Channel coding in wireless communication pdf

Popular topic for study. Ultrasonic waves can also be generated electronically. The following section gives an detailed description for an arrangement Read this topic. Thread Tools Show Printable Version. Branch: : Aeronautical Engineering. This thesis explores the structure and features of polar codes to improve their performance using Gaussian approximation-based construction of polar codes. We begin by studying partial orders POs for the synthesized bit Author : K.

Recently, polar codes take the place of turbo codes in the 5G eMBB control channels and on the physical broadcast On the other hand, Huawei Corporation announced the achievement of Gbps performance in 5G field trial tests using polar codes for channel coding. In the following month, 3GPP decided to adopt polar This n Figure 2.

An illustration of asynchronous bursts in the presence of frequency analogy illustrates the difficulty in suppressing hopping. CCI, particularly if only one receive antenna is available.

For more than two years, the Third Genera- es. A reuse factor of importance is channel estimation, which is more one is under discussion, in conjunction with a demanding than in single-user receivers, particu- fractional load as high as possible.

United States larly in asynchronous networks. Finally, numeri- network operators suggest a fractional load of cal results are presented before conclusions are up to 70 percent i. In such a scenario, CCI and adjacent channel interference will be severe. Due to ments for mobile terminals from Release 6 frequency hopping, physical layer signal process- onward. In field trails, significant gains due to ing has to be done on a burst-by-burst basis, interference cancellation have already been which prohibits excessive averaging.

Last but not demonstrated [2]. For example, EDGE interference may as follows. Then the channel model under con- should not be worse than that of a conventional sideration is presented. Subsequently, decoupled receiver.

Perhaps the most simple filter-based Compared to filter-based approaches, multi- multiuser detection approach is a linear filter, the coefficients of user-based techniques are, generally speaking, much attention which are designed so that CCI and ISI are mitigated jointly see, e.

In the context more sensitive with respect to model errors. This implies that in multi-user detection much atten- should be paid to of SAIC, linear CCI cancellation is capable of tion should be paid to channel estimation. In cancelling a single interferer if and only if the effect, channel estimation appears to be the bot- channel estimation. Gaussian minimum shift keying in GSM.

Vectors are regarded as column tion is not applicable because we run out of vectors and are written in lower case boldface. Given For matrices upper case boldface is used. The baud rate sampling, only a single complex-val- transpose, complex conjugate transpose, and ued data stream as in EDGE or two real-val- expected value are denoted.

Since the excess band- tified by. Any performance improve- written as ment observed with oversampling is due to the J L use of finite filter lengths. This imposes fewer constraints on the linear filter. All ran- interference plus noise process is modeled by an dom processes are assumed to be mutually auto-regressive model.

The corresponding receiv- independent. In case of Multi-user detection algorithms are different square-root Nyquist receive filtering and baud from filter-based approaches in the sense that rate sampling, the Gaussian noise process is the signals of all cochannels are estimated explic- white. In case of N-times oversampling, where itly. Either the data of the co-channels can be N is an integer, any of the N polyphase chan- estimated jointly, or the interference is subtract- nels can be represented by Eq.

The equiva- ed off the received signal in either a sequential lent discrete-time channel model is suitable for or parallel fashion. The detection. The remaining ISI of the desired user is canceled by the nonlinear equalizer.

This receiv- has to be matched to the ISI channel of the er structure is compatible with state-of-the-art desired user modified by the linear CCI cancel- TDMA receivers, where just the linear filter er only. Trellis-based, tree-based, and graph- called a prefilter is missing based equalizers are suitable, preferably In the following, a finite impulse response delivering reliability information in conjunction FIR filter w is considered.

The optimum filter with the detected data symbols. As mentioned coefficients in the sense of maximizing the sig- earlier, only one complex-valued data stream or nal-to-interference-plus-noise ratio SINR can two real-valued data streams can be resolved per be obtained as receive antenna by means of a linear filter.

