FFT/DWT/DCT OFDM CHANNEL ESTIMATION USING EM ALGORITHM IN THE PRESENCE OF CHAOTIC INTERLEAVING

Zahraa Abd El-Hamid1 , A. El-Henawy3 , F. Abd El-samie2 and H. El-Shenawy1, Moataz Samir

1Faculty of Engineering, El-Shorouk Academy, Cairo, Egypt Eng_Zahrra2@yahoo.com

2Faculty of Engineering, Menoufia University, Cairo, Egypt Fathi_Sayed@yahoo.com

3Faculty of Engineering, Ain Shams University, Cairo, Egypt

Abstract— This paper presents a new interleaving scheme for efficient data transmission with Orthogonal Frequency Division Multiplexing (OFDM) over fading channels. This approach is based on the chaotic Baker map. The binary data is interleaved with the proposed approach prior to the modulation step. In addition to improve the system performance in fading channel, the proposed chaotic interleaving approach adds a degree of encryption to the transmitted data. The performance of the proposed approach is tested on the conventional Fast Fourier Transform OFDM (FFT-OFDM), Discrete Wavelet Transform OFDM (DWT-OFDM), and Discrete Cosine Transform OFDM (DCT-OFDM) with and without chaotic interleaving. Expectation–Maximization (EM) algorithm is also proposed to efficiently estimate the channel impulse response (CIR) of such a system operating on a channel with multipath fading. Starting from the Maximum Likelihood (ML) principle, we derive an iterative estimation algorithm based on the (EM) algorithm. This algorithm is capable of improving the channel estimate. In the initialization phase of this iterative algorithm, the initial channel estimate is based on the observation of the pilot carriers only. Then the EM algorithm updates the channel estimates until convergence is reached, the resulting bit error rate essentially coincides with the case of the perfectly known channel. By simulations, the efficiency of these algorithms can be investigated with simulation and the results of estimation will come to a comparison.

KEYWORDS OFDM, chaotic interleaving, DWT, DCT, and EM.

1. INTRODUCTION

In recent years, OFDM communication systems have received a considerable amount of interest due to their high spectral efficiency obtained by orthogonality, robustness to frequency selective fading, and simple equalizer implementation. The choice of individual sub-carriers is such that they are orthogonal to each other, which allows overlapping of sub-carriers, because the orthogonality ensures the separation of sub-carriers at the

receiver end. Due to these advantages, OFDM has become the widely-recognized modulation technique for high data rate communications over wireless links [1]. Many researchers have investigated the use of DWT-OFDM and compared it with FFT-OFDM. They found that DWT-OFDM has more advantages than FFT-OFDM due to its excellent orthogonality between subcarriers and spectral containment [2], [3]. Also in the literature, [4] proposed using a DCT rather than a FFT, to implement multicarrier modulation (MCM), because of the bandwidth advantage a DCT-based system can achieve. The main problem in the design of a communication system over a wireless link is to deal with multipath fading, which causes a significant degradation in terms of both the reliability of the link and the data rate [5]even though those systems exhibit efficient bandwidth utilization. In a multipath environment, the orthogonality is lost and thus Inter-symbol Interference (ISI) and Inter-Carrier Interference (ICI) occur, which decreases the system performance. The evolution of Multiple Input Multiple Output (MIMO) and beam forming techniques in wireless communications has opened new field of research for efficient transmission with MIMO techniques. The use of multiple antennas at both ends of the wireless link can improve the efficiency of data transmission. Beam forming can be used for interference reduction. These new technologies can achieve a high efficiency in data transmission systems, but with increased complexity and cost [6]. Another approach, which will be considered in this paper, is the data interleaving, which causes performance enhancement of data transmission systems. It is known that data transmission over wireless channels may face severe adverse conditions, especially, burst errors, thus interleaving becomes an effective means to combat error bursts by converting bursts of errors into random errors [7], [8], [9]. Chaotic maps can be used to build strong interleaves. Thus, in this paper, a chaotic interleaving scheme based on chaotic Baker map is introduced. The chaotic randomization generates permuted sequences with

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lower correlation between their samples. Due to the strong interleaving ability of chaotic maps, the proposed approach can combat the channel effects without a need for complicated coding schemes for error detection. Since the proposed interleaving algorithm didn’t add any additional processing on the data stream, so there is no additional complexity added to compare with. Moreover, the chaotic Baker map adds a degree of encryption to the transmitted signal, which increases the security of the transmission process.

