In order that the definition is useful in code synthesis we require the codes

In order that the definition is useful in code

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In order that the definition is useful in code synthesis, we require the codes to satisfy certain properties. We shall intentionally take trivial examples in order to get a better understanding of the desired properties.
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Information Theory and Coding 10EC55 Dept. of ECE/SJBIT Page 158 1. Block codes : A block code is one in which a particular message of the source is always encoded into the same “ fixed sequence ” of the code symbol. Although, in general, block m eans ‘ a group having identical property ’ we shall use the word here to mean a ‘ fixed sequence ’ only. Accordingly, the code can be a ‘ fixed length code ’ or a “ variable length code ” and we shall be concentrating on the latter type in this chapter. To be more specific as to what we mean by a block code, consider a communication system with one transmitter and one receiver. Information is transmitted using certain set of code words. If the transmitter wants to change the code set, first thing to be done is to inform the receiver. Other wise the receiver will never be able to understand what is being transmitted. Thus, until and unless the receiver is informed about the changes made you are not permitted to change the code set. In this sense the code words we are seeking shall be always finite sequences of the code alphabet-they are fixed sequence codes . Example 6.1: Source alphabet is S = {s 1 , s 2 , s 3 , s 4 } , Code alphabet is X = {0, 1} and The Code words are: C = {0, 11, 10, 11} 2. Non singular codes: A block code is said to be non singular if all the words of the code set X 1 , are “distinct”. The codes given in Example 6.1 do not satisfy this property as the codes for s 2 and s 4 are not different. We can not distinguish the code words. If the codes are not distinguishable on a simple inspection we say the code set is “ singular in the small ”. We mo dify the code as below. Example 6.2: S = {s 1 , s 2 , s 3 , s 4 } , X = {0, 1} ; Codes, C = {0, 11, 10, 01} However, the codes given in Example 6.2 although appear to be non-singular, upon transmission would pose problems in decoding. For, if the transmitted sequence is 0011 , it might be interpreted as s 1 s 1 s 4 or s 2 s 4 . Thus there is an ambiguity about the code. No doubt, the code is non-singular in the small, but becomes Singular in the large ”. 3. Uniquely decodable codes: A non-singular code is uniquely decipherable, if every word immersed in a sequence of words can be uniquely identified. The n th extension of a code, that maps each message into the code words C , is defined as a code which maps the sequence of messages into a sequence of code words. This is also a block code, as illustrated in the following example. Example 6.3: Second extension of the code set given in Example 6.2. S 2 ={ s 1 s 1 ,s 1 s 2 ,s 1 s 3 ,s 1 s 4 ; s 2 s 1 ,s 2 s 2 ,s 2 s 3 ,s 2 s 4 ,s 3 s 1 ,s 3 s 2 ,s 3 s 3 ,s 3 s 4 ,s 4 s 1 ,s 4 s 2 ,s 4 s 3 ,s 4 s 4 } Source Codes Source Codes Source Codes Source Codes Symbols Symbols Symbols Symbols s 1 s 1 0 0 s 2 s 1 1 1 0 s 3 s 1 1 0 0 s 4 s 1 0 1 0 s 1 s 2 0 1 1 s 2 s 2 1 1 1 1 s 3 s 2
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