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In this paper, multilevel coded asymmetric modulation with multistage decoding and unequal error protection (UEP) is discussed. These results further emphasize the fact that unconventional signal set partitionings are more promising than traditional (Ungerboeck-type) partitionings, to achieve UEP capabilities with multilevel coding and multistage decoding. Three types of unconventional partitionings are analyzed for asymmetric 8-PSK and 16-QAM constellations over the additive white Gaussian noise channel to introduce design guidelines. Generalizations to other PSK and QAM type constellations follow the same lines. Upper bounds on the bit-error probability based on union bound arguments are first derived. In some cases, these bounds become loose due to the large overlappings of decision regions associated with asymmetric constellations and unconventional partitionings. To overcome this problem, simpler and tighter approximated bounds are derived. Based on these bounds, it is shown that additional refinements can be achieved in the construction of multilevel UEP codes, by introducing asymmetries in PSK and QAM signal constellations. Index Terms-Asymmetric constellations, multilevel coded modulation, multistage decoding, unequal error protection, union bound. I. INTRODUCTION U NCONVENTIONAL signal set partitionings are a good approach to achieve unequal error protection (UEP) capabilities in multilevel coded modulation and multistage decoding [1], as shown in the companion paper [2]. In general,

2) in the case where i '" j when T = (pm + 1)/.for () SIS pm -2 occurs (p,,-l + pm _ 2p"'-') times occurs (pn-l _ 2pm-l) times.for l S �' ::;p-l. V. CONCLUSIONIt was shown that the new family consisting of pn/ 2 (where n is even) balanced n onbinary sequences with period pn -1 can be obtained from the modified Kumar-Moreno sequences of the same period, and the distribution of correlation values for the family was shown to have 1'+ 2 distinct correlation values and the same maximum nontrivial correlation value of 1',,/ 2 + 1 as that of Kumar-Moreno sequences. On the other hand, it was shown that the cost of making sequences balanced is a decrease of family size in addition to the condition that n is an even number. The family size of the new sequences is pn/2 which is much smaller than p", that of Kumar-Moreno sequences. VI. ACKNOWLEDGMENTThe authors wish to thank Prof. P. V. Kumar for sending the draft of their paper [3] to one of the authors (K.I.), and would like to thank the anonymous referees for helpful comments useful for improving the readahility of the paper. codes. An example is given which illustrates TSD decoding of QPSK block-modulation codes for UEP. Finally, in Section IV, conclusions on the results are presented. REFERENCES II. BASIC CONCEPTS OF LUEP CODESWhen a code is used to provide multiple levels of error protection, the conventional definition of minimum distance must be generalized. Since different levels of error protection are possible with a UEP code, a vector of minimum distances, one for each level of error protection, needs to be defined. Let C be an (n. h') block code (not necessarily linear) over a finite alphabet ..t, n 2: �'. That is, C is a one-to-one mapping from 11' to .-1", i.e. where ,-'As usual, an element m from ..t k is called a message, and an element c( fl.) from C is called a codeword, .-1' is known as the message seT. Let A k be decomposed into the direct product of two disjoint message subsets, .-1"', i = 1,2, such thatA message m E .1 k can then be expressed aswhere each m, is called the ith message part, i = 1, 2, The separation vector of C is defined as the two-tuple ii = (8,,82), where i = 1. 2 and rI(x, x ' ) denotes the Hamming distance between x and x ' in A", Note that in the definition of Si above, there is no restriction on mj, m�, for j # i. Assume that C has both components of its separation vector distinct and arranged in decreasing order, i.e., s, > S2. such that C is an (n.�·) block code of minimum distance S2. We call m I the most important message part and m 2 the least important message part.Code C is said to be an (II, 1;) two-level UEP code of separation vector ii = (81,8"), for the message set A k] X A k,. This correspon dence concentrates on hinary linear two-level error correcting codes.That is, .-1. = {O, I}. For a binary linear two-level error correcting code, or binary LUEP code C, each element of the separation vector is given bywhere wt (x) denotes the Hamming weight of vector x. C is called an (II,k) two-level LCEP code, of separatiun vec...