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Vincent Y. F. Tan - One of the best experts on this subject based on the ideXlab platform.
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Scaling Exponent and moderate deviations asymptotics of polar codes for the awgn channel
Entropy, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent μ of polar codes for a memoryless channel q Y | X with capacity I ( q Y | X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed e ∈ ( 0 , 1 ) , the gap between the capacity I ( q Y | X ) and the maximum non-asymptotic rate R n * achieved by a length-n polar code with average error probability e scales as n - 1 / μ , i.e., I ( q Y | X ) - R n * = Θ ( n - 1 / μ ) . It is well known that the Scaling Exponent μ for any binary-input memoryless channel (BMC) with I ( q Y | X ) ∈ ( 0 , 1 ) is bounded above by 4 . 714 . Our main result shows that 4 . 714 remains a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O ( n - 1 / μ log n ) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O ( n - 1 / μ log n ) by using a superposition of log n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.
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Scaling Exponent and moderate deviations asymptotics of polar codes for the awgn channel
arXiv: Information Theory, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent $\mu$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/\mu}$, i.e., $I(q_{Y|X})-R_n^* = \Theta(n^{-1/\mu})$. It is well known that the Scaling Exponent $\mu$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/\mu}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/\mu}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.
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Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel
MDPI AG, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent μ of polar codes for a memoryless channel q Y | X with capacity I ( q Y | X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed ε ∈ ( 0 , 1 ) , the gap between the capacity I ( q Y | X ) and the maximum non-asymptotic rate R n * achieved by a length-n polar code with average error probability ε scales as n - 1 / μ , i.e., I ( q Y | X ) - R n * = Θ ( n - 1 / μ ) . It is well known that the Scaling Exponent μ for any binary-input memoryless channel (BMC) with I ( q Y | X ) ∈ ( 0 , 1 ) is bounded above by 4 . 714 . Our main result shows that 4 . 714 remains a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O ( n - 1 / μ log n ) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O ( n - 1 / μ log n ) by using a superposition of log n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel
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on the Scaling Exponent of polar codes for binary input energy harvesting channels
IEEE Journal on Selected Areas in Communications, 2016Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates the Scaling Exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of independent and identically distributed random variables with finite variances. The Scaling Exponent $\mu $ of polar codes for a binary-input memoryless channel (BMC) $q_{Y|X}$ with capacity $\mathrm {C}(q_{Y|X})$ characterizes the closest gap between the capacity and the non-asymptotic achievable rates in the following way. For a fixed average error probability $\varepsilon \in (0, 1)$ , the closest gap between the capacity $\mathrm {C}(q_{Y|X})$ and a non-asymptotic achievable rate $R_{n}$ for a length- $n$ polar code scales as $n^{-1/\mu }$ , i.e., $\min \{|\mathrm {C}(q_{Y|X})-R_{n}|\} = \Theta (n^{-1/\mu })$ . It has been shown that the Scaling Exponent $\mu $ for any binary-input memoryless symmetric channel with $\mathrm {C}(q_{Y|X})\in (0,1)$ lies between 3.579 and 4.714, where the upper bound 4.714 was shown by an explicit construction of polar codes. Our main result shows that 4.714 remains to be a valid upper bound on the Scaling Exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. An auxiliary contribution of this paper is that the upper bound on $\mu $ holds for binary-input memoryless asymmetric channels.
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On the Scaling Exponent of Polar Codes for Binary-Input Energy-Harvesting Channels
IEEE Journal on Selected Areas in Communications, 2016Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates the Scaling Exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. The Scaling Exponent $\mu$ of polar codes for a binary-input memoryless channel (BMC) characterizes the closest gap between the capacity and non-asymptotic rates achieved by polar codes with error probabilities no larger than some non-vanishing $\varepsilon\in(0,1)$. It has been shown that for any $\varepsilon\in(0,1)$, the Scaling Exponent $\mu$ for any binary-input memoryless symmetric channel (BMSC) with $I(q_{Y|X})\in(0,1)$ lies between 3.579 and 4.714 , where the upper bound $4.714$ was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the Scaling Exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. The main result is proved by leveraging the following three existing results: Scaling Exponent analyses for BMSCs, construction of polar codes designed for binary-input memoryless asymmetric channels, and the save-and-transmit strategy for EH channels.
