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In the Huffman code, the bit sequences that represent individual symbols can have differing lengths so the bitstream index does not increase in lock step with the symbol-valued signal's index . To capture how often bits must be transmitted to keep up with the source's productionof symbols, we can only compute averages. If our source code averages bits/symbol and symbols are produced at a rate , the average bit rate equals , and this quantity determines the bit interval duration .
Calculate what the relation between and the average bit rate is.
.
A subtlety of source coding is whether we need "commas" in the bitstream. When we use an unequal number of bits to representsymbols, 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 sinceyou 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 anunequal 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 rightone. Note that we must start at the beginning of the bit stream; jumping into the middle does not guarantee perfectdecoding. 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 efficientsource coding algorithms.
Show by example that a bitstream produced by a Huffman code is not necessarily self-synchronizing. Arefixed-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 wehad 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 losteven if the receiver started "in synch" with the source. Despite the small probabilities of error offered by good signalset design and the matched filter, an infrequent error can devastate the ability to translate a bitstream into a symbolicsignal. We need ways of reducing reception errors without demanding that be smaller.
The first electrical communications system—the telegraph—was digital. When first deployed in 1844, it communicated text over wirelineconnections using a binary code—the Morse code—to represent individual letters. To send a message from one placeto another, telegraph operators would tap the message using a telegraph key to another operator, who would relay the messageon 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 anunderstatement. It was also far ahead of some needed technologies, namely the Source Coding Theorem. The Morsecode, shown in , was not a prefix code. To separate codes for each letter, Morsecode required that a space—a pause—be inserted between each letter. In information theory, that space countsas another code letter, which means that the Morse code encoded text with a three-letter source code: dots, dashes andspace. The resulting source code is not within a bit of entropy, and is grossly inefficient (about 25%). shows a Huffman code for English text, which as we know is efficient.
| % | Morse Code | Huffman Code | |
|---|---|---|---|
| A | 6.22 | .- | 1011 |
| B | 1.32 | -... | 010100 |
| C | 3.11 | -.-. | 10101 |
| D | 2.97 | -.. | 01011 |
| E | 10.53 | . | 001 |
| F | 1.68 | ..-. | 110001 |
| G | 1.65 | --. | 110000 |
| H | 3.63 | .... | 11001 |
| I | 6.14 | .. | 1001 |
| J | 0.06 | .--- | 01010111011 |
| K | 0.31 | -.- | 01010110 |
| L | 3.07 | .-.. | 10100 |
| M | 2.48 | -- | 00011 |
| N | 5.73 | -. | 0100 |
| O | 6.06 | --- | 1000 |
| P | 1.87 | .--. | 00000 |
| Q | 0.10 | --.- | 0101011100 |
| R | 5.87 | .-. | 0111 |
| S | 5.81 | ... | 0110 |
| T | 7.68 | - | 1101 |
| U | 2.27 | ..- | 00010 |
| V | 0.70 | ...- | 0101010 |
| W | 1.13 | .-- | 000011 |
| X | 0.25 | -..- | 010101111 |
| Y | 1.07 | -.-- | 000010 |
| Z | 0.06 | --.. | 0101011101011 |
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