Hop rate It is the rate at which the system changes from one frequency to another. The hit probability is the probability that a number of interfering users are transmitting on the same frequency-hop channel as the reference user. Equation 2.

It is the exact inverse of the transmitter shown in Fig. By combing two or more such channels, fading can be reduced. This is called diversity.

On a fading channel, the SNR at the receiver is a random variable, the idea is to transmit the same signal through r separate fading channels. These are chosen so as to provide the receiver with r independent or close-to-independent replicas of the same signal, giving rise to independent SNRs. By suitably combining the received signals, the fading effect will be mitigated Fig. Among the categorized techniques, the most important ones are as follows: 1. Space diversity 2. Polarization diversity 3.

Frequency diversity 4. Time diversity 5. It does not require any extra spectrum occupancy and can be easily implemented. Polarization diversity: Over a wireless channel, multipath components polar- ized either horizontally or vertically have different propagation.

Diversity is pro- vided when the receiving signal uses two different polarized antennas. In another way, two cross-polarized antennas with no spacing between them also provide diversity. Cross-polarized are preferred since they are able to double the antenna numbers using half the spacing being used for co-polarized antennas.

Polarized diversity can achieve more gain than space diversity alone in reasonable scattering areas, and hence, it is deployed in more and more BSs. Frequency diversity: In order to obtain frequency diversity, the same signal over different carrier frequencies should be sent whose separation must be larger than the coherence bandwidth of the channel.

Time diversity: This is obtained by transmitting the same signal in different time slots separated by a longer interval than the coherence time of the channel. Cooperative diversity: This is obtained by sharing of resources by users or nodes in a wireless network and transmits cooperatively.

The users or nodes act like an antenna array and provide diversity. This type of diversity can be achieved by combining the signals transmitted from the direct and relay links. Selection combining 2.

Equal gain combining EGC 3. Example 2. Obtain an approximation to the outage probability for the parallel channel with M Rayleigh branches. Assume three-branch MRC diversity in a Rayleigh fading channel. Write a MATLAB program to simulate the performance of selection diversity, equal gain combiner, and maximum ratio combiner and compare the perfor- mance with the theoretical results.

References 1. Lu, J. IEEE Trans. Proakis, J. McGraw-Hill, New York 3. Rappaport, T. Lindsey, W. Theory IT 4 , — 5.

Simon, M. Wiley, New York 7. Cheng, J. Wireless Commun. Geraniotis, E. Com- 30 5 , 9. Goh, J. Theory 44 3 , — Shi, Q. IEEE Commun. Barry, J. Kluwer Academic Publishers, Massachusetts Zhang, Q. An empty set contains zero elements.

A set Y is called a subset of a set X if and only if every element Y is in X. If the inter- section of two sets is empty, they are said to be disjoint. F forms a commutative group under addition 2. F forms a commutative group under multiplication 3. Addition and multiplications are distributive. The modulo 2 addition and multipli- cation are shown in Table 3. The integers f0; 1; 2;.

The value of q must be equal to pm , Table 3. Example 3. The addition and multiplication over GF 7 will be modulo 7 as shown in Table 3. Vector spaces 1. V is a commutative group under addition operation on V 2. Examples 1. All primitive polynomials are irreducible polynomials, but all irreducible polynomials are not primitive. Table 3. The set 1; a; a2 is used as a basis for the vector space representation of GF 8. All the above values are tabulated in the following Table 3.

Similarly, for the values from x5 to x15, follow the same procedure to get the polynomial representations. The above values are tabulated in the following Table 3. Similarly, for the values from x6 to x31, follow the same procedure to get the polynomial representations.

Similarly, for the values from x7 to x63, follow the same procedure to get the polynomial representations. Theorem 3. Then, all its conjugates l 1 a; a2 ;. Step 2: Find the groups of the conjugate roots. Step 3: The construction of minimal polynomial of each elements is by using Eq.