Silas L. Fong - One of the best experts on this subject based on the ideXlab platform.
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Scaling Exponent and moderate deviations asymptotics of polar codes for the awgn channel
Entropy, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent μ of polar codes for a memoryless channel q Y | X with capacity I ( q Y | X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed e ∈ ( 0 , 1 ) , the gap between the capacity I ( q Y | X ) and the maximum non-asymptotic rate R n * achieved by a length-n polar code with average error probability e scales as n - 1 / μ , i.e., I ( q Y | X ) - R n * = Θ ( n - 1 / μ ) . It is well known that the Scaling Exponent μ for any binary-input memoryless channel (BMC) with I ( q Y | X ) ∈ ( 0 , 1 ) is bounded above by 4 . 714 . Our main result shows that 4 . 714 remains a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O ( n - 1 / μ log n ) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O ( n - 1 / μ log n ) by using a superposition of log n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.
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Scaling Exponent and moderate deviations asymptotics of polar codes for the awgn channel
arXiv: Information Theory, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent $\mu$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/\mu}$, i.e., $I(q_{Y|X})-R_n^* = \Theta(n^{-1/\mu})$. It is well known that the Scaling Exponent $\mu$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/\mu}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/\mu}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.
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Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel
MDPI AG, 2017Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The Scaling Exponent μ of polar codes for a memoryless channel q Y | X with capacity I ( q Y | X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed ε ∈ ( 0 , 1 ) , the gap between the capacity I ( q Y | X ) and the maximum non-asymptotic rate R n * achieved by a length-n polar code with average error probability ε scales as n - 1 / μ , i.e., I ( q Y | X ) - R n * = Θ ( n - 1 / μ ) . It is well known that the Scaling Exponent μ for any binary-input memoryless channel (BMC) with I ( q Y | X ) ∈ ( 0 , 1 ) is bounded above by 4 . 714 . Our main result shows that 4 . 714 remains a valid upper bound on the Scaling Exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O ( n - 1 / μ log n ) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O ( n - 1 / μ log n ) by using a superposition of log n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel
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on the Scaling Exponent of polar codes for binary input energy harvesting channels
IEEE Journal on Selected Areas in Communications, 2016Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates the Scaling Exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of independent and identically distributed random variables with finite variances. The Scaling Exponent $\mu $ of polar codes for a binary-input memoryless channel (BMC) $q_{Y|X}$ with capacity $\mathrm {C}(q_{Y|X})$ characterizes the closest gap between the capacity and the non-asymptotic achievable rates in the following way. For a fixed average error probability $\varepsilon \in (0, 1)$ , the closest gap between the capacity $\mathrm {C}(q_{Y|X})$ and a non-asymptotic achievable rate $R_{n}$ for a length- $n$ polar code scales as $n^{-1/\mu }$ , i.e., $\min \{|\mathrm {C}(q_{Y|X})-R_{n}|\} = \Theta (n^{-1/\mu })$ . It has been shown that the Scaling Exponent $\mu $ for any binary-input memoryless symmetric channel with $\mathrm {C}(q_{Y|X})\in (0,1)$ lies between 3.579 and 4.714, where the upper bound 4.714 was shown by an explicit construction of polar codes. Our main result shows that 4.714 remains to be a valid upper bound on the Scaling Exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. An auxiliary contribution of this paper is that the upper bound on $\mu $ holds for binary-input memoryless asymmetric channels.
