Since the ratio of the total good indices of BEC Z P is 1 Z P then the total

Since the ratio of the total good indices of bec z p

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Since the ratio of the total good indices of BEC( Z ( P )) is 1 Z ( P ), then the total error that we make by replacing P with BEC( Z ( P )) is at most Z ( P ) which in the above algorithm is less that the parameter δ . Now, for a fixed level n , according to Theorem 1 if we make k large enough, the ratio of the quantized sub-channels that their Bhattacharyya value is less that δ approaches to its original value (with no quantization), and for these sub-channels as explained above the total error made with the algorithm is δ . Now from the polarization theorem and by sending δ to zero we deduce that as k → ∞ the number of good indices approaches the capacity of the original channel. 3.5 Simulation Results In order to evaluate the performance of our quantization algorithm, we compare the perfor- mance of the degraded quantized channel with the performance of an upgraded quantized channel. Similarly to Section 3.2 Algorithm 2, we introduce an algorithm which this time splits the masses between its two neighbors. To clarify, consider three neighbor masses in positions ( x i 1 , x i , x i +1 ) with probabilities ( p i 1 , p i , p i +1 ). Let t = x i x i - 1 x i +1 x i - 1 . Then, we split the middle mass at x i to the other two masses such that the final probabilities will be ( p i 1 + (1 t ) p i , p i +1 + tp i ) at positions ( x i 1 , x i +1 ). The greedy algorithm is shown in Algorithm 3. An upper bound on the error of this algorithm can be provided similarly to Section 3.3
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3.5. Simulation Results 31 Algorithm 3 Splitting Masses Algorithm 1: Start from the list ( p 1 , x 1 ) , · · · , ( p n , x n ). 2: Repeat n k times 3: Find j = argmin { p i ( f ( x i ) tf ( x i 1 ) (1 t ) f ( x i 1 )) : i negationslash = 1 , n } 4: Add (1 t ) p j to p j 1 and tp j to p j +1 . 5: Delete ( p j , x j ) from the list. with a little bit of modification. Consider the error made in each step of the algorithm: e i = p i ( f ( x i ) tf ( x i +1 ) (1 t ) f ( x i 1 )) (3.32) = tp i ( f ( x i +1 f ( x i )) + (1 t ) p i ( f ( x i ) f ( x i 1 )) (3.33) ≤ − tp i (1 t x i f ( x i +1 ) + (1 t ) p i t Δ x i f ( x i 1 ) (3.34) = p i t (1 t x 2 i | f ′′ ( c i ) | , (3.35) where x i 1 c i x i +1 and Δ x i defines x i +1 x i 1 . The difference with Section 3.3 is that now i Δ x i 1 (not 1 / 2). On the other hand, for 0 t 1 we have t (1 t ) 1 4 . Therefore, exactly the same results of Section 3.3 can be applied here, and the total error of the algorithm can be upper bounded by O parenleftBig log( k ) k k parenrightBig for the entropy function, and O ( 1 k 2 ) for functions that | f ′′ ( x ) | is bounded. In the simulations, we measure the maximum achievable rate while keeping the prob- ability of error less than 10 3 by finding maximum possible number of channels with the smallest Bhattacharyya parameters such that the sum of their Bhattacharyya pa- rameters is upper bounded by 10 3 . The channel is a binary symmetric channel with capacity 0 . 5. First, we compare 3 different functions f 1 ( x ) = h ( x ) (entropy function), f 2 ( x ) = 2 radicalbig x (1 x ) (Bhattacharrya function), and f 3 ( x ) = x (1 x
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