A Compensation Method Based on NN at the Transmitter and the Receiver Level for Nonlinear Distortion in MIMO OFDM Sytems Using MMSE Receiver

Maha CHERlF DAKHLl, Rafik ZA Y ANI and Ridha BOUALLEGUE

Innov'Com Lab, Sup'Com, University of Carthage, Tunis, Tunisia

Abstract-We propose a Non-Linear (NL) distortion

compensator based on Neural Network (NN) accompanied

with MMSE (Minimum Mean Square Error) which corrects

the nonlinearity introduced by HP A (High Power Amplifier).

The neural network consists on a feed-forward Multi-Layer

Perceptron (MLP) associated with Levenberg-Marquardt

learning algorithm. In this paper, the correction is done at the

transmitter after HPA and at the receiver level after OFDM

demodulation. We present the structure of the NN adapted to

both the predistortion (NN-PD) and frequency domain. Finally

some simulation results are presented in a VB LAST MIMO

OFDM (Vertical Bell Laboratories Layered Space-Time

Multiple-Input Multiple-Output Orthogonal Frequency

Division Multiplexing) system running under a Rayleigh fading

channel.

Keywords- VBLAST, MIMO, OFDM, HPA, NEURAL

NETWORK, Predistortion, CONNPD, MLP, MMSE.

I. INTRODlJCTION

The combination of MIMO with OFOM is a promising technique for high performance with very high data rates in 4G broadband wireless communications [1].

In order to propagate the signal on the channel, an amplifier (HPA) is needed. However, like in classical MIMO OFDM exhibits large Peak-to-Average Power Ratios (PAPR), "i.e.", large fluctuations in their signal envelopes [2]. Indeed, the performance of the transceiver is very sensitive to nonlinear distortions caused by the High Power Amplifiers (HP As) when operating near their nonlinear saturation regions [3].

Recent several research efforts have dealt with the issue of HPA nonlinearity in OFOM and MIMO systems. For instance, the impact of HP A nonlinearity on the symbol error probability (SEP) was studied in [4] for MIMO systems employing space-time trellis codes. The above-mentioned work used the Saleh nonlinearity model [5], which is useful for Travelling Wave Tube Amplifiers (TWTA). On the other hand, HP A nonlinearity in OFDM systems was investigated in [3].

In the fact, nonlinear distortion can be compensated either at the transmitter side or the receiver side. For the former case, the signal to be transmitted is modified before the HPA, and among the well-known methods for this purpose are predistortion with NN (NN-PD) and PAPR reduction techniques. SISO predistortion methods aim to model the inverse of the HP A nonlinear response [3]. The NN-PD is placed before the HPA such that the cascade NN-

978-1-4673-5604-6112/$31.00 ©20 12 IEEE

PO-HPA produces a linearly amplified signal. On the other hand, in MIMO systems, multiple transmission paths are implemented in the same chipset. These implementations cause the crosstalk between those paths and it affects the signal quality and the predistorter performance [6]. For solved this problem, we propose to use a learning algorithm for each branch and therefore we implement a NN-PD in each branch also.

At receiver-side, we propose a compensating NN in frequency domain, which corrects the nonlinearity distortion accompanied with MMSE receiver. Indeed, we have placed the compensator after OFOM demodulator. The aim is to adapt the NN functionality with the FFT (Fast Fourrier Transform) output.

Results were supported with Matlab simulation of a VBLAST MIMO OFOM system under Rayleigh fading channel. The remainder of this paper is organized as follows: In section II the VBLAST MIMO OFDM system with nonlinear HP A and the MMSE receiver is introduced. In Section III, the architecture ofNN compensator, the NN-PD and NN in frequency domain is presented. Numerical results and discussion are presented in Section IV, while the conclusion is given in Section V.

'" .." o lJ C 'LU a 2: 2:

H

+ './ w � Y2� � � :;;

Figure 1. Non linear MMO OFDM system.

II. SYSTEM MODEL

The V-BLAST MIMO-OFDM system under consideration consists of one Base Station BS equipped with K = 2 transmits antennas and M = 2 receives antennas that result in a MXK MIMO OFOM system. Figure I shows a block diagram of the system. The frequency domain FO symbol Xk is assumed to contain the source information and to belong to a BPSK alphabet. In this work, the Fast Fourrier Transform (FFT) matrix of dimension Nc is denoted by V E rr.NcXNc. The transmitted signal xk(n), at time instant n is obtained by means of an

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Invers fast Fourrier Transform (IFFT) of the Xk(n), and is given by:

(1)

The multi carrier signal after passing the NL HP A consists of two terms [7]:

Where the first term xk(n) is the distortion-free discrete­time input signal vector in time domain of equation (1) and

Ko is the gain of the linear part. The second term dk(n) is the nonlinear distortion which is a function of the modulated

x k (n) and the P A transfer function 9 [. ] . Before reaching the M receiving antennas, the

transmitted signals are affected by the propagation channel

which can be modeled by a K propagation matrix H m,k' The recei ved signal of a nonlinear HP A is represented by:

K

Ym(n) = I Hm,k(n) (KOXk (n) + dk(n)) + nm(n), (3) k=l

where nm(n) is the Additive White Gaussian Noise

AWGN and Hm,k(n) is a NcXNc circulant Time Domain

TO channel matrix at time instant n, which is formed by the

channel response vector hm,k (n). The frequency domain expression by taking the FFT of the received signal is obtained of equation (3):

(4)

The received signal vector at each antenna on subcarrier can be written as:

Y(n, 0 = KoH(n, oxen, 0 + H(n, Od(n, 0 + n(n, 0, (5)

where H(n,O E [MXK is the channel transfer matrix, x(n,O is the transmitted signal, contains the NL distortion of each antenna on subcarrier i and n(n, 0 is the noise with

[n nH] = (J�l . The FD channel transfer matrix H(n, i) in equation (5) is given by [7]:

[ hll (n, 0 h12 (n, 0 ... hlL (n, 0 1 H(n, 0 = hz,l;(n,O hziz (n, 0 ... hZ,L (�' 0 ,

hM,l(n,O hM,z(n, i) hM, L(n, i)

(6)

where hm,k(n, i) denotes the channel response on

subcarrier i at time n between antenna element m of the BS and antenna element k.

A. Nonlinear HPA model

For the HPA model, we have chosen Saleh's well­established TWT A model. In this model, AMI AM

(amplitude modulation to amplitude modulation) and AMIPM (amplitude modulation to phase modulation) conversion of the TWTA can be represented as follow:

A(I ( ) I) aalx(n)1 (I ( ) I)

_ aplx(n)12 x n = 1+j5alx(n)12 ' P x n

- 1+j5plx(n)l2' (7)

where x(n) is the input modulus of the TWTA, aa and

f3a are the parameters to decide the non linear level, and ap and f3p are phase displacements. The values for these

parameters are assumed to be: aa = 2, f3a = 1, ap =

4 and f3p = 9 which assumed typical TWTA employed in

real systems [3]. The output of the TWTA can be represented as:

zen) = g[x(n)] = A(lx(n)l)expU(arg[lx(n)I] + P[lx(n)I])} , (8)

where A(.) and P(') denote the HPA amplitude conversion (AMI AM) and phase conversion (AM/PM), respectively.

We need a criterion to show how much power back-off is needed for optimum power efficiency. In the simulations, we define the input back-off (IBO) as:

IBO = l O loglO ( A5 ), Pm

(9)

where Ao is the maximum output amplitude and Pin is the input average power.

B. Minimum Mean Square Error (MMSE)

The signals transmitted by different users on subcarrier i can be estimated with the aid of a suitable linear combiner W E [MXK. The Minimum Mean Square Error

(MMSE) approach tries to find a coefficient W(MMSE) which

minimizes the criterion, E {[W(MMSE)Y - X][W(MMSE)Y­

X]H} [8, 9], solving:

where, (J� is the Noise variance and I is the matrix identity. After equalization the approximated signal can be expressed as:

X(n,O = W(MMSE)Y(n, 0· (11 )

III. EQUALIZATION OF NONLINEAR HPA DISTORTIONS IN FREQUENCY DOMAIN

It was shown that neural networks (NNs), which are nonlinear in their nature in addition to their very developed aspect, could be a good tool to compensate for nonlinearity.

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More recently, the NN was proposed as a compensator technique for communications systems r3, 101-

These compensations are realized with Multi Layer Perceptron (MLP) neural networks, to linearize stationary HP A, associated with learning algorithm. In a digital communication system, the compensator can be inserted on various locations.

In this work, we study the performance of neural network compensator at the transmitter side (NN-PD) and at the receiver side (in the frequency domain).

A. Architecture of the applied neural network

I

Q

�nput layer Output layer

H i dden layers

Figure 2. Multilayer neural network: this network has 2 layers, I input signal, 4 neurons in the first layer, 3 neurons in the second layer and 2

neurons in the output layer (I output signal).