Coherent OFDM detection requires knowledge of the channel impulse response, to this purpose, CIR have to be estimated accurately, we can apply the expectation maximization (EM) algorithm [10], which is an iterative algorithm that converges to the ML estimate. Since the channel estimation approaches can be divided into two categories: data-aided, e.g., [11], and non-data-aided (blind), e.g., [12] to initialize the algorithm. In this paper EM algorithm employing an OFDM preamble symbol to perform channel estimation iteratively, at each iteration the EM procedure updates the CIR estimate.

2. TRADITIONAL OFDM SYSTEM MODEL

2.1. FFT-based OFDM

An OFDM transceiver system is shown in figure 1. The inverse and forward transform blocks are concerned in more attentions since they can be FFT-based or DWT-based or DCT-based OFDM. In the transmitter, frames consisting of datablocks are transmitted. Each data frame is preceded by pilot blocks, based on which channel estimation is performed, as we will see in section 5, Each block

consists of symbols where belongs to an point constellation . The ( + ) data blocks are

buffered and converted, one at a time, to OFDM blocks by using an pointinverse Fast Fourier transform (IFFT),

The OFDM signal fed into cyclic prefix v insertion before transmission to minimize the inter-symbol interference at the receiver side [13], the output sequences are , . As long as the cyclic prefix duration is longer than that of the CIR, the ISI effect can be ignored. The transmitted signal propagate to the receiver through a wireless channel, hence The channel in OFDM is characterized by various obstacles and reflections which have a large influence on the signals during the propagation of radio waves from the base station to the mobile station. The radio waves transmitted from a base station radiate in all directions, and hence the receiver receives many reflected waves from various obstacles, diffracted waves, scattered waves and the direct waves from the base station. This wireless channel characterized

Figure 1: Block diagram of the (DWT/DCT/FFT)-OFDM

system model.

By where L is the channel length, the received signal can be expressed as

, where w is noise vector. 2.2. DWT-based OFDM

In the DWT-OFDM, the IDWT and DWT take place of the IFFT and the FFT respectively. Note that the low pass filter (LPF) contains coefficients namely as approximated coefficients, where the high pass filter (HPF) contains detailed coefficients or wavelet coefficients. Different wavelet families have different filter length and values of approximated and detailed coefficients. Both of these filters have to satisfy orthonormal bases in order to operate as wavelet transform [14]. The output of the (IDWT) can be represented as:

Where are the wavelet coefficients and is the wavelet function with compressed factor k times and shifted m times for each subcarrier. The output of (DWT) is[15],

2.3. DCT-based OFDM

Not only the complex exponential functions set is orthogonal basis that can be used to construct multicarrier signals, but also a set of co sinusoidal functions can be used as an orthogonal basis to implement the MCM scheme, this can be synthesized using (DCT) The output signal of a IDCT can be written as,

Where

Where N is the IDCT length (number of subcarriers), the output after (DCT) is,

3. CHAOTIC BAKER MAP

The channel errors caused by the mobile wireless channels are bursty in nature, interleaving is a must in mobile communication systems. Several interleaving schemes have been proposed. The simplest and most popular of such schemes is the block interleaver scheme, but this interleaver is not efficient with 2-D error bursts. As a result, there is a need for advanced interleavers for this task. The 2-D chaotic Baker map in its discretized version is a good candidate for this purpose. In the system model after (IFFT/IDWT/IDCT), the signal samples can be arranged into a square matrix then randomized using the chaotic Baker map. The chaotic interleaving is summarized in algorithm 1. Algorithm 1 Chaotic interleaving algorithm Chaotic interleaving of a square matrix can be summarized as follows: 1) An square matrix is divided into vertical rectangles of height and width . Such that

. 2) These vertical rectangles are stretched in the horizontal direction and contracted vertically to obtain a horizontal rectangle. 3) These rectangles are stacked as shown in figure 2, where the left one is put at the bottom and the right one at the top. 4) Each vertical rectangle is divided into boxes of dimensions containing exactly points. 5) Each of these boxes is mapped column by column into a row of data items as shown in Figure (2-b).

