49865607 Trellis Coded Modulation

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Transcript of 49865607 Trellis Coded Modulation

Seminar report On Trellis coded modulation



ABSTRACTThis report describes basic framework of TCM, which have very good performance as comparison to conventional coding in case of bandlimited channels by just increasing the number of symbols instead of increasing symbol rate. And it also explain that in TCM coding and modulation occurs simultaneously while in conventional and convolutional coding first coding is done then modulation occurs. QAM and PSK are also briefly explained as TCM is widely used with them. There is a practical example of TCM in the end which briefly explains all key aspects of TCM that is trellis encoding, trellis decoding, mapper and set partitioning



1. TCM Trellis Code Modulation 2. ACG Asymptotic Coding Gain 3. AWGN Additive white Gaussian Noise 4. 5. 6. 7. SED Squared Euclidian Distance MSED Minimum Squared Distance SSED Sum of Squared Euclidean Distance SNR Signal To Noise Ratio

8. QAM Quadrature Amplitude modulation 9. PSK Phase Shift Key 10. BPSK Binary Phase Shift Key 11. QPSK Quadrature Phase Shift Key 12. ASK Amplitude Shift Key 13. FSK Frequency Shift Key 14. BER Bit Error Rate


1 Introduction........................................5 1.1 Features of Channel Coding5 1.2 TCM Introduction...6 2 TCM....8 2.1 History of TCM.8 2.2 Basic Concepts.....9 2.2.1 Euclidean Distance9 2.2.2 Hamming Distance9 2.3 Basics Principles of TCM....11 2.4QAM....12 2.5 PSK.13 2.5.1 Binary Phase Shift Keying (BPSK) ....13 2.5.2 Quadrature Phase Shift Keying (QPSK) .....15 2.6 TCM in QAM and PSK..16 2.7 Coding Gain18 3 Implementation...19 3.1 Basic Philosophy.19 3.2 Trellis Encoding 3.2.1 Trellis Encoder.19 3.2.2 Trellis State Diagram20 3.2.3 Set Partitioning..21 3.2 Trellis Decoding..22 3.2.1 Viterbi Decoding..24 4. Applications of Trellis Coded Modulation...25 4.1 Applications in Wireline Communications.25 4.1.1 Adsl..25 4.1.2 Data/Fax Modem..25 4.1.3 Cable Modem25 4.1.4 Ethernet.25 4.2 Applications In Wireless Communications.26 4.2.1 Wi-Fi..26 4.2.2 Wimax26 4.2.3 Wpan..26 4.2.4 Uwb...26 4.2.5 Satellite Communications..26 5 Conclusion...27 6 References...28


1 INTRODUCTIONThe aim of channel encoding theory is to find codes which transmit quickly, contain many valid code words and can correct or at least detect many errors. These aims are mutually exclusive however, so different codes are optimal for different applications. The needed properties of this code mainly depend on the probability of errors happening during transmission.[1] The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then further researched. Algebraic Coding theory, is basically divided into two major types of codes 1. Linear block codes 2. Convolution codes 1.1 FEATURES OF CHANNEL CODING Information theory tells us that for optimal communications we should design long sequences of signals, with maximum separation among them; and at the receiver we should perform decision making over such long signals rather than individual bits or symbols. If this process is done properly, then the message error probability will decrease exponentially with sequence length, n provided that the rate R is less than R0, which in turn is less than the Shannon Capacity.

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2 d min

2 2

. 1.1

This is the idea behind coding. In conventional coding, the coding is separate from modulation. Coding occurs at the digital level, before modulation and generally involves adding bits to the input sequence. The resultant redundancy requires added bandwidth. At the receiver, hard decoding occurs after demodulation. The decoding operation is based on hard decisions, since a digital bit (or symbol) stream fees the decoder and is either in error or not. Decoding can also be done based on the analog received samples, and this is called soft decoding. The theoretical loss due to hard [vs. soft] decoding leads to a ~2dB performance loss.[5]


