A Class of Transformations that Polarize Symmetric Binary-Input Memoryless Channels

arXiv:0811.1770v1 [cs.IT] 11 Nov 2008 A Class of Transformations that Polarize Symmetric Binary-Input Memoryless Channels Satish Babu Korada and Eren S¸a¸so˘glu October 24, 2018 Abstract A generalization of Arıkan’s polar code construction using transformations of the form G⊗n where G is an ℓ× ℓmatrix is considered. Necessary and sufficient conditions are given for these transformations to ensure channel polarization. It is shown that a large class of such transformations polarize symmetric binary-input memoryless channels. 1 Introduction Polar codes, introduced by Arıkan in [1], are the first provably capacity achieving codes for arbitrary symmetric binary-input discrete memoryless channels (B-DMC) with low encoding and decoding complexity. Polar code construction is based on the following observation: Let G2 =  1 0 1 1  . (1) Consider applying the transform G⊗n 2 (where “⊗n” denotes the nth Kronecker power) to a block of N = 2n bits and transmitting the output through independent copies of a B-DMC W (see Figure 1). As n grows large, the channels seen by individual bits (suitably defined in [1]) start polarizing: they approach either a noiseless channel or a pure-noise channel, where the fraction of channels becoming noiseless is close to the symmetric mutual information I(W). It was conjectured in [1] that polarization is a general phemonenon, and is not restricted to the particular transformation G⊗n 2 . In this note we give a partial affirmation to this conjecture. In particular, we consider transformations of the form G⊗n where G is an ℓ× ℓmatrix for ℓ≥3 and provide necessary and sufficient conditions for such Gs to polarize symmetric B-DMCs. 2 Preliminaries Let W : {0, 1} →Y be a B-DMC. Let I(W) ∈[0, 1] denote the mutual information between the input and output of W with uniform distribution on the inputs. Also let Z(W) ∈[0, 1] denote the Bhattacharyya parameter of W, i.e., Z(W) = P y∈Y p W(y|0)W(y|1). Fix an ℓ≥3 and an invertible ℓ× ℓ{0, 1} matrix G. Consider a random ℓ-vector U ℓ 1 that is uniformly distributed over {0, 1}ℓ. Let Xℓ 1 = U ℓ 1G, where the multiplication is performed over 1 W · · · W G⊗n bit1 bit2 · · · bitN Figure 1: GF(2). Also let Y ℓ 1 be the output of ℓuses of W with the input Xℓ 1. Observe now that the channel between U ℓ 1 and Y ℓ 1 is defined by the transition probabilities Wℓ(yℓ 1 | uℓ 1) ≜ ℓY i=1 W(yi | xi) = ℓ Y i=1 W(yi | (uℓ 1G)i). Define W (i) : {0, 1} →Yℓ× {0, 1}i−1 as the channel with input ui, output (yℓ 1, ui−1 1 ) and transition probabilities W (i)(yℓ 1, ui−1 1 | ui) = 1 2ℓ−1 X uℓ i+1 Wℓ(yℓ 1 | uℓ 1), and let Z(i) denote its Bhattacharyya parameter, i.e., Z(i) = X yℓ 1,ui−1 1 q W (i)(yℓ 1, ui−1 1 | 0)W (i)(yℓ 1, ui−1 1 | 1). For k ≥1, let W k : {0, 1} →Yk denote the B-DMC with transition probabilities W k(yk 1 | x) = k Y j=1 W(yj | x). Also let ˜W (i) : {0, 1} →Yℓdenote the B-DMC with transition probabilities ˜W (i)(yℓ 1 | ui) = 1 2ℓ−i X uℓ i+1 Wℓ(yℓ 1 | 0i−1 1 , uℓ i). (2) Observation 1. If W is symmetric, then the channels W (i) and ˜W (i) are equivalent in the sense that for any fixed ui−1 1 there exists a permutation πui−1 1 : Yℓ→Yℓsuch that W (i)(yℓ 1, ui−1 1 | ui) = 1 2i−1 ˜W (i)(πui−1 1 (yℓ 1) | ui). 2 Finally, let I(i) denote the mutual information between the input and output of channel W (i). Since G is invertible, it is easy to check that ℓ X i=1 I(i) = ℓI(W). 3 Polarization We will say that G is a polarizing matrix if there exists an i ∈{1, . . . , ℓ} for which ˜W (i) is equivalent to W k for some k ≥2, in the sense that ˜W (i)(yℓ 1 | ui) = c Y j∈A W(yj | ui) (3) for some constant c and A ⊆{1, . . . , ℓ} with |A| = k. If W is symmetric, then Observation 1 implies the equivalence of W (i) and W k (which we denote by W (i) ≡W k) in the sense that W (i)(yℓ 1, ui−1 1 | ui) = c 2i−1 Y j∈A W((πui−1 1 (yℓ 1))j | ui). (4) Note that the equivalence W (i) ≡W k implies I(i) = I(W k) and Z(i) = Z(W k). It will be shown that channel transformations of the form G⊗n polarize symmetric channels if and only if G is polarizing. This statement is made precise in the following theorem: Theorem 1. Fix a symmetric B-DMC W. Let G⊗n denote the nth Kronecker power of G and consider the transformation G⊗n : W →(W (i) : i = 1, . . . , ℓn). i. If G is polarizing, then for any δ > 0 lim n→∞ #  i ∈{1, . . . , ℓn} : I(W (i)) ∈(δ, 1 −δ)

ℓn = 0. ii. If G is not polarizing, then I(W (i)) = I(W) for all n and i ∈{1, . . . , ℓn}. Theorem 1 is a direct consequence of Lemmas 1 and 2 below. Note that any invertible {0, 1} matrix G can be written as a (real) sum G = P + P ′, where P is a permutation matrix, and P ′ is a {0, 1} matrix. This fact can be inferred from Hall’s Theorem [3, Theorem 16.4.]. Therefore, for any such matrix G, there exists a column permutation that results in Gii = 1 for all i. Since the transition probabilities defining W (i) are invariant (up to a permutation of the outputs yℓ

1. under column permutations on G, we only consider matrices with 1s on the diagonal. The following

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