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In Wireless Communication, many techniques are emerging to give high data rate which is need of today. Some techniques are quite efficient while other possess some drawbacks. There are many modulation schemes adapted to send high data rate, which has become the need of today. Initially, to get high data rate, single carrier modulation schemes were used.
Single carrier modulation scheme was well implemented in the early stages of wireless communication. It allowed one carrier to send data over the channel. This provided many distortions of the data when it was send over a specific channel and even allowed the entire chunk of data to be collapse. The following figure depicts the distortion of data (Fig1.1).
Since the single carrier modulation technique was unable to cater heavy traffic, there was a need of high speed communication. One way to solve this heavy traffic flow was to send data in the form of multiple carriers spaced equally. This was a traditional FDM technique being used, where multiple carriers send data over the channel. These carriers are termed as sub channels. Therefore, there was an advent of orthogonal carriers needed to send data over a channel. Because of its rapid transmission of data, easy realizations in countering interference of multi path and high utilization of frequency spectrum, OFDM has got increasing great interest in the radio communication and has been widely used in many fields, such as Digital Video Broadcasting (DVB), Digital Audio Broadcasting (DVB),wireless local area networks (WLAN) and power line carrier modulation, etc. OFDM will be one of the most important researches for the next generation mobile communication system.
Multi carrier Modulation
Fig 1.2 Multi carrier modulation Block Diagram, where c1, c2, c3 are the carrier frequencies
Contribution of the project:
Our primary focus of the project is to estimate the channel using Singular Value Decomposition (SVD). The SVD is implemented over the OFDM system generated in Matlab. Numerous channels are taken and then we will estimate the given channel by a method termed as LS estimation. Moreover, we will derive the rank of the transmitter matrix and then put this rank to compute the MMSE of the signals, which means the how much the signal deviated from the original signal. At the end we will demonstrate graphs relating to BER versus SNR, and also the improvement SVD put on the data which was received.
Overview of the chapters:
Chapter 2 highlights on the basics of OFDM system, its advantages, disadvantages, block diagrams, mathematical flow, advantages and disadvantages
Chapter 3 illustrates about the basics of singular value decomposition, its types analytically used, types, and an example for the proof of SVD.
Chapter 4 focuses on the basis of channel estimation, different types of channel estimation, LS estimation, low rank approximation, etc.
Chapter 5 draws the results of the various channel estimation and simulation results.
2.1 Introduction to OFDM
Orthogonal frequency division multiplexing (OFDM) is a modulation/multiplexing as well as multicarrier transmission technique, where multiple sub-carriers are used to transmit data instead of using single wide band carrier. Sub-carriers are made by dividing whole frequency spectrum into many narrow bands, which occupy different frequencies. OFDM uses available frequency band efficiently by spacing channels very close. This is done by making all sub-carriers orthogonal to one another. Data is divided into many data streams which are transmitted using different sub-carriers. Each subcarrier carries different frequency. Each sub-carrier is modulated using Quadrature Amplitude Modulation (QAM) technique. OFDM technology is based on frequency division multiplexing i.e. (FDD), where multiple data streams are transmitted over broadband medium. Medium can be optical fiber, co-axial cable, twisted pair or radio spectrum. Within the frequency spectrum of medium, every data stream is modulated over adjacent multiple sub-carriers. Cable TV is a good example of such system where multiple audio video channels are transmitted over single co-axial or fiber optic cable. Available spectrum of bandwidth can be used so efficiently by using this technique. OFDM technique was developed in between early 1960s and 1970s .OFDM offers much efficiency and better performance over old traditional single carrier modulation technique. If there comes error in modulation, whole data cannot be affected. Because data has been divided into many data streams, where each sub-carrier transmit every data stream at different frequency. OFDM has solved the multipath problem due to its ability to transmit data in parallel way. When data is transmitted in parallel way, symbol period becomes long. Hence Intersymbol Interference (ISI) reduces. OFDM provides high data rate transmission due to its efficiency to transmit data at various sub-carriers operating at different frequencies. Even in the presence of noise, data can be transmitted fast. OFDM provides spectral efficiency and it is measured in units of bits per second per hertz. OFDM was quite difficult to implement on electronic circuit, but with the invention of semiconductor and enhanced technology of computer, OFDM technology arises and became the need of the modern world.
2.2 BASEBAND OFDM SYSTEM:
Cyclic Prefix removal
High speed data rate transmission is the main purpose of using OFDM. At transmitter side, High speed data is input to S/P conversion block. QAM modulation is done before passing data at input of IFFT. IFFT transforms data from frequency domain into time domain. Samples of modulated and multiplexed signals are collected at output of IFFT. IFFT performs multiplexing and modulation in single step. Data is transformed back into serial way by passing through P/S conversion block. Low pass filtering is done at end of transmission and at reception of data at receiver side. Data is converted into digital form through A/D convertor. Then, data is passed to FFT, to transform data into frequency domain i.e. in baseband signal form followed by S/P conversion block. Output of FFT is passed through demapping block. Finally, data is again reset in form of serial by passing data through P/S conversion block. That's the mechanism of OFDM system, where high speed data is transmitted and received through this way.
