扩频通信基础：从编码到低密度奇偶校验码

扩频通信原理

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"Principles of Spread-Spectrum Communication Systems 3rd" 是一本由Don Torrieri撰写的关于扩频通信基础技术的经典书籍，详细介绍了扩频通信的关键概念和技术。这本书是扩频通信领域的入门必读，适合对通信技术感兴趣的读者。在本书中，作者深入探讨了多种编码和调制技术，包括： 1. **块码**：这部分讲解了块码的基本原理，以及在硬判决解码中的错误概率。此外，还涉及软判决解码和脉冲幅度调制中的代码度量，以及正交信号的代码度量。书中还介绍了未编码FSK符号的检测方法和性能示例。 2. **卷积码与格状码**：卷积码和格状码是重要的纠错编码技术。书中详细阐述了Chernoff边界，并讨论了格状编码调制。这些技术在提高通信系统的可靠性方面起着关键作用。 3. **交织器**：交织器用于分散数据序列中的错误，以增强抗干扰能力。书中详细讨论了交织器的作用和设计。 4. **经典级联码**：这部分介绍了传统的级联编码系统，这些系统通过结合不同类型的错误校验码来提高性能。 5. **涡轮码**：涡轮码是现代通信中的一种高效编码方式，书中有详细描述MAP解码算法以及并行和串行级联的涡轮码结构，这两种结构都具有卓越的纠错性能。 6. **迭代解调与解码**：迭代解调和解码是一种结合解调和解码过程的技术，可以显著提升系统性能。书中介绍了比特交织编码调制及其模拟例子。 7. **低密度奇偶校验检查码（LDPC）**：LDPC码是一种强大的纠错码，包括不规则重复累加码的介绍，以及其性能分析。每个章节都包含了相关的问题，以帮助读者理解和应用所学知识。第二章进一步讨论了直接序列扩频系统，包括定义、概念、扩频序列和波形，特别是随机二进制序列、移位寄存器序列和最大序列，以及它们的自相关性和功率谱。此外，还涵盖了BPSK调制和四元系统的应用，以及噪声和干扰环境下的系统性能分析。该书全面覆盖了扩频通信的基础理论和技术，是学习扩频通信系统不可多得的参考资料。

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xviii Contents

Selection of Modulation Index.............................. 552

Partial-Band Interference .................................. 554

Asynchronous Multiple-Access Interference .................. 558

9.5 Problems ............................................... 562

10 Detection of Spread-Spectrum Signals ........................... 565

10.1 Detection of Direct-Sequence Signals........................ 565

10.2 Radiometer ............................................. 569

Rayleigh Fading ......................................... 575

Estimation of Noise Power................................. 577

Other Implementation Issues ............................... 580

10.3 Detection of Frequency-Hopping Signals..................... 584

10.4 Channelized Radiometer .................................. 588

10.5 Problems ............................................... 594

Appendix A: Gaussian Random Variables and Processes ............... 595

A.1 General Characteristics .................................... 595

A.2 Central Limit Theorem ................................... 599

Appendix B: Special Functions ...................................... 605

B.1 Gamma Functions ........................................ 605

B.2 Beta Function ........................................... 606

B.3 Bessel Functions of the First Kind .......................... 606

B.4 Q-function .............................................. 607

B.5 Marcum Q-function ...................................... 607

B.6 Hypergeometric Function .................................. 608

Appendix C: Signal Characteristics .................................. 609

C.1 Bandpass Signals......................................... 609

C.2 Stationary Stochastic Processes ............................. 612

C.3 Direct-Conversion Downconverter .......................... 616

C.4 Sampling Theorem ....................................... 618

Appendix D: Probability Distributions ............................... 621

D.1 Chi-Square Distribution ................................... 621

D.2 Central Chi-Square Distribution ............................ 623

D.3 Rice Distribution ........................................ 623

D.4 Rayleigh Distribution ..................................... 625

D.5 Exponential and Gamma Distributions ....................... 626

References ........................................................ 629

Index ............................................................ 635

Chapter 1

Channel Codes and Modulation

This chapter reviews fundamental results of coding and modulation theory that are

essential to a full understanding of spread-spectrum systems. Channel codes, which

are also called error-correction or error-control codes, are vital in fully exploiting the

potential capabilities of spread-spectrum communication systems. Although direct-

sequence systems greatly suppress interference, practical systems require channel

codes to deal with the residual interference and channel impairments such as fading.

