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| ARTICLE |
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| Year : 2011 | Volume
: 57
| Issue : 5 | Page : 478-486 |
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Practical Blind Approach for Joint Detection and Parameter Assessment in Direct Sequence Code Division Multiple Access Systems
Javad Afshar Jahanshahi, Mohammad Eslami, Seyed Ali Ghorashi
Cognitive Telecommunications Research Group, Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Shahid Beheshti University G.C., Evin, Tehran, Iran
| Date of Web Publication | 24-Nov-2011 |
Correspondence Address:
Javad Afshar Jahanshahi
Cognitive Telecommunications Research Group, Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Shahid Beheshti University G.C., Evin, Tehran
Iran
 DOI: 10.4103/0377-2063.90177
 Abstract | | |
In this paper, the blind code extraction of Direct Sequence Code - Division Multiple Access signals is considered based on independent component analysis (ICA), some propositions are defined in order to estimate the number of active users, to distinguish between correct and incorrect extracted codes, and also to determine the quality of detection along with the ICA-based blind detection procedure. These propositions are used to improve the performance of the ICA blind detection-based method. Principal component analysis and gaussian mixture model are employed as analyzing tools in proposed criteria. Experimental results show that the defined criteria are verified. Keywords: Active user enumeration, Blind detection, Code division multiple access, Gaussian mixture model, Independent component analysis, Principal component analysis
How to cite this article:
Jahanshahi JA, Eslami M, Ghorashi SA. Practical Blind Approach for Joint Detection and Parameter Assessment in Direct Sequence Code Division Multiple Access Systems. IETE J Res 2011;57:478-86 |
How to cite this URL:
Jahanshahi JA, Eslami M, Ghorashi SA. Practical Blind Approach for Joint Detection and Parameter Assessment in Direct Sequence Code Division Multiple Access Systems. IETE J Res [serial online] 2011 [cited 2013 Jun 19];57:478-86. Available from: http://www.jr.ietejournals.org/text.asp?2011/57/5/478/90177 |
| 1. Introduction | |  |
In a Direct Sequence - Code Division Multiple Access (DS-CDMA) system, users share the same frequency bands and the same time slots, but they are separated in codes. Conventional multiuser detection algorithms are designed based on some priori information of active users [1] . In blind user detection algorithms, however, the receiver does not have sufficient information about the number of active users or codes, in order to detect their signals efficiently [2] . Most of the active users' enumeration algorithms proposed in literature use some information theory criteria [3],[4] and Eigen Value Decomposition [5] . Independent component analysis (ICA) is a statistical technique based on higher order statistics, whose goal is to represent a set of random variables as a linear transformation of statistically independent components [6] . ICA-based techniques assume that sources are Non-Gaussian and independent. Fast-ICA algorithm is applied for detection of DS-CDMA in a previous study [7] , but the convergence is not guaranteed. The RAKE-ICA proposed by a study [8] needs the information of multi-path delay time of the desired user, which is difficult to be estimated. Two types of receivers, RAKE-ICA and MMSE-ICA, are proposed in a study [9] , in which a Rayleigh fading channel is assumed. The MMSE-ICA normally requires training sequences, but one study [10] proposed a new blind multiuser detector that requires no training data sequences. ICA has been applied using SAND algorithm [11] and in addition, JADE and RADICAL ICA algorithms are applied for DS-CDMA detection [12] . In general, the mentioned methods in literature have not considered the two following items:
- Estimation of the number of active users, which can be useful for modifying the ICA algorithm
- The working states of the detection in respect with the number of users, number of observed signals and signal to noise ratio (SNR). In this paper, we show how these states which are labeled as excellent, good, and bad can improve the performance of ICA algorithm.
In the present research, some propositions are defined to estimate the number of active users and determine the working situation along with the ICA-based blind detection. The paper is organized as follows. In section II, we present the background and system model in details. Section III explains several propositions for active user enumeration, determination of the blind detection situations, and removing undesired detected signals based on some discovered facts. In section IV, simulation results and discussion are presented. Finally, in section V, we provide our concluding remarks.
| 2. Background and System Model | | |
2.1 Independent Component Analysis
The ICA is a statistical technique in order to represent a set of random variables as a linear transformation of statistically independent component variables [6] . The main application of ICA is in the Blind Source Separation problem. In other words, ICA involves the task of computing the matrix projection of a set of components onto another set of so called independent component.