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On the Scaling Exponent of Polar Codes for Binary-Input Energy-Harvesting Channels
IEEE Journal on Selected Areas in Communications, 2016Co-Authors: Silas L. Fong, Vincent Y. F. TanAbstract:This paper investigates the Scaling Exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. The Scaling Exponent $\mu$ of polar codes for a binary-input memoryless channel (BMC) characterizes the closest gap between the capacity and non-asymptotic rates achieved by polar codes with error probabilities no larger than some non-vanishing $\varepsilon\in(0,1)$. It has been shown that for any $\varepsilon\in(0,1)$, the Scaling Exponent $\mu$ for any binary-input memoryless symmetric channel (BMSC) with $I(q_{Y|X})\in(0,1)$ lies between 3.579 and 4.714 , where the upper bound $4.714$ was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the Scaling Exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. The main result is proved by leveraging the following three existing results: Scaling Exponent analyses for BMSCs, construction of polar codes designed for binary-input memoryless asymmetric channels, and the save-and-transmit strategy for EH channels.
Rudiger Urbanke - One of the best experts on this subject based on the ideXlab platform.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
IEEE Transactions on Information Theory, 2016Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_{\mathrm{ e}}$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$ , $N$ , $P_{\mathrm{ e}}$ , and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$ . In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R are fixed, and it was proved that the error probability $P_{\mathrm{ e}}$ scales roughly as $2^{-\sqrt {N}}$ . In the Scaling Exponent approach, the channel $W$ and the error probability $P_{\mathrm{ e}}$ are fixed and it was proved that the gap to capacity $I(W)-R$ scales as $N^{-1/\mu }$ . Here, $\mu $ is called Scaling Exponent and this Scaling Exponent depends on the channel $W$ . A heuristic computation for the binary erasure channel (BEC) gives $\mu =3.627$ and it was shown that, for any channel $W$ , $3.579 \le \mu \le 5.702$ . Our contributions are as follows. First, we provide the tighter upper bound $\mu \le 4.714$ valid for any $W$ . With the same technique, we obtain the upper bound $\mu \le 3.639$ for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the Scaling Exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity $I(W)-R$ and the error probability $P_{\mathrm{ e}}$ as the functions of the block length $N$ . In other words, we neither fix the gap to capacity (error Exponent regime) nor the error probability (Scaling Exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length $N$ , simultaneously go to 0. Third, we prove that polar codes are not affected by error floors . To do so, we fix a polar code of block length $N$ and rate $R$ . Then, we vary the channel $W$ and study the impact of this variation on the error probability. We show that the error probability $P_{\mathrm{ e}}$ scales as the Bhattacharyya parameter $Z(W)$ raised to a power that scales roughly like ${\sqrt {N}}$ . This agrees with the Scaling in the error Exponent regime.
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Scaling Exponent of list decoders with applications to polar codes
IEEE Transactions on Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their Scaling Exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under $\rm MAP$ decoding with list size $L$ for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under $\rm MAP$ decoding, although the introduction of a list can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the Scaling Exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the Scaling Exponent of the latter remains open.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
International Symposium on Information Theory, 2015Co-Authors: Marco Mondelli, Rudiger Urbanke, Hamed S HassaniAbstract:Consider transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W with capacity I(W) and Bhattacharyya parameter Z(W) and let P e be the error probability under successive cancellation decoding. Recall that in the error Exponent regime, the channel W and R e scales roughly as 2−√N. In the Scaling Exponent regime, the channel W and P e are fixed, while the gap to capacity I(W) − R scales as N−1/μ, with 3.579 ≤ μ ≤ 5.702 for any W. We develop a unified framework to characterize the relationship between R, N, P e , and W. First, we provide the tighter upper bound μ ≤ 4.714, valid for any W. Furthermore, when W is a binary erasure channel, we obtain an upper bound approaching very closely the value which was previously derived in a heuristic manner. Secondly, we consider a moderate deviations regime and we study how fast both the gap to capacity I(W) − R and the error probability P e simultaneously go to 0 as N goes large. Thirdly, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N and rate R, we let the channel W vary, and we show that P e scales roughly as Z(W)√N.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
arXiv: Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_e$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$, $N$, $P_e$, and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$. In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R
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Scaling Exponent of list decoders with applications to polar codes
Information Theory Workshop, 2013Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience when they are decoded with successive cancellation list decoders, we study how the Scaling Exponent changes as a function of the list size L. In particular, we fix the block error probability Pe and we analyze the tradeoff between the blocklength N and the back-off from capacity C-R using Scaling laws. By means of a Divide and Intersect procedure, we provide a lower bound on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the blocklength grows large. We show that, although list decoding can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected. This result applies in particular to polar codes, since their minimum distance tends to infinity as N increases. Some considerations are also pointed out for the genie-aided successive cancellation decoder when transmission takes place over the binary erasure channel.