2.3 OFDM GENERATION:
In OFDM, we know that whole data is divided into multiple data streams and every sub-carrier is used to modulate unique data stream. OFDM signals are generated when data streams are input to IFFT and at output, IFFT produce signal samples which are added together. Filtration and D/A conversion results into baseband OFDM signal. Modulation which is used in OFDM is Quadrature Amplitude Modulation (QAM). Cyclic prefix and Guard insertion is done to get rid of Intersymbol Interference (ISI).
2.3.1 Quadrature Amplitude Modulation (QAM):
Quadrature Amplitude Modulation is a combination of two techniques i.e. phases shift keying and amplitude modulation. In technical terms, in this type of modulation data is transmitted by varying/modulating amplitude of two different carrier waves, which are out of phase by 90 degrees. i.e. (waves are sine and cosine). They are known as Quadrature carriers due to their phase difference between them. The two carriers , which are used in QAM have same frequency. For transmission, two modulated carriers are combined at the source. At the receiver side, modulated carriers are separated and data is extracted from each carrier. Then, data is combined into real modulating information. If offers many advantages over other modulation scheme such as PSK. That's why it is being widely used. A signal obtained by adding the phase and amplitude modulation of carrier signal is used for data transmission.
Information which is transmitted with each symbol is increased by increasing value of M. Expression is given below:
If M=16, its mean 4 bits per 16-ary symbol is being send. The figure given below shows 16-ary QAM with 4 bits/symbol.
Constellation diagram is used in QAM. In constellation diagram, constellation points are placed in a square grid with equal vertical and horizontal spacing. The number of constellation points are usually in a power of 2 (2, 4, 8, 16, etc). Most common are 16-QAM, 64 -QAM, 128-QAM and 256- QAM.
2.3.2 Inverse Fast Fourier Transform (IFFT):
Inverse of FFT is known as IFFT. It is a technique where data from frequency domain is converted into time domain. By taking Inverse FFT of message signal, OFDM signal is obtained. In OFDM system, IFFT is used also to do multiplexing and modulation in one step. Mathematical equation of IFFT is given below
Where n=0, 1â€¦ N-1
2.3.3 Cyclic Prefix:
Cyclic prefix means to fix the end part of the symbol with the starting of symbol. That is prefixing of data symbol with repetition of the ending of symbol. Length of multipath channel must be at least equal to the length of cyclic prefix for the effectiveness of cyclic prefix. Cyclic prefix is assumed as a traditional part of OFDM system, but it is also being used in single carrier modulation system. Cyclic prefix is used in OFDM system in order to fight against multipath effect by making channel estimation simple. As example, take an OFDM system which has N sub-carriers. Message signals can be denoted as:
Now if we take IDFT of message signal, OFDM symbols are obtained, followed by cyclic prefix. Signals which are obtained by IDFT are shown below
With a cyclic prefix of length L-1, prefix it. Then, we obtain OFDM signals which are as:
Cyclic prefix has a main purpose to eliminate ISI.
2.4 Mathematical Description of OFDM:
As we are using 16-QAM modulation, hence mathematical description of OFDM according to 16 QAM is given below
Bits per symbol =
Each symbol is denoted by m (i)
Where 0 < i < M-1, and M is number of symbols. QAM is subjected to each data symbol. Output shows the constellation plotting of symbol m (i).
Let be the constellation point which is received after QAM. The output of QAM is then passed to the input of IFFT where data is transformed from Frequency domain into time domain.
Where f (k) is the k'th carrier frequency
Equation (1) shows the OFDM signal for i'th symbol.
2.5 Applications of OFDM:
Wireless local area network (WLAN)
Metropolitan area network (MAN)
Wireless personal area network (PAN)
Digital radio and television broadcasting
Digital video broadcasting (DVB-T)
Digital audio radio services (SDARS)
2.6 Advantages of OFDM:
OFDM works efficiently even in severe channel conditions.
OFDM fight against narrowband co-channel interference.
Efficient against intersymbol interference (ISI) and fading caused by multipath propagation.
High spectral efficiency in comparison with conventional modulation schemes, spread spectrum.
OFDM is efficient implementation using Fast Fourier Transform (FFT).
Low sensitivity to time synchronization errors.
Tuned sub-channel receiver filters are not required (unlike conventional FDM).
Single frequency networks (SFNs) are facilitated; i.e., transmitter macro-diversity.
2.7 Disadvantages of OFDM:
Sensitivity to Doppler shift.
