Now we use the result of Lemma 1 to provide bounds on the total error made in estimating the mutual information of a channel after n levels of operations (2.5) and (2.6). Theorem 1. Assume W is a symmetric B-DMC and using Algorithm 1 or 2 we quantize the channel W to a channel ˜ W . Taking k = n 2 is sufficient to give an approximation error that decays to zero. Proof. First notice that for any two symmetric B-DMCs W and V , doing the polarization operations (2.5) and (2.6), the following is true: ( I ( W − ) − I ( V − )) + ( I ( W + ) − I ( V + )) = 2( I ( W ) − I ( V )) (3.30) Let S n be the sum of the errors in approximating the mutual information of the N channels after n level of polarizations. Replacing V with ˜ W in (3.30) and using the result of Lemma 1, we get the following recursive relation for the sequence s n . s 0 ≤ 1 k s n ≤ 2 s n − 1 + 2 n k (3.31) Solving (3.31) explicitly, we have s n ≤ ( n +1)2 n k . Therefore, we conclude that after n levels of polarization the sum of the errors in approximating the mutual information of the 2 n channels is upper-bounded by O ( n 2 n k ). In particular, taking k = n 2 , one can say that the “average” approximation error of the 2 n channels at level n is upper-bounded by O ( 1 n ). Therefore, at least a fraction 1 − 1 √ n of the channels are distorted by at most 1 √ n i.e., except for a negligible fraction of the channels the error in approximating the mutual information decays to zero. The theorem above shows that the algorithm finds all the “good” channels except a neg- ligible fraction of 1 − 1 √ n . The computational complexity of construction is O ( k 2 N ) which for k = n 2 yields to an almost linear complexity in blocklength except a polylogarithmic factor of (log N ) 4 .
30 Polar Codes Construction 3.4 Exchange of Limits In this section, we show that there are admissible schemes such that as k → ∞ , the limit in (3.3) approaches I ( W ) for any BMS channel W . In contrast to the previous section, k is a constant that grows, unlike k = n 2 . We use definition stated in equation (3.2) for the admissibility of the quantization procedure. Theorem 2. Given a BMS channel W and for large enough k , there exist admissible quantization schemes Q k such that ρ ( Q k , W ) is arbitrarily close to I ( W ) . Proof. Consider the following algorithm: The algorithm starts with a quantized version of W and it does the normal channel splitting transformation followed by quantization according to Algorithms 1 or 2, but once a sub-channel is sufficiently good, in the sense that its Bhattacharyya parameter is less than an appropriately chosen parameter δ , the algorithm replaces the sub-channel with a binary erasure channel which is degraded (polar degradation) with respect to it (As the operations (2.5) and (2.6) over an erasure channel also yields and erasure channel, no further quantization is need for the children of this sub-channel).
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- Winter '18
- Gaurav Beniwal
- Information Theory, polar codes