We follow the notation of Griffiths and Tavaré (1995) and use several operators to denote the changes of sample configuration: S is the shift operator that can be operated on a specific haplotype h k or the entire haplotype set T. Specifically, S h k represents the haplotype obtained by deleting the first mutation coordinate of haplotype h k.
We use the operator row to denote a row vector, col to denote a column vector, and diag to denote a diagonal matrix.
The operator tr is used to denote the trace of a square matrix.
where y, h, x, and r denote the N-dimensional row vectors of the received signals yi, the channel coefficients hi, the transmitted symbols xi, and the reliabilities ri within one codeword, respectively, ∥·∥ is the Euclidean norm of a vector, ⊙is the component-wise (Hadamard) product of vectors, and the binary operator '·' is used to denote the dot-product of vectors.
Notation: We denote by T the transpose and * the transpose complex conjugate operator, while we use * to denote convolution.
If F t) = F(f(t),g(t)), then we use F(A) to denote matrix operator of F(f(A), g(A)), it means that F(A) = F(f(A),g(A)), while F A,B) denotes the matrix function of two variables as it was well defined in some known articles.
Also, we use to denote the differential operator and use to denote the Hodge codifferential operator defined by on,.
E is used to denote the expectation operator.
Throughout this paper, ℰ stands for the mathematical expectation operator, ∥ x ∥ 2 is used to denote a vector norm defined by ∥ x ∥ 2 = ∑ i = 1 n x i 2, 'T' represents the transpose of a matrix or a vector, I n is an n-dimensional identical matrix.
| to denote the absolute value operator.