Using the above procedure, the following examples illustrate the construction of the minimal polynomial for GF 8 , GF 16 , and GF 32 with respect to GF 2. Construct modulo- 5 addition and multiplication tables for GF 5. Find whether each of the following polynomial is irreducible in GF 2.

Find whether each of the following polynomial is primitive in GF 2. Construct GF as a vector space over GF 2. When the 64 elements in GF 26 are arranged in conjugacy classes and their associated minimal polynomials. Chapter 4 Linear Block Codes This chapter deals with linear block codes covering their fundamental concepts, generator and parity check matrices, error-correcting capabilities, encoding and decoding, and performance analysis.

The n-bit block of the channel block encoder is called the code word. Hamming Distance The Hamming distance between two blocks v and w is the number of coordinates in which the two blocks differ. The Minimum Distance of a Block Code The minimum distance of a block code C is the minimum Hamming distance between all distinct pairs of code words in C. The code word is said to be systematic linear code word, if each of the 2k code words is represented as linear combination of k linearly independent code words.

Property 2: The minimum distance of a linear block code is equal to the minimum weight of any nonzero word in the codeThe two well- known bounds on the minimum distance are 1. Hamming Bound An n, k block code can correct up to tec errors per code word, provided that n and k satisfy the Hamming bound.

It follows from Theorem 3. A matrix G is constructed by taking the vectors in the basis as its rows. It can be used to directly encode k-bit blocks in the following manner.

Example 4. Solution It follows from Eq. Thus, the transpose of the parity check matrix H will be 4 by 15 matrix. The last eleven rows are arbitrarily chosen, with the restrictions that no row is zero, and all the rows are distinct.

The syndrome table gives the syndrome value based on the simple relationship with parity check matrix. The single error-correcting codes, i. Consider a code word c corrupted by e an error pattern with a single one in the jth coordinate position results a received vector r. Let fh0 ; h1 ;. When the syndrome is computed, we obtain the transposition of the jth column of H. Compute the syndrome s for the received word. Find the position j of the column of H that is the transposition of the syndrome.

Complement the jth bit in the received code word to obtain the corrected code word. The syndrome is the transposition of 7th column of H.

Determine the dimensions of C1 , and compute the number of code words in C1. Equation 4. Verify that the message has been systematically encoded. Solution The systematic generator matrix is obtained by selecting as rows those code words associated with the message blocks , , , , , , , , , and The rectangular boxes represent flip-flops which reside either in 0 or 1 state.

The encoder operation is as follows. The switches are placed in position in 1. The k message bits are sent to the modulator and placed at the end of the systematic code word. The switches are moved to the position 2 to break the feedback connection. The parity bits in the shift register are shifted out into the transmitter to form the parity bits of the systematic code word.

The given message bits are The contents of the shift register are shown in Table 4. Hence, the four parity check bits are Therefore, the code word output is Syndrome Calculator The syndrome calculator shown in Fig. At the end of the last received bit shifting, the contents of the shift register contain the desired syndrome s.

If the syndrome is zero, there are no transmission errors in the received word or else the received code word contains transmission error. By knowing the value of syn- drome, we can determine the corresponding error pattern and also make the appropriate correction. At the end of the seventh shift, the contents of the shift register syndrome is The nonzero value of the syndrome indicates the error, and the error pattern for the syndrome is from the Table 4.

CRC codes are implemented from cyclic codes and hence the name, even when they are generally not cyclic. The following three CRC codes given in Table 4. From Theorem 3. Select 2tec consecutive powers of a. Obtain the minimal polynomials for all the 2tec consecutive powers of a having the same minimal polynomial for the roots in the same conjugacy class. Solution Since 15 is of the form 2m 1, the BCH codes are primitive. Since the code is to be double error correcting, the generator polynomial thus must have a; a2 ; a3 ; a4 as roots.