Frequency-hopping systems are designed to avoid interference, but the possibility

of hopping into an unfavorable spectral region usually requires a channel code to

maintain adequate performance. In this chapter, coding and modulation theory [7, 52,

61] are used to derive the required receiver computations and the error probabilities of

the decoded information bits. The emphasis is on the types of codes and modulation

that have proved most useful in spread-spectrum systems.

1.1 Block Codes

A channel code for forward error control or error correction is a set of codewords

that are used to improve communication reliability. An (n, k) block code uses a

codeword of n code symbols to represent k information symbols. Each symbol is

selected from an alphabet of q symbols that belong to the Galois Field GF(q), and

there are q

codewords. If q = 2

, then a q-ary symbol may be represented by

m bits, and a nonbinary codeword of n symbols may be mapped into an (mn, mk)

binary codeword. A block encoder can be implemented by using logic elements or

memory to map a k-symbol information word into an n-symbol codeword. After

the waveform representing a codeword is received and demodulated, the decoder

uses the demodulator output to determine the information symbols corresponding to

the codeword. If the demodulator produces a sequence of discrete symbols and the

decoding is based on these symbols, the demodulator is said to make hard decisions.

Electronic supplementary material The online version of this chapter (doi: 10.1007/978-3-319-

14096-4_1) contains supplementary material, which is available to authorized users.

D. Torrieri, Principles of Spread-Spectrum Communication Systems,

DOI 10.1007/978-3-319-14096-4_1

2 1 Channel Codes and Modulation

Fig. 1.1 Conceptual

representation of

n-dimensional vector space

of sequences

Codewords

Received

words

Planar boundary

Conversely, if the demodulator produces analog or multilevel quantized samples of

the waveform, the demodulator is said to make soft decisions. The advantage of soft

decisions is that reliability or quality information is provided to the decoder, which

can use this information to improve its performance.

The number of symbol positions in which the symbol of one sequence differs from

the corresponding symbol of another equal-length sequence is called the Hamming

distance between the sequences. The minimum Hamming distance between any

two codewords is called the minimum distance of the code. When hard decisions are

made, the demodulator output symbols are called channel symbols, and the output

sequence is called the received sequence or the received word. Hard decisions imply

that the overall channel between the output and the decoder input is the classical

binary symmetric channel. A decoder that processes the received word is called a

hard-decision decoder. If the channel-symbol error probability is less than one-half,

then the maximum-likelihood criterion implies that the correct codeword is the one

that is the smallest Hamming distance from the received word. A complete decoder

is a hard-decision decoder that implements the maximum-likelihood criterion. An

incomplete decoder does not attempt to correct all received words.

The n-dimensional vector space of sequences is conceptually represented as a

three-dimensional space in Fig. 1.1. Each codeword occupies the center of a de-

coding sphere with radius t in Hamming distance, where t is a positive integer. A

complete decoder has decision regions deﬁned by planar boundaries surrounding

each codeword. A received word is assumed to be a corrupted version of the code-

word enclosed by the boundaries. A bounded-distance decoder is an incomplete

decoder that attempts to correct symbol errors in a received word if it lies within

one of the decoding spheres. Since unambiguous decoding requires that none of the

spheres may intersect, the maximum number of random errors that can be corrected

by a bounded-distance decoder is

t =(d

− 1)/2 (1.1)

1.1 Block Codes 3

where d

is the minimum Hamming distance between codewords and x denotes

the largest integer less than or equal to x. When more than t errors occur, the received

word may lie within a decoding sphere surrounding an incorrect codeword or it may

lie in the interstices (regions) outside the decoding spheres. If the received word lies

within a decoding sphere, the decoder selects the incorrect codeword at the center

of the sphere and produces an output word of information symbols with undetected

errors. If the received word lies in the interstices, the decoder cannot correct the

errors, but recognizes their existence. Thus, the decoder fails to decode the received

word.

Since there are





(q − 1)

words at exactly distance i from the center of the

sphere, the number of words in a decoding sphere of radius t is

V =



i=0





(q − 1)

. (1.2)

Since a block code has q

codewords, q

V words are enclosed in some sphere. The

number of possible received words is q

≥ q

V , which yields

n−k

≥



i=0





(q − 1)

. (1.3)

This inequality implies an upper bound on t and hence d

. The upper bound on d

is called the Hamming bound.