Marco Mondelli - One of the best experts on this subject based on the ideXlab platform.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
IEEE Transactions on Information Theory, 2016Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_{\mathrm{ e}}$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$ , $N$ , $P_{\mathrm{ e}}$ , and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$ . In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R are fixed, and it was proved that the error probability $P_{\mathrm{ e}}$ scales roughly as $2^{-\sqrt {N}}$ . In the Scaling Exponent approach, the channel $W$ and the error probability $P_{\mathrm{ e}}$ are fixed and it was proved that the gap to capacity $I(W)-R$ scales as $N^{-1/\mu }$ . Here, $\mu $ is called Scaling Exponent and this Scaling Exponent depends on the channel $W$ . A heuristic computation for the binary erasure channel (BEC) gives $\mu =3.627$ and it was shown that, for any channel $W$ , $3.579 \le \mu \le 5.702$ . Our contributions are as follows. First, we provide the tighter upper bound $\mu \le 4.714$ valid for any $W$ . With the same technique, we obtain the upper bound $\mu \le 3.639$ for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the Scaling Exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity $I(W)-R$ and the error probability $P_{\mathrm{ e}}$ as the functions of the block length $N$ . In other words, we neither fix the gap to capacity (error Exponent regime) nor the error probability (Scaling Exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length $N$ , simultaneously go to 0. Third, we prove that polar codes are not affected by error floors . To do so, we fix a polar code of block length $N$ and rate $R$ . Then, we vary the channel $W$ and study the impact of this variation on the error probability. We show that the error probability $P_{\mathrm{ e}}$ scales as the Bhattacharyya parameter $Z(W)$ raised to a power that scales roughly like ${\sqrt {N}}$ . This agrees with the Scaling in the error Exponent regime.
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Scaling Exponent of list decoders with applications to polar codes
IEEE Transactions on Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their Scaling Exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under $\rm MAP$ decoding with list size $L$ for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under $\rm MAP$ decoding, although the introduction of a list can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the Scaling Exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the Scaling Exponent of the latter remains open.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
International Symposium on Information Theory, 2015Co-Authors: Marco Mondelli, Rudiger Urbanke, Hamed S HassaniAbstract:Consider transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W with capacity I(W) and Bhattacharyya parameter Z(W) and let P e be the error probability under successive cancellation decoding. Recall that in the error Exponent regime, the channel W and R e scales roughly as 2−√N. In the Scaling Exponent regime, the channel W and P e are fixed, while the gap to capacity I(W) − R scales as N−1/μ, with 3.579 ≤ μ ≤ 5.702 for any W. We develop a unified framework to characterize the relationship between R, N, P e , and W. First, we provide the tighter upper bound μ ≤ 4.714, valid for any W. Furthermore, when W is a binary erasure channel, we obtain an upper bound approaching very closely the value which was previously derived in a heuristic manner. Secondly, we consider a moderate deviations regime and we study how fast both the gap to capacity I(W) − R and the error probability P e simultaneously go to 0 as N goes large. Thirdly, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N and rate R, we let the channel W vary, and we show that P e scales roughly as Z(W)√N.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
arXiv: Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_e$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$, $N$, $P_e$, and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$. In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R
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Scaling Exponent of list decoders with applications to polar codes
Information Theory Workshop, 2013Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience when they are decoded with successive cancellation list decoders, we study how the Scaling Exponent changes as a function of the list size L. In particular, we fix the block error probability Pe and we analyze the tradeoff between the blocklength N and the back-off from capacity C-R using Scaling laws. By means of a Divide and Intersect procedure, we provide a lower bound on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the blocklength grows large. We show that, although list decoding can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected. This result applies in particular to polar codes, since their minimum distance tends to infinity as N increases. Some considerations are also pointed out for the genie-aided successive cancellation decoder when transmission takes place over the binary erasure channel.