Sensitive to frequency synchronization problems.
Having high peak-to-average-power ratio, it requires linear transmitter circuitry, which suffers from poor power efficiency.
Loss of efficiency caused by cyclic prefix/guard interval.
3.1 Singular Value Decomposition
SVD is a technique well used in factorization of real or complex matrix. It is also known as Principle Component Analysis. In numerical Analysis, measure of the effective rank of given matrix provide by the Singular Value Decomposition. The SVD of matrix A is the factorization
Where V and U are orthogonal matrices having singular vectors and . Singular values are ordered by convention such as. The are the singular values of A. Singular values are all positive, if A has full rank. Basically SVD breaks matrix A into three major components. 
3.2 WHY SVD?
SVD is well known due to its two very important properties. First one is that, it works for any and all type of matrices i.e. small, large, non-singular, singular and rectangular etc. The second property of SVD is optimality property i.e. if we want to find rank-k approximation of our matrix, by using SVD we can find best one.
3.3 What are Singular Values?
If there is square matrix A, the square roots of the eigenvalues of are singular values, where is conjugate transposed.
3.4 Types of SVD
There are mainly three types of SVD which are given below.
3.4.1 Thin SVD
Thin SVD is significantly fast and more economical than full SVD because only n column vectors of U corresponding to the row vectors of V* are calculated but on one condition i.e. if n << m. mathematically it is written as
Where matrix is thus m*n, is n*n diagonal and V is n*n
3.4.2 Compact SVD
Corresponding to the non-zero singular values, only the r column vectors of and r row vectors of V* are calculated. Remaining vectors of U and V* don't need to calculate.
If r << n, compact SVD is faster and economical than Thin SVD. Mathematically it is written as
Where is m x r, is r x n and is r x r diagonal
3.4.3 Truncated SVD
Only t row vectors of V* and r column vectors of U are calculated corresponding to singular values. If t << r, it is quicker than Compact SVD
3.5 Analysis of Channel using SVD
Where X= transmitted Signal
Substitute equations 5 and 6 in equation 4
Substitute Eq. 2 in Eq. 7
Substitute Eq. 8 in Eq. 3
Resulting received signals
X= Transmitted Signal
V= noise vector
3.6 EXAMPLE OF SVD
Let's compute SVD for the following matrix
STEP 1ïƒ Find its transpose and
STEP 2ïƒ Find eigenvalues of and sort in descending order. Taking Square root of eigenvalues results singular values of A.
The Equation gives two values. In descending order these are ïƒ >
Eigen Values ïƒ
Singular values ïƒ
STEP 3ïƒ By placing singular values in descending order along its diagonal, Construct diagonal matrix S. Also compute.
STEP 4 ïƒ From step 2 use ordered eigenvalues and compute eigenvectors of. Write these eigenvectors along the columns of V and compute its transpose.
STEP 5 ïƒ Compute U as
If we compute, we will get back the original matrix.
3.5 Applications of SVD
Widely used in signal processing.
Widely used in Image processing.
Low rank approximation via SVD.
SVD is used in estimation/Inversion.
4.1 CHANNEL ESTIMATION
As the words exhibit the meaning of channel estimation is to estimate the channel response, i.e. how much change in signal came after passing through the channel? Complex channel estimation is done with the help of known transmitted pilot symbols. Modulation can be of two type's i.e. Differential or Coherent. Differential modulation doesn't need to estimate channel. In the difference between two consecutive symbols, the information is encoded. In design of channel estimator for wireless OFDM systems, two problems mostly come to face. How to transmit pilot information, that's the first problem. Pilot information act as a reference for channel estimation. Second problem relates to design an estimator with both qualities such as low complexity and good channel tracking ability, which is difficult to achieve.
4.1.1 SYSTEM DESCRIPTION
Pilot based channel estimation of OFDM system is shown in figure. In signal mapper, binary data is grouped and mapped according to the modulation being used. Pilots are inserted in all sub-carriers with a specific interval/period. IDFT transform the data sequence into time domain from frequency domain. After IDFT, guard insertion is done and it is assumed that guard time is larger than the expected delay spread. This is done to prevent data from Intersymbol Interference (ISI). The signal is thus transmitted. The transmitted signal is passed through the time varying fading channel with addition of noise. At the receiver side, Guard insertion is removed and DFT transforms data sequence into frequency domain. Pilot signals are separated/extracted and the estimated channel is obtained in channel estimation block. The estimation of the transmitted data is analyzed by
At signal demapper block, binary data is obtained back.
4.2 Types of Channel Estimation
There are two ways of Channel estimation i.e. Block- Type pilot channel estimation and Comb -Type pilot channel estimation.