A multilayer neural network (see figure 2) is composed of neurons connected to each other. Connections are allowed from input to the hidden layers and them to output layer. It is well known that each neuron in the network is composed of a linear combiner and an activation function which gives the neuron output of NN-PD and NN in frequency domain:

(12)

(13)

where Wl,k is the weight which connects the k neuron in

layer I ,b1k is the biais terms, Xij,m and Xij,mdenotes

respectively the input signal to the neuron of NN-PD and NN in frequency domain from the j - th component of the i - th input vector of the m - th receive antenna. In general, the activation function is a nonlinear function (sigmoid function or hyperbolic tangent) [3]. For our NN, activation functions of the hidden layer are nonlinear and given by:

eX_e-X [(x) = tanh(x) = ---x--=x . e +e (14)

The activation function of the output neurons is linear in our implementation.

B. Training and generalization

The basic idea proposed is to identify the TWT inverse transfer function with a feed-forward neural network. Therefore, by using this structure (see Figure 3 (a-b)), we aim at obtaining direct estimation of the amplitude and phase nonlinearities.

Training Gen eralizat i o n

Tr,aining Genera li.zation

�L_N_N_-M_MS---.:-E-----,�

.. y

Figure 3. Block diagram for training and generalization of the PD (a) and the compensation in Frequency Domain (b) with HPA.

Training

Using the structure illustrated in figure 3(a-b) we aim to identify the HP A inverse transfer functions, the complex envelope signals are differentiated and the error sent to (learning algorithm) bloc reacts on coefficients of NNl.

Generalization (see Figure 3 (a-b))

Coefficients of the NNl are recopied on NN-PD and NN that achieves the equalization. The training procedure can be done off line because the HP A is stationary.

C. Predistortion ofNL HPA

Figures 3(a) and 4 show the detailed scheme of predistortion system.

The NN-PD forms an adaptive nonlinear device that its response can approximate the inverse transfer functions of the HPA. Then, in MIMO system we can't ignore the crosstalk effects, which are caused by the implementation of many transmission/reception paths in the same chipset. This crosstalk affects the NNPD performance, that's why, a

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Crossover Neural network Predistorter (CONNPD) was proposed in [6] in order to eliminate HPA nonlinearity and crosstalk effects simultaneously,

In this work, we propose another method for solved the problem of crosstalk using a learning algorithm for each branch and therefore we implement a NN-PD in each branch also,

Figure 4. NN-PD for compensate the HPA nonlinearity in V -BLAST MIMO OFDM transmitter.

D, Compensation of NL HP A in Frequency Domain (FD)

M IMO Encoder

'" "'C

o U C

W o � ::;'

Training H

Training

Generalizat i on

H Generalizatio n

yYz � v, �

- --

Figure 5. Equalization with NN in frequency domain of HPA in V­BLAST MIMO OFDM system.

To compensate for the non-linearities at the receiver, the proposed system uses a neural network based MMSE receiver, as shown in Figures 3(b) and 5, Indeed, we have a problem when we manipulate a large number of sub-carrier, such as 64, the size of the neural network must also be 64, to solve this problem with numerical simulation, we have adapted the compensator to operate like in the case of single-carrier system, which means that it treats each input independently,

We propose in Figure 5 two different conditions (a and b) for equalizing the HPA nonlinearity, For case (a), we present a learning algorithm for each branch and in case (b),

we take for two branch a single learning algorithm,

IV. SIMULATION RESULTS AND DISCUSSION

In this section we present numerical results illustrating the neural network compensator in the case predistortion and in the case in frequency domain accompanied with MMSE which corrects successively at the transmitter and receiver level, the nonlinear distortions due to the HPA in a SISO OFOM and VBLAST MIMO OFOM systems running under a Rayleigh fading channel. The figures below show the comparison results for the Bit Error Rate (BER) performance of the system considered as linear (L ), nonlinear (NL), NN( 2, x, 2) represents a NN with hidden layer of x neurons, NN( 2, x -y , 2) represents NN with two hidden layers of x and y neurons, NN( 2, x -Y -Z, 2) represents NN with three hidden layers of x, y and Z neurons with (2 inputs and 2 outputs ), NN( 4 , x , 4 ), NN(4 ,x -y ,4 ) and NN(4 ,x -y -z ,4 ) with (4 inputs and 4 outputs ), respectively on the Rayleigh channel.

10-2 ::::::::::::.: :::::::::::;:: : , - -. . _ . . . . . . . - . . . . . . . _ . . _ . . . . . . . . . . _ .. _ . .•

. . . . . . . . . . . . . . . . . .

:: -- L MMSE : ....... NLMMSE

10" ., - .. - NN-PD(2,9,2) : - e- NN-PD(2,15,2)

. : - • - NN-PD(2,5-5,2)

10-5 - - - NN-PD(2,9-9,2)

.. - A- NN-PD(2,5-5-5,2) : -.- NN-PD(2,15-15-15,2 :.