A block code of length n over GF(q) is called a linear block code if its q

codewords form a k-dimensional subspace of the vector space of sequences with n

symbols. Thus, the vector sum of two codewords or the vector difference between

them is a codeword. If a binary block code, which uses an alphabet of symbols 0

and 1, is linear, the symbols of a codeword are modulo-2 sums of information bits.

Since a linear block code is a subspace of a vector space, it must contain the additive

identity. Thus, the all-zero sequence is always a codeword in any linear block code.

Since nearly all practical block codes are linear, henceforth block codes are assumed

to be linear.

Let m denote a row vector of k message bits and c denote a row vector of n binary

codeword symbols. Let G denote a k×n generator matrix, each row of which is a

basis vector of the subspace of codewords. The deﬁnitions imply that

c = mG (1.4)

where modulo-2 additions are used in the matrix multiplication. The orthogonal

complement of the row space of G is an (n − k)-dimensional subspace of the

n-dimensional binary space such that each of its linearly independent vectors is or-

thogonal to the row space, and hence to the codewords. An (n −k) ×n parity-check

matrix H has row vectors that span the orthogonal complement. Therefore,

= 0. (1.5)

4 1 Channel Codes and Modulation

Table 1.1 Binary BCH codes

n k t D

7 4 1 1 63 45 3 0.1592 127 92 5 0.0077

7 1 3 1 63 39 4 0.0380 127 85 6 0.0012

15 11 1 1 63 36 5 0.0571 127 78 7 1.68 · 10

−4

15 7 2 0.4727 63 30 6 0.0088 127 71 9 2.66 · 10

−4

15 5 3 0.5625 63 24 7 0.0011 127 64 10 2.48 · 10

−5

15 1 7 1 63 18 10 0.0044 127 57 11 2.08 · 10

−6

31 26 1 1 63 16 11 0.0055 127 50 13 1.42 · 10

−6

31 21 2 0.4854 63 10 13 0.0015 127 43 14 9.11 · 10

−8

16 3 0.1523 63 7 15 0.0024 127 36 15 5.42 · 10

−9

31 11 5 0.1968 63 1 31 1 127 29 21 2.01 · 10

−6

31 6 7 0.1065 127 120 1 1 127 22 23 3.56 · 10

−7

31 1 15 1 127 113 2 0.4962 127 15 27 7.75 · 10

−7

63 57 1 1 127 106 3 0.1628 127 8 31 8.10 · 10

−7

63 51 2 0.4924 127 99 4 0.0398 127 1 63 1

A cyclic code is a linear block code in which a cyclic shift of the symbols of

a codeword produces another codeword. This characteristic allows the implemen-

tation of encoders and decoders that use linear feedback shift registers. Relatively

simple encoding and hard-decision decoding techniques are known for cyclic codes

belonging to the class of Bose-Chaudhuri-Hocquenghem (BCH) codes. A BCH code

has a length that is a divisor of q

− 1, where m ≥ 2, and is designed to have an

error-correction capability of t =(δ − 1)/2, where δ is the design distance. Al-

though the minimum distance may exceed the design distance, the standard BCH

decoding algorithms cannot correct more than t errors. The parameters (n, k, t) for

binary BCH codes with 7 ≤ n ≤ 127 are listed in Table 1.1.

A perfect code is a block code such that every n-symbol sequence is at a distance

of at most t from some n-symbol codeword, and the sets of all sequences at distance

t or less from each codeword are disjoint. Thus, the Hamming bound is satisﬁed with

equality, and a complete decoder is also a bounded-distance decoder. The only perfect

codes are the binary repetition codes of odd length, the Hamming codes, the binary

(23,12) Golay code, and the ternary (11,6) Golay code. Repetition codes represent

each information bit by n binary code symbols. When n is odd, the (n, 1) repetition

code is a perfect code with d

= n and t = (n − 1)/2. A hard-decision decoder

makes a decision based on the state of the majority of the demodulated symbols.

Although repetition codes are not efﬁcient for the additive-white-Gaussian-noise

(AWGN) channel, they can improve the system performance for fading channels if

the number of repetitions is properly chosen. An (n, k) Hamming code is a perfect

BCH code with d

= 3 and n = (q

n−k

− 1)/(q − 1). Since t = 1, a Hamming

code is capable of correcting all single errors. Binary Hamming codes with n ≤ 127

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