7.2 Source coding

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An introduction to the concept of typical sequences, which lie at the heart of source coding. The idea of typical sequences leads to Shannon's source-coding Theorem.

As mentioned earlier, how much a source can be compressed should be related to its entropy . In 1948, Claude E. Shannon introduced three theorems and developed very rigorousmathematics for digital communications. In one of the three theorems, Shannon relates entropy to the minimum number of bitsper second required to represent a source without much loss (or distortion).

Consider a source that is modeled by a discrete-time and discrete-valued random process ${X}_{1}$ , ${X}_{2}$ ,…, ${X}_{n}$ ,…where ${x}_{i}\in \{{a}_{1}, {a}_{2}, \dots , {a}_{N}\}$ and define ${p}_{{X}_{i}}({x}_{i}={a}_{j})={p}_{j}$ for $j=1,2,\dots ,N$ , where it is assumed that ${X}_{1}$ , ${X}_{2}$ ,… ${X}_{n}$ are mutually independent and identically distributed.

Consider a sequence of length $n$

$X=\left(\begin{array}{c}{X}_{1}\\ {X}_{2}\\ ⋮\\ {X}_{n}\end{array}\right)$
The symbol ${a}_{1}$ can occur with probability ${p}_{1}$ . Therefore, in a sequence of length $n$ , on the average, ${a}_{1}$ will appear $n{p}_{1}$ times with high probabilities if $n$ is very large.

Therefore,

$P(X=x)={p}_{{X}_{1}}({x}_{1}){p}_{{X}_{2}}({x}_{2})\dots {p}_{{X}_{n}}({x}_{n})$
$P(X=x)\approx {p}_{1}^{(n{p}_{1})}{p}_{2}^{(n{p}_{2})}\dots {p}_{N}^{(n{p}_{N})}=\prod_{i=1}^{N} {p}_{i}^{(n{p}_{i})}$
where ${p}_{i}=P({X}_{j}={a}_{i})$ for all $j$ and for all $i$ .

A typical sequence $X$ may look like

$X=\left(\begin{array}{c}{a}_{2}\\ ⋮\\ {a}_{1}\\ {a}_{N}\\ {a}_{2}\\ {a}_{5}\\ ⋮\\ {a}_{1}\\ ⋮\\ {a}_{N}\\ {a}_{6}\end{array}\right)$
where ${a}_{i}$ appears $n{p}_{i}$ times with large probability. This is referred to as a typical sequence . The probability of $X$ being a typical sequence is
$P(X=x)\approx \prod_{i=1}^{N} {p}_{i}^{(n{p}_{i})}=\prod_{i=1}^{N} 2^{\log_{2}{p}_{i}}^{(n{p}_{i})}=\prod_{i=1}^{N} 2^{(n{p}_{i}\log_{2}{p}_{i})}=2^{(n\sum_{i=1}^{N} {p}_{i}\log_{2}{p}_{i})}=2^{-(nH(X))}$
where $H(X)$ is the entropy of the random variables ${X}_{1}$ , ${X}_{2}$ ,…, ${X}_{n}$ .

For large $n$ , almost all the output sequences of length $n$ of the source are equally probably with $\mathrm{probability}\approx 2^{-(nH(X))}$ . These are typical sequences. The probability of nontypical sequences arenegligible. There are $N^{n}$ different sequences of length $n$ with alphabet of size $N$ . The probability of typical sequences is almost 1.

Consider a source with alphabet {A,B,C,D} with probabilities { $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{8}$ }. Assume ${X}_{1}$ , ${X}_{2}$ ,…, ${X}_{8}$ is an independent and identically distributed sequence with ${X}_{i}\in \{A, B, C, D\}$ with the above probabilities.

$H(X)=-(\frac{1}{2}\log_{2}\left(\frac{1}{2}\right))-\frac{1}{4}\log_{2}\left(\frac{1}{4}\right)-\frac{1}{8}\log_{2}\left(\frac{1}{8}\right)-\frac{1}{8}\log_{2}\left(\frac{1}{8}\right)=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{3}{8}=\frac{4+4+6}{8}=\frac{14}{8}$

The number of typical sequences of length 8

$2^{(8\frac{14}{8})}=2^{14}$

The number of nontypical sequences $4^{8}-2^{14}=2^{16}-2^{14}=2^{14}(4-1)=32^{14}$

Examples of typical sequences include those with A appearing $8\frac{1}{2}=4$ times, B appearing $8\frac{1}{4}=2$ times, etc. {A,D,B,B,A,A,C,A}, {A,A,A,A,C,D,B,B} and much more.

Examples of nontypical sequences of length 8: {D,D,B,C,C,A,B,D}, {C,C,C,C,C,B,C,C} and much more. Indeed, these definitions andarguments are valid when n is very large. The probability of a source output to be in the set of typical sequences is 1 when $n\to$ . The probability of a source output to be in the set of nontypicalsequences approaches 0 as $n\to$ .

The essence of source coding or data compression is that as $n\to$ , nontypical sequences never appear as the output of the source. Therefore, one only needs to be able to representtypical sequences as binary codes and ignore nontypical sequences. Since there are only $2^{(nH(X))}$ typical sequences of length $n$ , it takes $nH(X)$ bits to represent them on the average. On the average it takes $H(X)$ bits per source output to represent a simple source that produces independent and identically distributed outputs.

Shannon's source-coding

A source that produced independent and identically distributed random variables with entropy $H$ can be encoded with arbitrarily small error probability at any rate $R$ in bits per source output if $R\ge H$ . Conversely, if $R< H$ , the error probability will be bounded away from zero, independent ofthe complexity of coder and decoder.

The source coding theorem proves existence of source coding techniques that achieve rates close to the entropy but does not provide anyalgorithms or ways to construct such codes.

If the source is not i.i.d. (independent and identically distributed), but it is stationary with memory, then a similar theorem applies withthe entropy $H(X)$ replaced with the entropy rate $H=\lim_{n\to }n\to$ H X n | X 1 X 2 X n-1

In the case of a source with memory, the more the source produces outputs the more one knows about the source and themore one can compress.

The English language has 26 letters, with space it becomes an alphabet of size 27. If modeled as a memoryless source (nodependency between letters in a word) then the entropy is $H(X)=4.03$ bits/letter.

If the dependency between letters in a text is captured in a model the entropy rate can be derived to be $H=1.3$ bits/letter. Note that a non-information theoretic representation of a text may require 5 bits/letter since $2^{5}$ is the closest power of 2 to 27. Shannon's results indicate that there may be a compression algorithm with the rate of 1.3bits/letter.

Although Shannon's results are not constructive, there are a number of source coding algorithms for discrete time discretevalued sources that come close to Shannon's bound. One such algorithm is the Huffman source coding algorithm . Another is the Lempel and Ziv algorithm.

Huffman codes and Lempel and Ziv apply to compression problems where the source produces discrete time and discrete valuedoutputs. For cases where the source is analog there are powerful compression algorithms that specify all the steps fromsampling, quantizations, and binary representation. These are referred to as waveform coders. JPEG, MPEG, vocoders are a fewexamples for image, video, and voice, respectively.

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