BCJR解码算法研究论文：可变长度编码探讨

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本论文《BCJR_decode_thesis.pdf》由罗伯特·G·毛德撰写，作者在南安普顿大学电子与计算机科学学院攻读博士学位，并于2007年12月提交此作品。论文的主题聚焦于“IRREGULAR VARIABLE LENGTH CODING”（不规则变长编码），这是一种在信息技术领域中的关键概念，特别在数据压缩、信源编码和通信系统设计中具有重要意义。 BCJR（Baum,郑,杰弗里斯,鲁滨逊）算法是论文的核心内容，这是一种基于前向后向算法（Forward-Backward Algorithm）的改进版本，用于解码那些包含状态转移概率和发射概率的隐马尔可夫模型（Hidden Markov Model, HMM）的变长编码。这种算法在诸如语音识别、自然语言处理、生物信息学等应用中展现出了强大的性能，因为它能够高效地处理不确定性和噪声，实现对序列数据的精确解码。论文可能详细阐述了BCJR算法的工作原理，包括如何通过计算前向概率和后向概率来估计给定观测序列最可能的状态路径。它可能介绍了如何利用维特比算法（Viterbi Algorithm）的原理，结合平滑技术（如贝叶斯后验平滑或加权似然性）来处理初始和结束状态概率，以及如何在实际编码和解码过程中优化性能。此外，论文可能会探讨了BCJR算法在不同应用场景中的复杂度分析，如时间和空间复杂度，以及与其他解码算法（如最大似然解码）的比较。对于不规则变长编码，论文可能会讨论如何适应非均匀概率分布，以及如何处理编码效率和解码性能之间的平衡。由于版权限制，未经许可不能全文复制或广泛引用，读者需要获得版权所有者书面许可才能在非商业研究或学习目的下使用论文内容。引用时必须给出完整的引用信息，包括作者、论文标题、提交学位的机构（南安普顿大学）及日期等。《BCJR_decode_thesis.pdf》是一篇深入探讨不规则变长编码和BCJR算法的重要学术论文，为理解这类复杂编码技术及其在实际问题中的应用提供了有价值的理论支持。

to(T ) The state that the transition T merges to.

nr(ˆe

) The set of all VBs that immediately surround the VB ˆe

Viterbi algorithm

d(T ) The distortion of the trellis transition T .

D(T ) The minimum cumulative distortion of all trellis paths between the trellis state

(0,0)

and the trellis transition T .

D(S) The minimum cumulative distortion of all trellis paths to the state S.

m(T ) The Viterbi algorithm metric of the trellis transition T .

M(T ) The maximum cumulative Viterbi algorithm metric of all trellis paths between

the trellis state S

(0,0)

and the trellis transition T .

M(S) The maximum cumulative Viterbi algorithm metric of all trellis paths to the

state S.

Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm

= b) A priori probability of the transmission sub-frame bit u

taking the value

P (k) Probability of occurrence of the codebook entry with index k.

P (S) Probability of occurrence of the trellis state S.

P (T |fr(T )) Conditional probability of the occurrence of the trellis transition T given

the occurrence of the trellis state that it emerges from.

(T ) A posteriori trellis transition probability.

A posteriori trellis transition probability normalisation factor.

γ(T ) A priori trellis transition probability.

(T ) Weighted a priori trellis transition probability.

(S) A priori trellis transition probability normalisation factor used for all trellis

transitions that emerge from the trellis state S.

α(S) Alpha value obtained for the trellis state S.

β(S) Beta value obtained for the trellis state S.

BCJR algorithm LLR pruning threshold.

BCJR algorithm a priori probability pruning threshold.

BCJR algorithm forward recursion pruning threshold.

BCJR algorithm backwards recursion pruning threshold.

xiv

1.1.1. Near-capacity operation 2

theoretically be accurately predicted, given the perfect knowledge of its unpredictable

information content. Provided that unimpaired knowledge of the source signal’s infor-

mation content can be reliably conveyed to the receiver by the channel, the transmitter

is allowed to employ source coding in order to remove all the source’s redundancy, with-

out jeopardising the receiver’s ability to reconstruct it. When communicating over a

perfectly noiseless channel, Shannon [1] showed that the minimum number of bits that

are required to reliably convey perfect knowledge of the source signal’s information

content to the receiver is given by the source’s entropy. Note however that the com-

putational complexity and latency imposed by a source codec typically escalates in

optimal entropy coding, where all redundancy is removed from the source signal.