Suppose, X=AS where S is an n - dimensional random vector whose components are mutually independent, A is the constant mixing matrix by size m × n and X is an m - dimensional product vector. ICA goal is to estimate matrix A and vector S from vector X that is called one observed vector. The result of the separation process is the demixing matrix W which can be used to obtain the estimated statistical independent sources, Ŝ , from the mixtures: Ŝ =WX. This process is described by (1) and a schematic illustration of the mathematical model is shown in [Figure 1].
 | Figure 1: Schematic illustration of the mathematical model used to perform ICA decomposition.
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With the matrix W, the components Ŝ i and Ŝ j of any i ≠ j are observed as uncorrelated. The fundamental restriction of ICA is that independent components must be non-Gaussian for ICA to be possible. In real applications, observed matrix commonly contains several observed vectors, and so, estimated matrix Ŝ will contain vectors.
Some preprocessing is useful before attempting to estimate W. The observed signals should be centered by subtracting their mean values E{X}. Then, they are whitened, which means they are linearly transformed so that the components become uncorrelated with unit variance. Whitening can be performed by using Eigen values.
Fast ICA is a practical ICA algorithm that is a fixed-point iteration scheme for finding a maximum of the non-Gaussianity and is purposed in a study [13] . The algorithm estimates just one of the independent components (ICA) once. To estimate several independent components, the Fast ICA needs to run several times. The Matlab toolbox for Fast ICA is available on the web [14] .
2.2 System Model
The continuous DS-CDMA received signal in multipath fading channel can be modeled as:
Where, d km is the mth symbol of the kth user, a lm is fading factor of the lth path corresponding to the mth symbol, which may vary from symbol to symbol, M is the number of observation symbols, L is the number of independent transmission paths for each of the K simultaneous users, T is the symbol duration, the d l denotes the delay of the lth path, which is assumed to be constant during the observation interval of M symbol bits, and the last term n(t) denotes the additive white Gaussian noise with zero mean and unit variance. The chip sequence length (i.e., processing gain) is G s=T/T c, where T c is chip duration, and it is assumed that T c=1 for the sake of simplicity [6] .
The continuous received signal is sampled at chip rate. Here, it is assumed that both code timing and channel estimation are already done. According to (2), the discrete data samples are described as:
The observed matrix R with the size of G S × N then consists of N vectors of y m, where R = [y m , y m+1 , ... , y m], and
where m'>m, N = m'-m and m', m ∈ {0, 1, ... , M-1}. Each column of matrix R is an observed vector or signal for training the ICA algorithm. In typical ICA, N should be much greater than G s. After code timing and channel estimation, the observed matrix can be modeled as:
where R is the G S×N observed matrix, G=[c 1 c 2 ... c k] is the mixing matrix by the size of G S×K, and its vectors are c i, for i = 1, ... , K (c i is the related code for the ith user), B = [d m, d m+1, ... d m] is the symbol matrix with the size of K×N and contains N separate vectors of d m, where d m =[d 1m a 1m , ..., d km a km ] T and finally No=[N m , N m+1 , ..., N m ] T, is the corresponding noise matrix by the size of G S×N.
The AWGN noise, No, can be treated as one of the independent components of matrix R; then, (5) can be rewritten as R=G.B. Therefore, comparing this recent equation with ICA model in (1), we can calculate Ĝ and  as the decomposition results of matrix R by using ICA algorithm. Extracted codes then would be the columns of extracted matrix Ĝ, and corresponding symbols would be the rows of matrix  .
| 3. Proposed Method | | |
3.1 Shape Pattern of Eigen-values
In order to explain the proposed method procedure in this paper, some shape patterns of Eigen-values in CDMA codes should be considered. In order to compute Eigen-values (λn ) of codes, Principal Component Analysis (PCA) [6] is used as a tool. Suppose matrix  is defined as follows:
Each G rj , j=1, ... , L is a matrix similar to G, in which users' codes c i , i=1, ... , K are randomly located in columns (e.g., G rj , j= [c k-1 c 3 ... , c 3 ). As mentioned before, Eigen-values (λn , n = 1, 2, ... , Gs × L) of matrix  are obtained using PCA algorithm. A typical sorted set of Eigen-values for Hadamard Code by length 64 is shown in [Figure 2], where L = 4 and K = 30. [Figure 3] shows two examples of possible patterns for the sorted Eigen-values of Gold codes by the length of 63. Two different behaviors can be inferred by comparing [Figure 2] and [Figure 3]. | Figure 2: Sorted Eigen values for Hadamard codes with the length of 64 and L = 4, K = 30.
