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Unformatted text preview: ECE 5670 : Digital Communications
ECE department, Cornell University, Spring 2011 Homework 4 Solutions Instructor: Salman Avestimehr
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the noise, many entries of the transmitted codeword will be ﬂipped and a large amount of
redundancy will be required in order to ameliorate this effect. This will result in a low rate
of reliable communication. On the other hand, if the distance between constellation points
is much greater than the noise standard deviation, few if any of the codewords entries will
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of ﬂipped codeWord entries, by increasing the constellation size. Thus in general we should
space the constellation points at a distance roughly equal to the standard deviation of
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'PU‘H'I‘j HI”: 716641;?" W'M‘ ﬂit GHQvi" 110 (E): we ram ("anti/("C 'tH'lk’l “ME Cuff<91} ‘31 Em ivc {Lilith‘5’ ,) ; L F. Q3) 3. (a) A simple and efﬁcient way of storing the data is in an N X (D + 1) matrix. The ﬁrst 03) (c) (61) entry of the nth row stores the number of data bits used in the generation of the nth
coded symbol. The remaining D entries of the nth row store the indices of the data bits
that are added together modulo two, to form the nth coded symbol. Typically no data bits can be decoded given only 100 coded symbols. In order to be
able to decode at least one bit, we require a seed, that is a coded symbol formed from
only one data bit. The expected number of such coded symbols is about 0.8(1001). The
maximum possible number of bits that can be decoded from the 100 symbols is 100.
This upper bound is very loose. Typically only a few data bits, if any, can be decoded given 200 coded symbols. Now the
average number of seeds is 1.6, but the problem becomes one of sending another coded symbol that connects to the same seed symbol. The upper bound is now 200, but is still
very loose. See the following graph (note the code may take many hours to run). There is a sharp
transition at around N = 5000. The overhead needed to decode the 5000 data bits is
very low when N is slightly larger than 5000. When N is much larger than 5000, this
overhead becomes signiﬁcant. As the graph below suggests, many additional symbols
are typically required before the last few data bits can be decoded. Often in practice,
an outer code is used (typically a simple block code) to circumvent this. number of bits decoded 4500 3500 ~.n....m;_____;__
3000 ._HH,
2500  1500 ~v~~~ 1000 ~u 500 1000 2000 70600 4000 5000 6000 7000 3000 9000 10000
number of coded symbols received ...
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).
 Spring '11
 SCAGLIONE

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