Answer to Question #222828 in Electrical Engineering for Jhony beristow

Question #222828

1) Develop an argument on the fact that Even parity (4,3) code is a linear block code whereas an Odd parity (4,3) is not based on the three basic conditions for linear block codes.

2) What is hamming distance? Write Cyclic codes of a (4,7) encoding scheme and calculate the minimum and maximum hamming distance.

3) Draw the look-up table, trellis diagram, tree diagram and state diagram for a (2,1,4) convolution encoding scheme.

Expert's answer

Part 1

All columns of H have odd parity, so the sum (exclusive-or) of an odd number of columns has odd parity and is therefore nonzero. Every code word selects a set of columns of H whose sum is zero, so code words must have an even number of nonzero components.

BLOCK CODE LINEAR The first k bits of a (n,k) linear block coding are always identical to the message sequence to be sent. The second component of (n-k ) bits is calculated from message bits using the encoding rule and is known as parity bits.

Part 2

The Hamming distance between two strings of similar length in information theory is the number of locations where the corresponding symbols differ.

The (7,4) Hamming code has a generator polynomial "{\\displaystyle g(x)=x^{3}+x+1}" . This polynomial has a zero in Galois extension field "{\\displaystyle GF(8)}"  at the primitive element "{\\displaystyle \\alpha }" , and all codewords satisfy "{\\displaystyle {\\mathcal {C}}(\\alpha )=0}" "{\\displaystyle GF(2)}" . Blocklength will be "{\\displaystyle n} equal to {\\displaystyle 2^{m}-1}"  and primitive elements "{\\displaystyle \\alpha } and {\\displaystyle \\alpha ^{3}}"  as zeros in the "{\\displaystyle GF(2^{m})}"  because we are considering the case of two errors here, so each will represent one error.

The received word is a polynomial of degree "{\\displaystyle n-1}"  given as "{\\displaystyle v(x)=a(x)g(x)+e(x)}"

where "{\\displaystyle e(x)}"  can have at most two nonzero coefficients corresponding to 2 errors.

We define the Syndrome Polynomial"{\\displaystyle S(x)}"  as the remainder of polynomial "{\\displaystyle v(x)}"  when divided by the generator polynomial "{\\displaystyle g(x)} i.e.\n\n{\\displaystyle S(x)\\equiv v(x)\\equiv (a(x)g(x)+e(x))\\equiv e(x)\\mod g(x)} as\\\\\n {\\displaystyle (a(x)g(x))\\equiv 0\\mod g(x)}."

Part 3

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