4.2.1 Block-Type pilot channel estimation
In block-type pilot channel estimation, we assume to have a slow fading channel. In a specific period, pilot tones are inserted into all available sub-carriers of OFDM symbols. It is assumed that channel transfer function is constant over transmission of OFDM symbols and not varying rapidly.  There is periodically transmission of OFDM channel estimation symbols. There will be no channel estimation error if the channel is constant during a block because pilots are being sent to all carriers. Channel conditions are specified by H, pilot signals with matrix X and received Signal with Y. Channel statistics may or may not be known. At the receiver side, data is decoded using estimated channel conditions in a block, until arrival of next pilot symbol. This type of estimation is based on usually two types of techniques i.e. Least Square (LS) and Minimum Mean-Square Error (MMSE) .
4.2.2 Comb-Type channel estimation
When channel changes from one OFDM block to another subsequent one, there comes the need for equalization. Comb-Type Channel Estimation is best suited for such type of situation. Channel is estimated at pilot frequencies. Algorithms are used in Comb-Type channel estimation. Comb-type channel based estimation is based on LS or MMSE. MMSE performs better than LS. Highlights on properties of LS and MMSE are given below.
126.96.36.199 LS Estimator
The least square method is being widely used to estimate or find the numerical values of the parameters to fit a function to a set of data . LS method has many versions; simplest version is known as ordinary least squares (OLS). A better version is weighted least squares (WLS). Recently latest versions include Alternating Least Squares (ALS) and Partial Least Squares (PLS). Let's elaborate this method with an equation given below:
Where X is an independent variable and Y is dependent variable. This equation involves two parameters i.e. a and b. The LS estimation method defines estimation of parameters as the values which tends to minimize the sum of the squares between the predicted values and measurements . That's why it is known as Least Squares. The expression is given below which is to be minimized
Where Îµ denotes "error" which is to be minimized. In our case, the term LS Estimators highlights a frequency used approach to solve inexactly specified systems or over determined systems. LS Estimators are widely used due to its ability to calculate with low complexity, even without prior knowledge of Channel. Due to this, High mean square error generates. This technique can be implemented in MATLAB. Mathematical description is shown below
188.8.131.52LOW RANK APPROXIMATION
Given an MÃ-N matrix A, we want to find out an MÃ-N matrix of rank at most k, where k is a positive integer. Frobenius norm of the matrix difference i.e. defined as
Discrepancy between matrices and A is measured by Frobenius norm of X. Our goal is to find out the that minimizes discrepancy while keeping to have rank at most k. If rank of A is r, and .Hence is zero in such case. When k << r, to refer to as a Low Rank Approximation.
Low Rank approximation problem can be easily solved by using SVD. As we know that our goal is to find, hence there are given three steps below through which we can find .
Given A, construct its singular value decomposition in the form i.e.
Derive matrix from matrix âˆ‘.
Computer and find as the rank k approximation to A.
Herve abdi. Least Squares. University of texas at Dillas.
5.1 Project Flow
CYCLIC PREFIX REMOVAL
Steps of Project Flow:
A: Firstly randomly input of the data is taken as binary data, which is finally converted to the decimal form, according to 16 QAM, as four bits are taken and converted. For example 192 bits are taken randomly, and then according to 16 QAM that is, where n is the number of bits taken.
B: the coded decimal data goes through the QAM, which makes an amplitude and phase plot of the different symbol. According to 16 QAM there are 16 constellations. Fig 5.1 depicts the constellation plotting.
Fig 5.1 shows the QAM plotting of the values
C: Then the IFFT (Inverse Fourier Transform) of the entire data is done.
D: Cyclic prefix is being added then to avoid the inter symbol interference, and allows data to be circularly convolved.
E: The data is then send over a specific channel
F: The output data passes through the several steps, and then channel estimation is done after FFT. Here we have done with the LS estimation and also low rank Approximation, to estimate the known channel.
5.2 LS estimation results:
Fig 5.2 shows the frequency domain plot of the channel and estimated after LS estimator
Fig 5.3 the above result shows the BER versus SNR of the channel and LS estimator
5.3 Low Rank Approximation:
5.3.1Frequency domain plots of estimated and original channel
Fig 5.4 the rank 5
Fig 5.5 rank 4
Fig 5.5 rank 3
Fig 5.6 rank 2
Fig 5.7 rank 1 result
BER vs SNR of ranks and channel:
Conclusions & future work:
The project aims to develop the ofdm system with the channel estimation done with SVD. There are various methods deployed in the estimation of channel, and we are focusing on two estimators. One is the LS estimator, and other is low rank approximation. Lower the rank of the specific matrix allows the complexity to be reduced but allows the bit error to increase. Therefore we will trade off between two key things in a communication system that is complexity and bit error.
Our future work is imposing of this complete system over a cluster cellular network of 4G technology. The clustered cellular approach will try to increase the capacity of the users in a specific cluster, with MIMO interfacing.