Assignment 1A Solution

In particular for any is a prex of however we can

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Unformatted text preview: 8©V¥£ p h  ¦ e PH W vuQIF  `  h ¦  ¡£ ( c £ GQ` W§ 7¥)wU )( h e 9 ' e PH QIF e ' p f t   ¦  ¡£ GQt  1¥6( ¦¨ 8©¦ £ p while the unconditional entropy is ¦  ¡£ ( ¦ 2!¥) 6( £ Therefore, knowing the previous character reduces entropy by a factor of f ¨ C©¦ £ ¢ fec' ¦  ¨ ¤gGWd&28¦ It follows that and ech ¦ " WiT%B&¦ by using the equations Problem 2 The unary code ( ) is uniquely decipherable but not prefix-free. In particular, for any , is a prefix of . However, we can easily decipher the code by counting the number of zeros between two consecutive ones. €£ „ ƒ 5…x 6u€  „£ GI…x ‚  h ¦ €£ QB§5yx 1 Problem 3 We have  ¤¨ ¥©¡ tb¦hbh`b  Wd§GiWaWe &¦ ¤¢ ¥£¡ h hh   BQh BQQh QBh   We decode the sequence using the following procedure: x=0.3076171875; // the message in decimal format\ low=0; // initial lower interval bound high=1; // initial upper interval bound n=4; // the number of characters in the code while (n>0) do // repeatedly narrow the interval around x { determine which of the following intervals contains x: (1): [low, 9/10*low+1/10*high), (2): [9/10*low+1/10*high, 7/10*low+3/10*high), (3): [7/10*low+3/10*high, high) ; output a,b,c for intervals (1),(2),(3) respectively; low=low boundary of the interval containing x; high=high boundary of the interval containing x; n--; } We implement this idea in Mathematica: Clear[Code] (* executes one step of the coding *) Code[low_, high_, x_] := {a, low, 9/10*low + 1/10*high} /; x &...
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