Unfortunately, when communicating over noisy channels, which impose uncertainty

on the received signal, the reliable transmission of a source signal’s information content

may never be guaranteed. However, Shannon [1] also showed that if a source signal’s

information content is conveyed over a noisy channel at a rate (expressed in bits per

second) that does not exceed the channel’s capacity, then it is theoretically possible

to reconstruct it with an inﬁnitesimally low probability of error. This motivates the

employment of channel coding which introduces redundancy into the transmitted signal

in a speciﬁcally designed manner. This redundancy may be exploited in the receiver

to mitigate any channel-induced errors within the original non-redundant information

content.

Similarly to optimal entropy coding, the computational complexity and latency im-

posed by a channel codec escalates, when approaching the channel’s capacity [2]. In-

deed, Jacobs and Berlekamp [3] showed that the decoding trees employed by the then

state-of-the-art sequential decoding algorithm [4–7] for Convolutional Codes (CCs) [8]

have an escalating computational complexity, when operating at an information rate

above the so-called cutoﬀ rate of the channel. This cutoﬀ rate has an upper bound,

which is equal to the channel’s capacity. Throughout the 1960s and 1970s, the cut-

toﬀ rate was regarded as a practical design objective, whilst approaching the channel

capacity was deemed to be an unrealistic design goal. The escalating complexity of

CC decoding trees may be avoided by employing the more convenient trellises [9] to

represent CCs and by applying the Bahl-Cocke-Jelinek-Raviv (BCJR) [10] algorithm.

This forms the basis of turbo codes designed by Berrou et al [11] in 1993, which

employ a pair of iteratively decoded CCs and facilitate operation near the channel’s

capacity without an escalating computational complexity, as will be described in Sec-

tion 1.3. The advent of turbo coding led to the ‘rediscovery’ of Low Density Parity

Check (LDPC) codes [12–14], which apply iterative decoding techniques to bipartite

1.1.2. Joint source and channel coding 3

graphs [15] in order to facilitate operation near the channel’s capacity. Recently, ir-

regular coding techniques [16–18] have enabled ‘very-near-capacity’ operation, as will

be described in Section 1.4. Indeed, in Chapters 3 – 5 of this thesis, we demonstrate a

novel application of IrVLCs to near-capacity operation in this manner.

Near-capacity operation is of particular interest in digital multimedia broadcast

applications. This is because the channel’s capacity reduces with the channel’s Signal

to Noise Ratio (SNR), which depends on the transmit power, on the distance to the

remote receiver and on the amount of noise that is imposed upon the signal. Hence,

multimedia broadcast schemes having a particular information transmission rate are

associated with a minimum threshold SNR at which the channel’s capacity drops below

the particular information transmission rate to be maintained. If a multimedia broad-

cast scheme can be designed to support near-capacity operation, then the threshold

SNR required to achieve high quality reception is reduced.

1.1.2 Joint source and channel coding

Shannon’s source and channel coding separation theorem [1] states that the removal

of undesirable redundancy during source coding and the reintroduction of speciﬁcally

design intentional redundancy during channel coding can be performed separately,

without jeopardising our ability to achieve an inﬁnitesimally low probability of trans-

mission errors, while maintaining a near-capacity information transmission rate. How-

ever, Shannon’s ﬁndings are only valid under a number of idealistic assumptions [19],

namely that the information is transmitted over an uncorrelated non-dispersive nar-

rowband Additive White Gaussian Noise (AWGN) channel, while potentially imposing

an inﬁnite decoding complexity and buﬀering latency. These assumptions clearly have

limited validity for practical ﬁnite-delay transmissions over realistic fading wireless

channels [20].

Additionally, Shannon assumed that the source is stationary and is losslessly en-

coded with the aid of entropy-encoded symbols having equal signiﬁcance and identical

error sensitivity. These assumptions have a limited validity in the case of multime-

dia transmission, since video, image, audio and speech information is typically non-

stationary, having characteristics that vary in time and/or space [20,21]. Furthermore,

typically lossy multimedia coding [20, 21] is employed in order to achieve a high level

of compression and a concomitant low bandwidth requirement, while exploiting the

psycho-visual properties of the human vision or hearing. Finally, the components of

the encoded multimedia information typically have varying perceptual signiﬁcance and

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