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Summary: Some subtleties of coding, including self synchronization and a comparison of the Huffman and Morse codes.
In the Huffman code, the bit sequences that represent individual
symbols can have differing lengths so the bitstream index
Calculate what the relation between
A subtlety of source coding is whether we need "commas" in the bitstream. When we use an unequal number of bits to represent symbols, how does the receiver determine when symbols begin and end? If you created a source code that required a separation marker in the bitstream between symbols, it would be very inefficient since you are essentially requiring an extra symbol in the transmission stream.
Sketch an argument that prefix coding, whether derived from a Huffman code or not, will provide unique decoding when an unequal number of bits/symbol are used in the code.
Because no codeword begins with another's codeword, the first codeword encountered in a bit stream must be the right one. Note that we must start at the beginning of the bit stream; jumping into the middle does not guarantee perfect decoding. The end of one codeword and the beginning of another could be a codeword, and we would get lost.
However, having a prefix code does not guarantee total synchronization: After hopping into the middle of a bitstream, can we always find the correct symbol boundaries? The self-synchronization issue does mitigate the use of efficient source coding algorithms.
Show by example that a bitstream produced by a Huffman code is not necessarily self-synchronizing. Are fixed-length codes self synchronizing?
Consider the bitstream …0110111… taken from the bitstream 0|10|110|110|111|…. We would decode the initial part incorrectly, then would synchronize. If we had a fixed-length code (say 00,01,10,11), the situation is much worse. Jumping into the middle leads to no synchronization at all!
Another issue is bit errors induced by the digital channel; if
they occur (and they will), synchronization can easily be lost
even if the receiver started "in synch" with the source.
Despite the small probabilities of error offered by good signal
set design and the matched filter, an infrequent error can
devastate the ability to translate a bitstream into a symbolic
signal. We need ways of reducing reception errors
without demanding that
The first electrical communications system—the telegraph—was digital. When first deployed in 1844, it communicated text over wireline connections using a binary code—the Morse code—to represent individual letters. To send a message from one place to another, telegraph operators would tap the message using a telegraph key to another operator, who would relay the message on to the next operator, presumably getting the message closer to its destination. In short, the telegraph relied on a network not unlike the basics of modern computer networks. To say it presaged modern communications would be an understatement. It was also far ahead of some needed technologies, namely the Source Coding Theorem. The Morse code, shown in Figure 1, was not a prefix code. To separate codes for each letter, Morse code required that a space—a pause—be inserted between each letter. In information theory, that space counts as another code letter, which means that the Morse code encoded text with a three-letter source code: dots, dashes and space. The resulting source code is not within a bit of entropy, and is grossly inefficient (about 25%). Figure 1 shows a Huffman code for English text, which as we know is efficient.
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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.
Last edited by Don Johnson on Dec 5, 2007 11:02 pm -0600.
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