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 | Figure 3: Two possible sets of sorted Eigen values for Gold codes with the length of 63 and L = 4, K = 30.
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In regular behavior [Figure 2] and [Figure 3]a, the number of nonzero Eigen-values are close to the number of users' codes, K = 30 whereas in irregular behavior [Figure 3]b, this is not true. Since in this paper, the proposed method for joint blind detection and active user enumeration is based on Eigen-values, the regular and irregular behavior should be distinguished.
In order to recognize irregular behavior, it is required to define a new criterion. The criterion is based on the pattern of the magnitudes of sorted Eigen-values. In irregular behavior, some of Eigen-values have much more significant amplitudes than others. For example, as it can be seen in [Figure 3]b, the maximum amount of Eigen-values is about six times larger than the second level. Based on the empirical results, the criterion can be defined as follows. Assume that λmax and λave and are the maximum and the average of nonzero Eigen-values, respectively.
Proposition 1: Irregular behaviors occur when the maximum of Eigen-values λmax is larger than the twice of λave, i.e., λmax > 2λave. Therefore, the regular behavior occurs when λmax < 2λave obviously.
3.2 Active User Enumeration and Blind Detection's Situations
According to the model described in the section II, three parameters, i.e., SNR of received signals, the number of active users K, and the number of observed signal vectors N in matrix R, are very effective in blind multi-user detection. Three detection quality states are possible (blind detection's situations) in respect with the relations between these parameters:
Situation 1: ICA cannot merge to any users' codes correctly. Then, extracted matrix Ĝ contains incorrect codes and therefore wrong extracted symbols would be in  .
Situation 2: ICA can converge to part of users' codes.
Situation 3: ICA converges completely and all users' codes are extracted correctly.
Note that in all of the above cases, the number of extracted codes may be larger than the real number of active users. Therefore, it will be helpful if the number of active users can be estimated, before multi-user detection process. In the following subsections, three propositions are proposed in order to do active user enumeration, estimate the correct codes, and recognize the blind detection's situations. It should be noted that the current discussion is validated for the regular behaviors which are defined in the proposition 1.
Suppose that the received signal is split into L segments, and each segment contains N ×G s samples in a noisy condition and Ĝ j is the extracted code matrix by ICA algorithm from the jth segment. Now consider matrix Ĝ as follows:
The Eigen-values of matrix Ĝ are computed by PCA algorithm [6] . [Figure 4], [Figure 5] and [Figure 6] show three examples of sorted Eigen-values of extracted matrix Ĝ for Gold and Hadamard codes by the length of 63 and 64, respectively. SNR is set to 5 dB, L = 4, and K = 30 for three different choices of the observed signal numbers: N = 1000, 4000, and 7000, respectively. | Figure 4: Achieved sorted Eigen values for L = 4, N = 1000, K = 30. (a) Gold sequence with GS = 63 (b) Hadamard code with GS = 64.
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 | Figure 5: Achieved sorted Eigen values for L = 4, N = 4000, K = 30 (a) Gold sequence with GS = 63 (b) Hadamard code with GS = 64.
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 | Figure 6: Achieved sorted Eigen values for L = 4, N = 7000, K = 30 (a) Gold sequence with GS = 63 (b) Hadamard code with GS = 64.