Figure 2: Chaotic interleaving. (a) Discretized Baker map.

(b) Randomization of an 8 × 8 block. The discretized Baker map is required to transfers each element in a square matrix into a new position according to this map. Let denote the discretized map, where the vector represents the secret key . This key ischosen such that each integer

divides and . This secret key is constant during the transmission process, which is predefined for transmitter and receiver pre-transmission.

Let . The data item at the indices (q, z), is moved to the indices [16],

Where , and . Figure 2 shows an example of chaotic interleaving of an (8×8) square matrix. The secret key,

. 4. PROPOSED SYSTEM MODEL

The block diagram of the proposed (FFT/DWT/DCT)-OFDM with chaotic interleaving is shown in figure 3. The conventional OFDM block is modified by adding an interleaving stage. Both the real in-phase and imaginary quadrature fields of the OFDM signal (the output of IFFT/IDWT/DCT) are interleaved using the chaotic Baker map. At the receiver, the process is reversed. Since the data is processed to the chaotic deinterleaving, the receiver is assumed to have an ideal knowledge of the secret key of the chaotic map. In order to apply our algorithm to estimate the CIR ( ), the received signal is reformulate as below

Where is Toeplitz matrix constructed from the transmitted data

, with length?

5. THE PROPOSED ESTIMATION ALGORITHM

In this section, we describe how the CIR, can be estimated at the receiver. Based on a few pilots, the Maximum Likelihood (ML) principle is used to estimate the CIR. Then use the expectation maximization (EM) algorithm [17], [18], the soft information resulting from the detector can be iteratively exploited to improve the estimation process. To reduce the complexity of the proposed algorithm, a sub-optimal estimation scheme is also introduced. 5.1. EM Channel Estimation

The proposed algorithm based on EM algorithm. The EM algorithm [18] is an iterative method to obtain a maximum likelihood (ML) estimate of the parameter based on an observation, The EM algorithm is an iterative two-step algorithm that consists of the expectation step and the maximization step. The EM algorithm iterates until the estimate converges. It is based on the

Figure 3: Block diagram of the proposed (FFT/DWT/DCT)-OFDM system model.

Concept of the so called missing (or unobserved data) a, such that, if the missing data were known, estimating would be easy. However, since we do not know the missing data, an iterative approach starting from an initial estimate of (say, (1)) is used. Consider r as the “incomplete“observation and z [r; a] as the “complete” observation. At iteration l, the EM algorithm consists of two steps:

1) E-step: given the current estimate (l) and

“incomplete“ observation r, we first take the expectation of the log-likelihood of the complete data z [r; a] with respect to the unknown data a:

2) M-step: we maximize with respect to to find a new estimate:

The EM algorithm iterates until the estimate has converged or a certain stopping criterion has been met. 5.2. Soft Information Aided Algorithm

Backing to our original problem, let us take as complete data z [r; a].In that case, since a and h are independent, so that the E-step becomes

Where

So that

Where and . Finally, the updated estimate of h is given by:

r

Assuming that data symbols are uncorrelated then can be approximated by [19]. Since a = [S], then = [ ], and can be found by replacing each entry with the corresponding expectation E[ ], from equations (1),(2) and (4) we found that,

E ], Where Fis the orthogonal function (IFFT/IDWT/IDCT), the detector computes the soft information of the detected symbol ,

Is approximately one for a point in the constellation diagram, and it equals approximately zero for other points. Accordingly, the detector hard output is very close to its soft output [20]. Initialization In order to start up the EM-algorithm, an initial estimate of the channel should be available ; such an estimate can be provided by some conventional blind or data-aided estimation technique. In this paper, we consider a data-aided estimation method to initialize the iterative algorithm, since a pilot symbols are inserted each OFDM frame. Ideally, the (ML) estimate of h is

Where log defined as the logarithm of the probability density function of the received sequence r given the CIR, h. We write

And

Where is the AWGN noise variance? But, the ML estimator in (17) is not easy to solve, as it requires the evaluation of (18) over all possible transmitted data. Data-Aided (DA) ML Estimator The summation in (18) can be avoided by ignoring the unknown transmitted symbols. In this case

and is obtained by considering only the contribution of pilot symbols and replacing all unknown data symbols in Y with zeros Now, the can be easily found in a closed form as

Reliable DA estimation algorithms can be achieved by increasing the number of pilot symbols. This results lead to a significant loss in terms of power and bandwidth, there is great interest in developing algorithms that are also able to exploit the soft information coming from the detector. In the previous section, we show how the EM algorithm uses this soft information to obtain the ML estimate of the CIR.

