be sufficiently large so that tolerable SINR is obtained at the BS.

Algorithm 2. Sum-rate of MC-MU-m-MIMO for different receive antennas

1. ITER: No. of iterations
2. M_n: Initialize range of number of receive antennas
3. R: Initialize rate for MRC
4. for i=1:ITER
5. A_1j; $j \in[1: L]$: Matrix corresponding to large-scale fading parameters
6. for m=1:length(M_n)
7. N_R=M_n(m): the fixed value of Receive antennas
8. N_T: transmitting antennas equal to number of cell users
9. H_1j=randn(NR₁,N_Tj)+j×randn(NR₁,N_Tj) $j \in[1: L]$: complex channel matrix
10. $W_{1 j}=H_{1 j} \sqrt{A_{1 j}}$$j \in[1: L]$: channel coefficients matrix corresponding to fast and slow fading.
11. $s_{u}=E_{u s e r} / \sqrt{N_{R}}$: average user power
12. P_pilot=N_SYMs_u: Power of pilot matrix  
13. M_P: matrix of pilot symbols
14. $N_{1}=\sqrt{1 / N_{R}}\left(\operatorname{randn}\left(N_{R}, N_{S Y M}\right)+\operatorname{jrandn}\left(N_{R}, N_{S Y M}\right)\right.$: Noise matrix at BS antennas of cell 1.
15. $Z_{1}=\sqrt{P_{\text {pilot }}} W_{11} M_{P}+\sqrt{P_{\text {pilot }}} W_{12} M_{P}+\cdots+$$\sqrt{P_{\text {pilot }}} W_{1 L} M_{P}+N_{1}$ : Received pilot for CE at cell 1.
16. $\hat{W}_{1}=\sqrt{\frac{1}{P_{\text {pilot }}}} \times Z_{1} \times \operatorname{conj}\left(M_{P}^{\prime}\right)$ : Matrix having estimates of channel coefficients at cell 1due to PC.
17.  For k=1:K
18. $\hat{W}_{k 1}=\hat{W}_{1}(:, k)$ : channel vector of the k^th user of cell 1
19. D_mrc=1/N_SYM: one term in the denominator of SNR
20.  For iu=1:K
21. E_rr: Find error expression
22. D_mrc= D_mrc+ E_rr
23. if (iu~=k)
24. D_mrc=D_mrc+k_u× variance calculation of channel vectors $\hat{W}_{k 1}^{\prime}$ and $\hat{W}_{i u 1}$ respectively: Intracell interference calculation
25.  end
26.  end
27.  for j=1:K
28. D_mrc=D_mrc+k_u×sum of variances of channel vectors $\hat{W}_{k 1}^{\prime}$ and $\hat{W}_{j l}$ for l=2:L, Intercell interference
29. end
30. $N_{m r c}=k_{u} \times\left\|W_{11}(:, k)\right\|^{2}$: Numerator of SNR
31. R(m)=R(m)+log₂(1+N_mrc/D_mrc)
32. end
33. end
34. end
35. R=R/ITER: Sum-rate expression

The algorithm executes for L=2,4,6&8

The paper studied two-stage processing of received signal for reducing the error at the BS. The ICSI due to CE error and PC results in huge interference of signals from L-1 cells at the desired BS which separates the signals belonging to K users of the desired cell. The segregated signals also consist of signals of corresponding users present in L-1 cells and therefore the output of the MIMO receiver is decoded using LDPC codes using which significant performance improvement has been shown as error probability has been reduced by (80-85)%. Further, improvement in the performance has been shown when the channel strengths between BS antennas of desired cell and users of remaining cells is less, and hence, the reduction in BER has been observed for both uncoded and LDPC coded system with QAM and 16-QAM respectively. Sum-rates for MC-MU-m-MIMO increase with the number of BS antennas and improve by 60% for the case of reduced strength of channel coefficients. It has been inferred that PC puts a limit on the number of cells which can be given the same frequency as for a higher number of co-channel cells (L= 6, 8) the performance is almost constant even with LPDC codes.