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The average number of extracted codes in matrix Ĝ j is 59. Although by setting the number of observed signals N = 7000, all of the users' codes Ĝ j are extracted, by N = 4000, just 28 and 8 users' codes are discovered correctly in Gold and Hadamard codes, respectively. In N = 1000, none of the users' codes are discovered and all of the extracted codes in matrix Ĝ j are incorrect. Therefore, N = 7000, 4000 and 1000 are related to situation 3, 2, and 1, respectively. By considering [Figure 4], [Figure 5], and [Figure 6], the following three features can be observed:
- There is a severe drop between the Kth and the K+1 th Eigen-values. The amount of drop is comparable with the value of significant Eigen-values
- There are two significant clusters around the drop. In each cluster, there is a uniform slope among sorted Eigen-values
- The amount of drop between clusters is dependent of N. On the other hand, if the drops amount are labeled as d, then we have d N = 7000 > d N = 4000 > d N = 1000 or we may say: d situation 3 > d situation 2 > d situation 1.
The above observations are verified for plenty of different codes' length and number of observed signals N. In order to summarize the above features, a simple formula is defined by using an auxiliary variable γi as follows:
[Figure 7], [Figure 8] and [Figure 9] show the γi vector of the Eigen-values of the [Figure 4], [Figure 5], and [Figure 6], respectively. The number of active users is located in the notch (V-shaped hill) of the vector, γi . This fact can be used in order to active user enumeration applications. The height of notch (HON) is related to the quality states or situations (in a sufficient SNR). On the other hand, in situation 3 [Figure 9], the HON is larger than that in situation 2 [Figure 8] and the same has happened between situation 2 and situation 1 [Figure 7]. | Figure 7: Achieved auxiliary variables γi, L = 4, N = 1000, K = 30 (a) Gold sequence with GS = 63 (b) Hadamard code with GS = 64.
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 | Figure 8: Achieved auxiliary variables γi, L = 4, N = 4000, K = 30 (a) Gold sequence with GS = 63 (b) Hadamard code with GS = 64.
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 | Figure 9: Achieved auxiliary variables γi, L = 4, N = 7000, K = 30 (a) Gold sequence by GS = 63 (b) Hadamard code by GS = 64.
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Proposition 2 - Active User Enumeration:
Suppose that 
is an auxiliary vector where γi , i = 1, 2, ..., Gs × L are the sorted Eigen-values of matrix, Ĝ . If  is the location of the first notch in γi vector, the number of active users K is one of these values 
3.3 Estimation of Correct Codes and Recognition of the Blind Detection's Situation
In this subsection, a novel criterion based on the extracted codes distribution is proposed in order to check out and verify the correctness of the extracted codes. Therefore, along with ICA-based blind detection procedure, the reliable symbols can be selected in the rows of matrix  . according to the correct codes only and the redundant symbols are ignored. Finally, based on the proposed criterion, a method for recognizing the situation in a Ĝ j will be described.
In common CDMA codes such as Gold and Hadamard ones, code samples have only -1 and +1 values and also the probability of their usage is close to 0.5. On the other hand, if we find the histogram of samples of a code sequence with the length of U, there will be just two impulses in +1 and -1 with magnitudes of U/2. This means that correct extracted codes in noise-free conditions should satisfy the above mentioned simple facts similarly. Since samples in extracted codes by ICA algorithm are not integers, we can propose that the distribution of samples in each correct extracted code should have two Gaussian distribution around +1 and -1 with equal variances. [Figure 10]a and b show a typical histogram of a correct extracted code and incorrect extracted code, respectively (when SNR is set to 5 dB and code length is 64). | Figure 10: Histogram of an extracted code (a) correct code (b) incorrect code.
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It means that an extracted code is incorrect if it does not satisfy the above criterion. Proposed criterion for determining the correctness of the extracted codes is based on gaussian mixture model (GMM) [15] . GMM algorithm tries to model an unknown distribution by superposition of some Gaussian distributions and fit their parameters such as means and variances. We have used GMM as a tool to model the distribution of symbols in extracted codes.
Proposition 3: Check out the correctness of extracted codes:
For a selected Ĝ j , the distribution of the correct extracted code samples should be a combination of two Gaussians distributions with equal variances around +1 and -1. In order to check the distribution of extracted codes, GMM clustering is used.