6. SIMULATION RESULTS

In this section, computer simulations are presented in order to verify performance of the channel estimation via EM algorithm for the proposed chaotic interleaving scheme and to compare it with normal OFDM. Three types of OFDM systems were simulated; FFT-OFDM, DCT-OFDM and DWT-OFDM with and without chaotic interleaver using MATLAB and the graphical results found show the Symbol Error Rate (SER) for these systems. The results presented show the variation of the SER with the for the systems. The channel model considered is Rayleigh fading channel. We assume that the number of sub-carrier equal 256, and the QPSK modulation is used, each transmitted frame consists of

= 7 data blocks and =1 pilot blocks. A chaotic map of size 64×64 is used with a secret key = [2, . . . . . . . . . . . . .,2] . Thechannel has a length L = 5and is modeled with independent components, each being a zero-mean complex Gaussian random variable with an exponential power delay profile [19].

Where is chosen such that the average energy per subcarrier is normalized to unity, to avoid inter-symbol interference, cyclic prefixes of lengths 6 is employed. The simulation results are shown in figures. 4 and 5. From these results, it is clear that chaotic interleaving achieves the best results for the three systems, specially, over fading channels. Assuming perfect channel estimation and using Minimum Mean Square Equalization (MMSE). As a function of the , where 12.5% of the sub-carriers in the pilot OFDM symbol are used for training, chaotic interleaving is considered at the three systems. Further, the SER for the case of perfect estimation is employed as a reference in all figures. As one can observe, we find that as the number of iterations increase as the performance improves. Where the data-aided estimation, first iteration yields to unacceptable SER performance. At third iteration, the EM estimator still result unacceptable SER performance. Strong improvements in the SER performance can be obtained at the fifth EM iteration. These can be explained as follows. In the First iteration, since the estimation process is based on few pilots, the reliability of the data symbols is low. However, when the number of iterations increases, the reliability of the data detection and estimation improves, as more information is used than in the data-aided case (i.e. the first iteration).

Figure 4: SER performance for FFT-OFDM, DCT-OFDM and DWT-OFDM over a Rayleigh channel

Figure 5: SER performance for FFT-OFDM, DCT-OFDM and DWT-OFDM with chaotic interleaving over a

Rayleigh channel Figures 6, 7, and 8 shows the (SER) performance for the proposed optimal EM-based algorithm

Figure 6: SER of the proposed FFT-OFDM with EM-based algorithm, using chaotic interleaving, and 12.5% of

the sub-carriers in the pilot OFDM symbol are used for training.

Figure 7: SER of the proposed DCT-OFDM with EM-based algorithm, using chaotic interleaving, and 12.5% of

the sub-carriers in the pilot OFDM symbol are used for training.

Figure 8: SER of the proposed DWT-OFDM with EM-based algorithm, using chaotic interleaving, and 12.5% of

the sub-carriers in the pilot OFDM symbol are used for training.

7. CONCLUSION

This paper presented a chaotic interleaving scheme to be used with OFDM systems to reduce the channel effect and estimate the CIR using EM algorithm. These schemes have been tested with FFT-OFDM, DCT-OFDM and DWT-OFDM. Simulation results have shown that the chaotic interleaving has enhanced the performance of all systems. In addition to performance Enhancement, it adds a degree of security to the communication systems. In the proposed estimation algorithm, the receiver iterates between estimation and data detection, with the exchange of soft information provided by the detector. A reduced complexity sub-optimal algorithm is also proposed which relies on hard decision at the detector. The SER performance of the proposed algorithms almost coincides with the SER of perfect known parameters case.

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