Since the code is to be triple error correcting, the generator polynomial thus must have a; a2 ; a3 ; a4 ; a5 ; a6 as roots. Solution A parity check matrix for this code is obtained by using Eq. Each element in GF 24 can be represented by 4 tuples over GF 2.

By evaluating the received poly- nomial at 2tec zeros, the syndromes S1 S2. S2tec can be obtained. These codes work with symbols that consist of several bits. A common symbol size for non-binary codes is 8 bits or a byte. The RS codes are good at correcting burst errors because the correction of these codes is done on the symbol level. The parameter n indicates the code word length in terms of the number of symbols in the code word.

Step 2: Divide the message polynomial by the code generator polynomial using GF algebra. Step 3: The parity symbols are the remainder of this division. These steps are accomplished in hardware using a shift register with feedback.

The architecture for the encoder is shown in Fig. Solution A 15,11 Reed—Solomon code has minimum distance 5. Thus, the 15,11 Reed—Solomon code is double error corrections. The generator polynomial is constructed as follows using the representation for GF 16 over GF 2.

A narrow-sense generator is con- structed as follows using the representation for GF RS codes are maximum distance separable MDS. The weight distribution polynomial of RS code is known. Syndrome generation is similar to parity calculation. A Reed—Solomon code word has 2tec syndromes that depend only on errors not on the transmitted code word.

Once the error locator polynomial is known, the error locations can be found by using the Chien search algorithm [4]. The flowchart of the Berlekamp—Massey iterative algorithm is shown in Fig. The Berlekemp— Massey algorithm initially i. Then, the length of the LFSR is to be tested. For triple error correction, the roots of the generator polynomial include a; a2 ; a3 ; a4 ; a5 ; a6. Step 2: Replace all erasures with ones in a received word, and decode it to a code word c1.

Step 6: Compute the error magnitudes using Eq. Step 7: Compute the erasure magnitudes using Eq. Program 4. This can be attributed to the highly imperfect nature of RS codes. However, the performance is not better as compared to that of BPSK modulation. Design a four-error-correcting binary BCH code of length Let the transmission code be the triple-error-correcting binary BCH code of length Let the transmission code be the double-error-correcting binary BCH code of length The f indicates erasure.

Construct a generator polynomial for a double-error-correcting Reed—Solomon code of length 7, and determine the number of code words it does have. Determine the weight distribution for the RS code of problem 1. Compute a generator polynomial for a triple-error-correcting Reed—Solomon code of length Construct a generator polynomial for a 63,57 RS code, and determine the code words it does have. Let the transmission code be the double-error-correcting RS code of length 7.

Let the transmission code be the triple-error-correcting RS code of length Let the transmission code be the double-error-correcting RS code of length Forney, G. Theory IT, — 2. Massey, J. Theory IT 1 , — 3. Berlekamp, E. Aegean Park Press, Laguna Hills 4. Chien, R. Theory IT 1 , — 5. The name convolutional codes are due to the fact that the redundant bits are generated by the use of modulo-2 convolutions in a convolutional encoder. The use of D transform is most common in the coding literature.

The delay operator D is equivalent to the indeterminate z 1 of the z-transform. Example 5. An encoder with j memory elements can assume any one of 2j possible states. The encoder can only move between states. The encoder shown in Fig. A catastrophic code can cause an unlimited number of data errors for a small number of errors in the received code word. The following Theorem [1] can be used to verify whether a convolutional code is catastrophic.

Theorem 5. Solution From the encoder diagram shown in Fig. For a given branch, we label Y i X j where j is the weight of the input vector X and i is the weight of the output vector Y the number of nonzero coordinates.

Solution The state diagram of the systematic convolutional encoder is shown in Fig. The signal flow graph of the above state diagram is shown in Fig. In this signal flow graph, the self loop at node S0 is eliminated as it contributes nothing to the distance properties of a code relative to the all zero code sequence.