Proposition 4: Recognize the situation of the detection system:
Suppose that P is the number of extracted codes that satisfy the proposition 3 and therefore they are correct. The type of the quality state of detection system can be recognized as follows:
3.4 Overall Algorithm
According to the defined propositions, the overall joint blind detection and active user enumeration is summarized as follows:
- Choose L×N×G S samples of the received signal, split it in L segments, and reshape each segment to obtain observed matrix R j , j=1, ..., L
- Apply ICA algorithm and compute extracted matrixes Ĝ j and
 for each R j , j=1, ..., L - Generate matrix
 - Apply PCA algorithm on matrix
 to find Eigen-values - Sort the Eigen-values in respect with their values λi , i = {1, 2, ... , L×G S}
- Compute
 - Determine the number of active users based on proposition 2
- Select one of Ĝ j , j = 1, 2, ... , L
- Check each extracted code by using proposition 3 and find P and index of correct codes
- Determine the state of blind detection based on proposition 4
- If this situation is desired, just use correct extracted codes and their corresponding detected symbols. Otherwise, go to step 1 with larger number of observed signals, N
| 4. Simulation Results | | |
In this section, some of the experimental results are reported in order to show the reliability of the proposed algorithm and clarify the mentioned facts. In all simulation results, three different kinds of Gold sequence codes with the lengths of 63 are used which these codes manner like the regular behavior (proposition 1). It should be noted that since ICA algorithm is a statistical method, the reported results are the average of ten times simulations for each kind of code and then, the results themselves are averaged on codes to achieve final reported values.
Some of the experimentally achieved user enumeration results  with different user numbers (K = 10, 25 and 41), SNRs (SNR = 2, 5 and 10) and number of observed signals (N = 500, 2000 and 7000) are categorized and reported in [Table 1].
It can be seen from [Table 1] that the Proposition 2 always happened regardless of the observed signal's number and SNR values. It will be shown that the state of the situation is determined by the relations between K, SNR, and N in the rest of paper. Moreover, [Table 1] includes all of the defined situations and implies that the proposition 2 is acceptable in all situations.
[Figure 11] shows the HoN in  location in respect with the variations of SNR and the number of observed signals, N. The curves of [Figure 11]a show the HONs vs N for three different number of Users, K = 10, 25 and 41 in a fixed SNR = 5 dB. It can be seen that the height on notch decreases for a greater number of users and also by increasing the number of observed signals, the HON increases.  | Figure 11: Height of notch (HoN) in K location in respect with the variations of (a) N (b) SNR.
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As mentioned before, the HONs increases when the performance of ICA algorithm is better. Therefore, two facts can be resulted from [Figure 11]a. First, for a fixed number of users, K, the performance of ICA improves by increasing the number of observed signals. Second, larger number of users, K, needs more number of observed signals, N, for a desirable performance. Similar facts are also seen in [Figure 11]b between SNR and the Number of users for a fixed number of signals, N = 4000. In addition, it can be inferred that the number of observed signals N = 4000 is suitable for SNR ≥ 32 dB, SNR ≥ 30 dB, and SNR ≥ 26 dB, when K=41, 25, and 10, respectively, and yields to the situation 3 (typical results show that complete convergence of ICA occur when HoN ≥0.96). These estimated SNR values for situation 3 will be mentioned and verified again in the following experimental results.
P/ vs SNR and N are shown in [Figure 12]a and b, respectively. P denotes the Number of extracted codes which satisfies GMM criteria in proposition 3 and therefore are labeled as the correct extracted codes.  is the estimated number of active users based on proposition 2.
According to the proposition 4, the situation 3 occurs when P/ ≥ 1.
As expected, in [Figure 12]a, the blind detection system with N=7000 enters to the situation 3 in lower SNRs than N=4000 for both K = 10 and K = 25 cases. For example, the curves of {K = 10, N = 7000} and {K = 10, N = 4000} intersect the line P/ =1 in SNR = 6 and 26, respectively. It is noticeable to mention that the condition {K = 10, N = 4000} have also great performance in SNR ≥ 26 in [Figure 11]b. The same can be distinguished in [Figure 12]b in respect with the number of observed signals. For example, {K = 25, SNR = 40} meets the situation 3 in N = 3000 while {K = 25, SNR = 20} merged to the situation 3 in larger number of observed signals N such as N = 6000 and it is caused by SNR considerations. According to the proposition 4, Situations 1, 2, and 3 are sketched in [Figure 12] by stripping regions for K = 25 in respect with two following conditions:
- SNR variations, while number of observed signals is fixed, N = 4000 [Figure 12]a
- N variations, while SNR is fixed, SNR = 20 [Figure 12]b.
