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### kiyavash_summery

Course: ECE 459, Fall 2008
School: University of Illinois,...
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Word Count: 935

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Codes Space-time and Fixed Point Free Groups Negar Kiyavash Dec 9, 2002 Abstract Space-time code constellations with low encoding and decoding complexity rely on nding a set of unitary matrices for which each pairwise dierence matrix has a nonzero determinant. The paper [1] explores the connection between the x-point free groups and their corresponding representations. It also tries to classify all such groups....

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Codes Space-time and Fixed Point Free Groups Negar Kiyavash Dec 9, 2002 Abstract Space-time code constellations with low encoding and decoding complexity rely on nding a set of unitary matrices for which each pairwise dierence matrix has a nonzero determinant. The paper [1] explores the connection between the x-point free groups and their corresponding representations. It also tries to classify all such groups. Further more some non-group extensions of the these groups are investigated. The design problem for unitary space-time code is the following: given M transmitter antennas and desired transmission rate R (in bits/channel-use), we want to select a set of L = 2RM unitary M M matrices such that for all A, B the quantity |det(A B)| is as large as possible. If all the pairs A, B have |det(A B)| > 0 then has full diversity. This paper [1] classies all matrices that achieve full diversity and form a group. The classication closely follows that of Zassenhaus [2]. Two main reasons for exploiting the group property are: Potential for better performance: If is not a group, |det(A B)| can have larger distance. In dierential space-time code modulation, the transmitted signal matrix is formed from multiplying the matrices in . If the matrices form a group, this becomes a simple table lookup. L(L1) 2 distinct values. However, for a group it can only have L 1 values, yielding a possibly 1 As stated earlier, we aim to maximize C = 1 minI=V |det(I V )| 2 , but for C (G) to be 2 non-zero the dierence determinant should have no eigenvalues at unity. This leads us to xed-point free groups and their representation. An M -dimensional representation of a group G is a group homomorphism () from G to group GLM (C). If a group has representation without an eigenvalue at unity, then it is called xed-point free. Theorem 1 [1] classies all nite xed-point free groups. Theorem 1 A nite group is xed-point free if and only if it is isomorphic to either G m,r , Dm,r,l , Em,r , Fm,r,l , Jm,r , or Km,r,l . The general format of each of these groups is given [1]. As an example, the group Gm,r is given: Gm,r =< , | m = 1, n = t , 1 = r > n is order of r (mod m), t = m . gcd(r1,m) 1 (1) The next challenge is to nd the corresponding xed-point free representations of these groups. Theorem 2 [1] states the representations for the above groups. It is not included due to its length. It is known that [3] cyclic groups have a xed-point free representation. It can be observed that cyclic groups are a specic form of group Gm,r , where n equals one. A cyclic group is known to be abelian and every abelian group has a one dimensional representation called its character. The characters are u ( l ) = e2iul/L for u = 0, 1, 2, ..., L 1, which are the roots of unity. Then, an M -dimensional representation of g is: u1 ( ) 0 . . . 0 l 0 u2 ( l ) . . . . . . 0 0 0 . . . uM ( l ) (2) If we look at some other simple constellation, we observe that the smallest group G m,r with xed-point free representation is G6,1 . The constellation is then 12 matrices As B k , 2 s = 0, ..., 5, k = 0, 1: A= and = e2i/6 . The smallest Dm,1,l is the quaternion group Q2 = D4,1,1 given by i 0 0 1 0 0 ,B = 0 1 (3) 1 0 P = 0 i ,Q = 1 0 . (4) The smallest Em,1 is the group E3,1 , which is isomorphic to SL(F3 ) and the constellation is given by matrices As P j Qp where s = 0, 1, 2, j = 0, ..., 3, p = 0, 1 and e 10i/8 A= 1 1 2 i 1 ,P = i 0 0 i ,Q = 0 1 (5) 1 0 Furthermore, the paper investigate some non-group structures that are inspired by groups. The non-group generalization of Gm,r is done by simply arbitrarily choosing n. For group Gm,r , n is the order of r (mod m). Then, by choosing and which are the respective sth and mth roots of unity, the set Sm,s of matrices Al B k with...

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University of Illinois, Urbana Champaign - ECE - 2002
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University of Illinois, Urbana Champaign - ECE - 459
On the Performance of Low Density Parity Check CodesXiangyu TangOutline Brief introduction of low density parity check codes Performance under optimal decoding Conclusion1Low Density Parity Check codes (LDPC codes) -Specified by matrix c
University of Illinois, Urbana Champaign - ECE - 461
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University of Illinois, Urbana Champaign - ECE - 567
Problem Set 3 ECE567 Each problem is worth 10 points Spring 2005 Due: March 17 The objective of this homework is to test your knowledge of reversibility and queueing networks, and basic large deviations techniques. Do all six problems. Note: There
University of Illinois, Urbana Champaign - ECE - 567
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University of Illinois, Urbana Champaign - ECE - 567
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University of Illinois, Urbana Champaign - ECE - 461
University of Illinois, Urbana Champaign - ECE - 461
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University of Illinois, Urbana Champaign - GE - 288
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University of Illinois, Urbana Champaign - ECE - 2002
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Washington - UWCH - 4
HOMER: Odyssey, trans. Samuel Butler From Book I:Tell me, O muse, of that ingenious hero who travelled far and wide after he had sacked the famous town of Troy. Many cities did he visit, and many were the nations with whose manners and customs he wa
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