10�L-----�----�------�----�------L-----� o 5 10 15

Eb/NO (dB) 20 25 30

Figure 6. BER of the SISO OFDM system with NN-PD versus SNR for Nc = 64 and IBO = 8dB,

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10-' ::::::::::::::: ....... . . . . . ... . . . . . . . - . . - . . , . ....... . . . . . ...

cr: -3 Mj10 , -LMMSE

: - NLMMSE .. - .. - NN-PD(2.9.2)

10-4 " _ • _ NN-PD(2.20.2) : - • - NN-PD(2.5-S.2)

•• --- NN-PD(2.15-15.2)

10-5 .. - ... - NN-PD(2.9-9-9.2) ..... : -.- NN-PD(2.15-15-1S.2::

10�L-----�-------L------�------L-----�------� o 10 15 20 25 30

Eb/NO (dB)

Figure 7. BER of the MIMO OFO M system with NN-PO versus SNR for Nc = 64, lBO = 8dB and Nt = Nr = 2.

--L _NL

10"". - e - NN(2.15.2) - • - NN(2,20.2) - A - NN(2.9--9.2) - • - NN(2,20-20,2)

10-', - *- NN(2.15-15-15,2 - .. - NN(2,20-20-20,2

10�0�----�-------

1�

0------�

1�5

------�20

�----�2�5------�30

Eb/NO (dS)

Figure 8. BER of the SISO OF O M system with NN in frequency domain versus SNR for Nc = 64 and lBO = 8dB.

For the reason to study the performance of the neural network compensator we have treated many structures and we have conserved those which gives the best results.

Figures 6 and 7 present the tests performed on the NN­PO structure for several compensators, successively on SISO and MIMO OFOM systems. All neural predistorters can reduce the BER compared on to the one without any predi storter.

1if �,�,�, �,,�,,�,,�,,",,� .. ����������====�====� --L

10-' , " "''''''''" . : :: :::::::: � . . . . . . . . - _ . . _ .

:::: ::::::::; : .. . . . .. . . . - _ . . .. . . . . .. . . . - _··"r . . . . . . . _ _ ...• .... _ _ ... . ... . .. . _ . .

. . . . . . . . - _ . ... . . . • . • • . • • • - - " " r '

. . . .. . . . _ _ . . . . . . . . . . . . . . .. . . . _ _ . . . . . . . . . . .

. . . . - - : : � . . . . - . : : : : : : :: : :: : ::: : : : : ::: � � � >: : : : :: . . . :: . . . ..

10� �----�------�-------L------�------�----� a 5 10 15 20 25 30

EblNO (dB)

Figure 9. Comparison of two conditions (a and b) of MIMO OF O M systems for Nc = 64, lBO = 8dB and Nt = Nr = 2.

_NL - • - NNb(4,15.4)

10-'.

- . - NNb(4,20.4) - A- NNb(4,15-15,4) - • - NNb(4,20-20.4) - *- NNb(4,15-15-15.4).

·

- .. - NNb(4 ,20-20- 20,4) 10-'

" :'

10�L-----�-------L------�----��----��----� o 25 30

Figure 10. BER of the MIMO OFO M system with NN in frequency domain versus SNR for Nc = 64, lBO = 8dB and Nt = Nr = 2.

In figure 6, we can say that the one that gets the best performances in the NN-PO is the NN -PD(2,9 -9 ,2) .

In figure 7, simulation results shown that NN­PD(2,5 - 5,2) presents the best performance in comparison to other structures.

Figure 8 shows the best performance of each neural network compensator structure on SISO OFOM for an IBO = 8 dB. For an SNR more than 20 dB, we proove that

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'" -3 � 10

10-4 - e - NNb(lbo=6dB) ,,,:,,. - • - NNb(ibo=7dB) : .. - * - NNb(lbo=8dB)

:

__ NL(ibo=6dB)

10-' ........ NL(ibo=7dB) ",:",,, ,, ,, ;" ..

�NL(ibo=8dB) ",.", " . " . ,' ", .. . . .. ... . . . .. , ............ . .. .. . .. . . . , . . . . . . . . . . . . . . , . . . . . . . . . . . . ". . . . . . . . . . . � . .

10 ... 0'--------':----1--':O----'-15=-----::'c-------:2L.5--- ---"30 EblNO (dB)

Figure 11. BER of the MIMO OFDM system with NN in frequency domain versus SNR for Nc = 64, Nt = Nr = 2 and diflerent values of

IBG.

the NN(2,20 - 20,2) gives good performance compared to other structures.

