Measured and estimated data of non-linear BRAN channels using HOS in 4G wireless communications

The aim of this research is to develop a non-linear blind estimator able to represents a Broadband Radio Access Networks (BRAN) channels. In the one hand, we have used Higher Order Statistics (HOS) theory to build our algorithm. Indeed, we develop a non-linear method based only on fourth order cumulants for identifying the diagonal parameters of quadratic systems. In the other hand, the developed approach is applied to estimate the experimental channels, BRAN A, C and E data normalized for MC-CDMA, in non-linear case. However, the estimated data will be used in the blind equalization. The simulation results in noisy environment and for different signal to noise ratio (SNR) show the accuracy of develop estimator blindly (i.e., without any information about the input) with non-Gaussian signal input. Furthermore, in part of blind equalization problem the obtained results, using Zero forcing (ZF) and Minimum Mean Square Error (MMSE) equalizers, demonstrate that the proposed algorithm is very adequate to correct channel distortion in term the Bit Error Rate (BER). Finally, these estimated data present a necessary asset for conducting validation experiments, and can be also used as a baseline.


a b s t r a c t
The aim of this research is to develop a non-linear blind estimator able to represents a Broadband Radio Access Networks (BRAN) channels. In the one hand, we have used Higher Order Statistics (HOS) theory to build our algorithm. Indeed, we develop a nonlinear method based only on fourth order cumulants for identifying the diagonal parameters of quadratic systems. In the other hand, the developed approach is applied to estimate the experimental channels, BRAN A, C and E data normalized for MC-CDMA, in non-linear case. However, the estimated data will be used in the blind equalization. The simulation results in noisy environment and for different signal to noise ratio (SNR) show the accuracy of develop estimator blindly (i.e., without any information about the input) with non-Gaussian signal input. Furthermore, in part of blind equalization problem the obtained results, using Zero forcing (ZF) and Minimum Mean Square Error (MMSE) equalizers, demonstrate that the proposed algorithm is very adequate to correct channel distortion in term the Bit Error Rate (BER). Finally, these estimated data present a necessary asset for conducting validation experiments, and can be also used as a baseline.
& The estimated data provides information about the efficiencies of develop method; An analysis of the influence of the noise to estimated data of BRAN channels; The estimated data can be also used as a baseline; Exploiting the estimation data of BRAN channels in blind equalization; Can be used for wireless communications in order to compensate the fading channel in term the BER in 4G MC-CDMA systems.

Data
Three models, BRAN A, C and BRAN E, [1,2] are used in this investigation. These models correspond to typical large open space indoor and outdoor environments with large delay spread. The data presented in Tables 1, 2 and 3 represent the delay and magnitudes of 18 targets of BRAN A, C and E channels respectively.

Non-linear channels representation: A problem formulation
The BRAN channel is modeled as the output of a non-linear quadratic system that is excited by an non-Gaussian signal input and is corrupted at its output by an additive Gaussian noise.
This system can be represented as follows (Fig. 1): The output of this model is described by the following relationships: For this system we assume that: The considered noise sequence wðkÞ is assumed to be zero mean, independent and identically distributed ði:i:dÞ, Gaussian, independent of xðkÞ with unknown variance; The order q is known. Unknown quadratic kernels include fhði; iÞ ∀i ¼ 1; …; qg; The input sequence xðkÞ is ði:i:dÞ zero mean, stationary, non-Gaussian and with: where γ n;x denotes the n th order cumulants of the input signal xðkÞ at origin, with γ 2; The system is supposed causal and truncated, i.e. hði; iÞ ¼ 0 if i o 0 and i4 q with hð0; 0Þ ¼ 1; The system is supposed stable, i.e. jhði; iÞj o ∞.
In this section, we give the relationship linking the higher order statistics or cumulants with the diagonal parameters of quadratic systems.
The second and third order cumulant of the process fzðkÞg are described by the following expressions respectively [3]: The fourth order cumulants of the process fzðkÞg is defined by [4]:

Proposed non-linear blind estimator
In this subsection we develop a blind method for identifying non-linear BRAN channels and downlink MC-CDMA equalization.