TCM uses many diverse concepts from signal processing. In simplest terms it is a combination of coding and modulation, hence it name TCM where the word trellis stands for the use of trellis (also called convolutional) codes. Whereas we normally talk about coding and modulation as two independent aspects of the communications link, in TCM they are combined. TCM is a complex concept to understand particularly due to the nonlinear nature of the performance. It uses ideas from modulation and coding as well as dynamic programming, lattice structures and matrix math. A convolutional code that has optimum performance when used independently may not be optimum in TCM. Gray coding is helpful in uncoded signaling and constellation mapping, but not always so in TCM.[3] It uses concepts of convolutional codes, trellis, lattice, cossets, and cosset generators. Communications theory says that it is best to design codes in long sequences of messages. The allowed sequences should be very different from each other. The receiver can then make a decision between sequences using their statistics rather than on symbolby-symbol basis. When decoding this way, the probability of error is an inverse function of the sequence length. In general form the probability of error between sequences is given by the expression2 d min

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2 2


where dmin is the sequence Euclidean distance between sequences and is 2 the noise power. We measure the performance of TCM (and many other schemes) by ACG. This is the gain obtained over some baseline performance at high SNR in a Gaussian environment. ACG is not achievable in practice because we do not transmit signals at high SNRs, have hardware and channel imperfections that depart from AWGN assumptions. So recognize that all gains quoted herein are maximum possible only in theory.[3] The functions of a TCM consist of a Trellis code and a constellation mapper as shown in Figure TCM combines the functions of a convolutional coder of rate and a M-ary signal mapper that maps input points into a larger constellation of constellation points.[3]


FIGURE 1.1 A general trellis coded modulation



FIGURE 2.1 History of TCM


2.2 BASIC CONCEPTS 2.2.1 EUCLIDEAN DISTANCE While representing two signals in the Euclidean space as linear combination of M orthonormal basis functions.[4] A straight line distance between any two points is called the Euclidean distance. For a point p1 at (x1, y1) and another point p2 at (x2, y2), the Euclidean distance is given by the familiar formula .2.1 It is implicit that this distance is the shortest distance between these points. The Euclidean distance is an analog concept, the very concept of distance that we normally use day-to-day in the world of real numbers. For signals, we define this distance in the IQ plane. In Figure 1 we have a 8PSK signal constellation. The radius is equal to 1 and represents the maximum amplitude. Each point of the constellation is a certain combination of a particular amplitude and phase. The distance between these points is can be measured in the manner described above and these are given in the Figure below. The distances given in Figure 1 are squared and are called SED. The smallest of these distances is called the MSED, designated as for a particular constellation.[3]( X 1 X 2 ) 2 + (Y1 Y2 ) 2

FIGURE 2.2 8 PSK constellation and squared Euclidean distances between symbols

2.2.2 HAMMING DISTANCE Just as real numbers have a concept of distance, so do the binary numbers. Take two binary numbers, 011011 and 101101. The distance between these is the number of places these two numbers differ. And that number is 4. This distance is called the Hamming distance between these numbers. The distance would be zero, if these two numbers were the same. A zero distance means the numbers are the same, same interpretation as in Euclidean concept of distance. We distinguish these two types of distances by recognizing that one belongs to the analog world of real numbers and the other to the binary world. Both concepts are useful in signal processing.


In coding Hamming distance is most often used as a performance metric whereas it is Euclidean distance in the analog world.[3] Distance between sequences We can also talk about Euclidean distances between sequences by comparing distances between corresponding points of the sequences. Lets take for example an 8PSK signal that consists of a sequence of these symbols.[3] S0 S3 S2 S1 S0 In bits, we can map these as: 000 011 010 101 100 000

FIGURE 2.3 Euclidean and sequence Hamming distance The Euclidean distance for this sequence is the distance between each symbol in this sequence and a reference sequence. If we designate the all-zero-symbols as the reference sequence, then the squared Euclidean distance (SED) is the distance between each one of these symbols and the symbol S0. s0 to s0 = 0.0, s0 to s1 = .586, s0 to s2 = 2.0, s0 to s3 = 3.414 The SSED, also called d2 free of this sequence, from the all zero sequence is 3.414 + 2.0 + 0.586 = 6.0 This cumulative distance gives a feeling of how easy or difficult it would be to mistake one sequence for another. For the reference sequence we could have used any other sequence than the all zero, and the results would be the same. However using an all-zero sequence is convenient and conventional.[3]


2.3 BASIC PRINCIPLES OF TCM The key idea is that the operations of [baseband] modulation and coding are combined. The bandwidth is not expanded: same symbol rate, but redundancy is intro