Figure 9 shows BER performance of neural network technique in V-BLAST MIMO OFOM system. We approve that the NN structure using a learning algorithm for two branchs (NNb(4 ,15 - 15,4 ) gives best performance compared to the NN structure using a learning algorithm for each branch (NNa(2,15 - 15,2).

Figure 10 shows the performance of several compensators in MIMO OFOM system for an IBO =

8 dB. From this results we can say that the NNb(4 ,15-15 - 15,4 ) presents a good performance in comparison with other structures, because for a BER roughly equals 10-3,we note a gain of IdB compared to NNb(4 ,15-15,4 ), 2.5dB to NNb(4 ,15,4 ) and 3.5dB to NNb(4 ,20,4 ).

We note from results shown in Figure I I , the effect of 180 on compensator performance in the case N N b ( 4 ,15 -15 - 15,4 ). It indicates that SNR imposes a great impact on the BER performance when the 180 varies. Indeed, for a BER equal 10-3, we note a gain of 3.5 dB to NN with IBO = 8 dB, 1.5 dB to NN with IBO = 7 dB and 1 dB to NN with IBO = 6 dB

V. CONCLUSION

In this paper, we have presented the performance of neural network compensators regardless in the case of predistortion or at receiver level in frequency domain, accompanied with MMSE, which correct the nonlinearity distortion due to HPA. We have studied successfully the

predistortion in MIMO OFOM systems avoiding the problem of crosstalk and we have adapted at receiver, the NN functionality with the FFT output.

In this fact, NN compensator has been evaluated in term of BER performance in a complete V-BLAST MIMO OFOM system, using 64 carriers, a BPS\( modulation, running under in Rayleigh fading channel. According to simulation results, we approve that the NN -PD(2,5 -5,2) in MTMO OFDM has presented the best performance

comparing to other structures, and the proposed NNb(4 ,15 - 15 - 15,4 ) in the case of compensation in frequency domain allows a gain of 4 dB compared to the NL for a fixed BER of 10-4.

Finally, we have simulated the effect of 180 on the compensator performance in MIMO OFOM whose we have demonstrated that when the value of 180 increases the compensation of nonlinearities improves.

REFERENCES

[1] Sami Ahmed Haider, Khalida Noori, "System Design end Performance Analysis of Layered MIMO OFDM communication Systems", Published in the IEEE Feb. 15-18,2009 ICACT 2009.

[2] Byung Moo Lee, Rui J.P. de Figueiredo, "MIMO-OFDM PAPR redaction by selected mapping using side information power allocation", ELSEIVER, Digital Signal Processing 20 (2010) 462-471.

[3] Rafik Zayani, Ridha Bouallegue, Daniel Roviras, "Adaptative Predistortions Based on Neural Networks Associated with Levenberg-Marquardt Algorithm for Satellite Down Links", EURASIP Journal on Wireless Communications and Networking, volume 2008, Article ID 132729, 15 Pages.

[4] Jian Qi, Sonia Aissa, "On the Eflect of Power Amplifier Nonlinearity on MIMO Transmit Diversity Systems", Published in the IEEE ICCC 2009 Proceedings.

[5] AAM. Saleh, "Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers", IEEE Trans. Commun. 29 (1981) 1715-1720.

[6] Rafik Zayani, Ridha Bouallegue, Daniel Roviras, "Predistorter for the Compensation of Crosstalk and Nonlinearity in MIMO OFDM Systems", Published in the IEEE PIMRC 2010, Istanbul, Turkey.

[7] Fernardo H. Gregorio, Stefan Werner, Juan E. Cousseau and Timo I. Laakso, "Iterative Channel Estimation for Multiuser OFDM Systems in the Presence of Power Amplifier Nonlinearities", Published in the IEEE, 2006 PIMRC'06.

[8] Pierre-Jean Bouvet, Vincent Le Nir, Maryline H'elard, Rodolphe Le Gouable, "Spatial Multiplexed Coded MC-CDMA With Iterative Receiver", IEEE 15th International Symposium on Personal, Indoor, and Mobile Radio Communications, proceedings, Barcelona, Spain, 5-8 September, 2004.

[9] Maha Dakhli, Rafik Zayani, Ridha Bouallegue, "On the effect Of High Power Amplifier Nolinearity on MIMO MC-CDMA Systems", IEEE Signal Processing Society, ISSPIT( Decembre 20 I 0).

[10] M. Ibnkahla, "Adaptative Predistortion Techniqueq for Satellite Channel Equalization", in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP'OO), Istanbul, Turkey, June 2000.

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