Overview an MC-CDMA systems
In the purpose to support the anticipated multi-media intensive applications, we have needed a high data rate, this we pushed to use the future 4G communication systems. However, the demand for high data rates in 4G systems causes the transmitted signals to be subjected to frequency-selective fading. Indeed, Orthogonal Frequency Division Multiplexing (OFDM) and MC-CDMA are multicarrier modulation schemes that have been proposed for 4G systems due to their ability to achieve high spectral efficiency by using minimally spaced orthogonal subcarriers and provide robustness against frequency selectivity in wireless channels, without increasing the transceiver complexity [6].
The MC-CDMA signal can be generated by a inverse Fourier transform (IFFT) performed on the spreading code chips. Thus, the choice of spreading codes is fundamental [5], [7,8]. Indeed, the complex symbol a i of each user i is, firstly, multiplied by each chip c i;k of Walsh-Hadamard spreading code, and then applied to the modulator of multicarriers. Each subcarrier transmits an element of information multiply by a code chip of that subcarrier. Under the hypothesis of L c equal to N p , the expression of the signal transmitted at the output of the modulator is given by the following equation: where, the matrix C represent the spreading codes: where, c i ¼ ½c i;0 ; c i;1 ; …; c i;Lc−1 T . When N u users are active, the multi-user downlink MC-CDMA signal received at the input of the receiver, denoted by rðtÞ, is given by the following expression: After the equalization operation, the expression of the signal s k is given in vector form by the following expression: where, H ¼ diag ½h 0 ; h 1 ; …; h Np−1 represents the complex channel frequency response. The matrix G ¼ diag ½g 0 ; …; g Np−1 represent the diagonal matrix composed of the coefficients g k equalization.
Or, in scalar form by the following expression: After despreading and threshold detection, the data symbol of the user detected corresponds to the sign of the scalar produced between the vector of the received equalized signals, s, and the userspecific spreading code i, c T i , that is: Using Eqs. (23) and (24) the general expression of the symbol detected for i user is given by the following equation: where the term U, M and N of Eq. (25) are, respectively, the signal of the considered user, a signals of the others users (multiple access interferences) and the noise pondered by the equalization coefficient and by spreading code of the chip.
If we suppose that the spreading code are orthogonal, i.e., Eq. (25) will become:

ZF equalizer
The goal of ZF minimize the peak distortion of the equalized channel, it is applying of the inverse of channel to the received signal and restores signal, defined as: The estimated received symbol, b a i of symbol a i of the user i is described by: The goal of the equalization is to extract a i .

MMSE equalizer
The minimization of the function E½jεj 2 ¼ E½jx k −g k r k j 2 , gives us the optimal equalizer coefficient. The equalization coefficients based on this MMSE criterion applied independently per carrier are equal to: where ζ k ¼ E½jx k h k j 2 E½jn k j 2 .
The estimated received symbol, b a i of symbol a i of the user i is described by:

Numerical simulations results
The work presented in this paper is structured around two neighboring themes. Identification of BRAN channels in the one hand, and downlink MC-CDMA equalization in the other hand. In the part of identification impulse response parameters of BRAN channels using HOS method we use the non-Gaussian input, and the additive noise is Gaussian, with symmetric distribution, zero mean, with the m th order cumulants vanishes for m 4 2. Hence the utility to use the higher order cumulants domain. In the part of equalization problem of MC-CDMA systems we use the BPSK symbol constellation.
In this section, we present the numerical simulations results of the estimated data of BRAN A, C and E. The simulation is performed with MATLAB software in noise environment. To measure the strength of noise, we define the SNR by the following relationship: To measure the accuracy of the diagonal parameter estimation with respect to the real values, we define the Mean Square Error (MSE) for each run as: where b hði; iÞ and hði; iÞ, i ¼ 1; …; q, are respectively the estimated and the real parameters in each run. The Figs. 2, 3 and 4 represent the zeros of BRAN A, C and E channels respectively.        The non-linear BRAN A channel, presented in the Table 1, is described by the following model (Eq. 34):    Table 5 Estimation of the BRAN C radio channel impulse response for different SNR and data length N¼ 4800.

BER performance of MC-CDMA systems
In this subsection we present numerical simulation results of the measured and estimated BER in MC-CDMA systems . These estimation are preformed using MMSE and ZF equalizers.
Performances quality is evaluated under the following conditions (Table 7): In the Tables 8, 9 and 10 we represents the real and estimated BER of the BRAN A, C and E channels respectively, using ZF and MMSE equalizers.

Concluding remarks
In this paper we have proposed a blind non-linear approach, based on fourth order cumulants, for estimation of the non-linear BRAN channels excited by non-Gaussian and independent identically distributed signal. According to the analytic study and numerical simulations results we can draw the following remarks: Table 6 Estimation of the BRAN E radio channel impulse response for different SNR and data length N ¼4800. b hði; iÞ 7 std       The proposed estimator able to estimate the diagonal parameters of the quadratic BRAN channels from the output signal without knowing the system input i.e. blindly; The developed approach is based on simple equations that we have used only ðq þ 1Þ equations to estimate q parameters; The measured values of the BRAN channels are very close to the estimated in different SNR, same in very noise environment SNR ¼0 dB; Not affected by the presence of Gaussian noise, because it is vanish in the higher order cumulants domain; Gives a good result for a standard deviation.
In the part of blind equalization problem, we consider the 4G MC-CDMA systems. However, we use two equalizers, ZF and MMSE, after the channel identification to correct the channel distortion. According BER simulation results we can conclude the following: Using ZF equalizer, the BER simulation for various SNR between 0 dB and 28 dB, demonstrates that the estimated values are more close to the real value of all data BRAN, and we have a best accorded; Using MMSE equalizer, the obtained results show a prefect accorded between the real and estimated BER. Indeed, for example using BRAN A data, if the SNR Z 20 dB we have 1 bit error if we receive 10 5 bit.
Finally, the proposed estimator combining with the MMSE equalizer can compensate the distortion introduite by radio channel in noise environment.