Rogowski coil

2013-05-30T06:13:38Z

202.0.58.154: /* Advantages */

'''Unary coding''', sometimes called '''thermometer code''', is an [[entropy encoding]] that represents a [[natural number]], ''n'', with ''n'' ones followed by a zero (if ''natural number'' is understood as ''non-negative integer'') or with ''n'' − 1 ones followed by a zero (if ''natural number'' is understood as ''strictly positive integer''). For example 5 is represented as 111110 or 11110. Some representations use ''n'' or ''n'' − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality. Unary coding is both a [[Prefix-free code]] and a [[Self-synchronizing code]].

<table border="1" cellpadding="2">
<tr><th>n (non-negative)<th>n (strictly positive)<th>Unary code<th>Alternative
<tr><td>0<td>1<td>0<td>1
<tr><td>1<td>2<td>10<td>01
<tr><td>2<td>3<td>110<td>001
<tr><td>3<td>4<td>1110<td>0001
<tr><td>4<td>5<td>11110<td>00001
<tr><td>5<td>6<td>111110<td>000001
<tr><td>6<td>7<td>1111110<td>0000001
<tr><td>7<td>8<td>11111110<td>00000001
<tr><td>8<td>9<td>111111110<td>000000001
<tr><td>9<td>10<td>1111111110<td>0000000001
</table>

Unary coding is an optimally efficient encoding for the following discrete [[probability distribution]]

:<math>\operatorname{P}(n) = 2^{-n}\,</math>

for <math>n=1,2,3,...</math>.

In symbol-by-symbol coding, it is optimal for any [[geometric distribution]]

:<math>\operatorname{P}(n) = (k-1)k^{-n}\,</math>

for which ''k'' ≥ φ = 1.61803398879…, the [[golden ratio]], or, more generally, for any discrete distribution for which

:<math>\operatorname{P}(n) \ge \operatorname{P}(n+1) + \operatorname{P}(n+2)\, </math>

for <math>n=1,2,3,...</math>. Although it is the optimal symbol-by-symbol coding for such probability distributions, [[Golomb coding]] achieves better compression capability for the geometric distribution because it does not consider input symbols independently, but rather implicitly groups the inputs. For the same reason, [[arithmetic encoding]] performs better for general probability distributions, as in the last case above.

==Unary code in use today==
Examples of unary code uses include:
* In [[Golomb Rice code]], unary encoding is used to encode the quotient part of the Golomb code word.
* In [[UTF-8]], unary encoding is used in the leading byte of a multi-byte sequence to indicates the number of bytes in the sequence, so that the length of the sequence can be determined without examining the continuation bytes.

==See also==
*[[Unary numeral system]]

==References==

*Khalid Sayood, ''Data Compression'', 3rd ed, Morgan Kaufmann.
*Professor K.R Rao, EE5359:''Principles of Digital Video Coding''.

{{Compression Methods}}
[[Category:Coding theory]]
[[Category:Data compression]]
[[Category:Lossless compression algorithms]]

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