The P and H ave (the average number of extracted codes in matrix Ĝ j) curves vs SNR variations for two different number of users K = 10 and 25 in the fixed number of observed signals N = 4000 are shown in [Figure 13].  | Figure 13: P and Have curves vs SNR variations and efficiency plot for two different number of users K = 10 and 25 in the fixed number of observed signals N = 4000.
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The following properties can be inferred from [Figure 13]:
- In high SNRs, such as SNR >38, the number of observed signals N = 4000 is sufficient for complete convergence of ICA algorithm, so that there is no redundant extracted code. On the other hand, all of the extracted codes satisfy the GMM criterion and are exactly users' codes.
- However, in lower SNRs, there is no clear behavior for H ave, by increasing SNRs, the H ave, and P curves merge to each other.
- The efficiency of the defined criterion in proposition 3 with the GMM method is the gap between H ave curve and P curve. For example, the efficiency for K = 25 and SNR = 12 is depicted in [Figure 13], where the average number of extracted codes is 53 and the number of acceptable codes through proposition 3 is just 8. It means that there is 45 redundant and incorrect extracted codes and proposed criterion in proposition 3 causes to remove these wrong codes.
As mentioned before, blind detection system for K = 10 and N = 4000 works in situation 3 for SNR > 26. In [Figure 13], it can be seen that although the system with SNR = 26 works in situation 3 and all of the users' codes are extracted correctly, there are still some redundant and incorrect codes which proposition 3 will be removed them.
| 5. Conclusion | | |
Enumeration of active users is very helpful in blind detection methods such as ICA-based multi-user detection algorithms. In this paper, some propositions are defined to estimate the number of active users, to distinguish between correct and incorrect extracted codes, and to determine the quality of detection along with the ICA-based blind detection procedure. These propositions are used to improve the performance of the ICA blind detection-based method. Experimental results verified the defined criteria.
| 6. Acknowledgments | | |
Javad Afshar Jahanshahi and Mohammad Eslami would like to express special thanks of gratitude to Dr. Abouzar Eslami as well as Dr. Farid Atry who helped us in doing a lot of Research and we came to know about so many new things. We are really thankful to them.
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| Authors | | |
 Javad Afshar Jahanshahi received his B.Sc. and M.Sc. Degrees both in Electrical Engineering from the University of the Sistan & Baluchestan and Shahid Beheshti University, in 2006 and 2010, respectively. Now he is a member of Cognitive Telecommunications Research Group, Department of Electrical Engineering, Shahid Beheshti University G.C., at Tehran, Iran. His research interests are in Wireless Cognitive Radio, Wireless Communications, including Compressive Sensing PSD Map Construction, Cognitive Radio Spectrum Sensing.
 Mohammad Eslami is currently pursuing PhD. program in Communication Engineering in Shahid Beheshti University G.C., at Tehran, Iran. He is a member of Cognitive Telecommunication Research Group, Department of Electrical Engineering, Shahid Beheshti University G.C. His focus is on mathematical image and Video processing, Stereo and Multi view Vision systems. His research interests includes: Mathematical Image & Video Processing, Cognitive Radio & Spectrum Sensing, Stereo & Multi-view Vision.
 Seyed Ali Ghorashi received his B.Sc. and M.Sc. degrees in Electrical Enginering from the University of Tehran, Iran, in 1992 and 1995, respectively. Then, he joined SANA Pro Inc., where he worked on modelling and simulation of OFDM based wireless LAN systems and interference cancellation methods in W-CDMA systems. Since 2000, he worked as a research associate at King's College London on "capacity enhancement methods in multi-layer W-CDMA systems" sponsored by Mobile VCE. In 2003, He received his PhD at King's College and since then he worked at Kings College as a research fellow. In 2006 he joined Samsung Electronics (UK) Ltd as a senior researcher and now he is a faculty member of Cognitive Telecommunication Research Group, Department of Electrical Engineering, Shahid Beheshti University G.C., at Tehran, Iran, working on wireless communications.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13]
[Table 1]
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