Hamed S Hassani - One of the best experts on this subject based on the ideXlab platform.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
IEEE Transactions on Information Theory, 2016Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_{\mathrm{ e}}$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$ , $N$ , $P_{\mathrm{ e}}$ , and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$ . In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R are fixed, and it was proved that the error probability $P_{\mathrm{ e}}$ scales roughly as $2^{-\sqrt {N}}$ . In the Scaling Exponent approach, the channel $W$ and the error probability $P_{\mathrm{ e}}$ are fixed and it was proved that the gap to capacity $I(W)-R$ scales as $N^{-1/\mu }$ . Here, $\mu $ is called Scaling Exponent and this Scaling Exponent depends on the channel $W$ . A heuristic computation for the binary erasure channel (BEC) gives $\mu =3.627$ and it was shown that, for any channel $W$ , $3.579 \le \mu \le 5.702$ . Our contributions are as follows. First, we provide the tighter upper bound $\mu \le 4.714$ valid for any $W$ . With the same technique, we obtain the upper bound $\mu \le 3.639$ for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the Scaling Exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity $I(W)-R$ and the error probability $P_{\mathrm{ e}}$ as the functions of the block length $N$ . In other words, we neither fix the gap to capacity (error Exponent regime) nor the error probability (Scaling Exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length $N$ , simultaneously go to 0. Third, we prove that polar codes are not affected by error floors . To do so, we fix a polar code of block length $N$ and rate $R$ . Then, we vary the channel $W$ and study the impact of this variation on the error probability. We show that the error probability $P_{\mathrm{ e}}$ scales as the Bhattacharyya parameter $Z(W)$ raised to a power that scales roughly like ${\sqrt {N}}$ . This agrees with the Scaling in the error Exponent regime.
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Scaling Exponent of list decoders with applications to polar codes
IEEE Transactions on Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their Scaling Exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under $\rm MAP$ decoding with list size $L$ for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under $\rm MAP$ decoding, although the introduction of a list can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the Scaling Exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the Scaling Exponent of the latter remains open.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
International Symposium on Information Theory, 2015Co-Authors: Marco Mondelli, Rudiger Urbanke, Hamed S HassaniAbstract:Consider transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W with capacity I(W) and Bhattacharyya parameter Z(W) and let P e be the error probability under successive cancellation decoding. Recall that in the error Exponent regime, the channel W and R e scales roughly as 2−√N. In the Scaling Exponent regime, the channel W and P e are fixed, while the gap to capacity I(W) − R scales as N−1/μ, with 3.579 ≤ μ ≤ 5.702 for any W. We develop a unified framework to characterize the relationship between R, N, P e , and W. First, we provide the tighter upper bound μ ≤ 4.714, valid for any W. Furthermore, when W is a binary erasure channel, we obtain an upper bound approaching very closely the value which was previously derived in a heuristic manner. Secondly, we consider a moderate deviations regime and we study how fast both the gap to capacity I(W) − R and the error probability P e simultaneously go to 0 as N goes large. Thirdly, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N and rate R, we let the channel W vary, and we show that P e scales roughly as Z(W)√N.
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unified Scaling of polar codes error Exponent Scaling Exponent moderate deviations and error floors
arXiv: Information Theory, 2015Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Consider the transmission of a polar code of block length $N$ and rate $R$ over a binary memoryless symmetric channel $W$ and let $P_e$ be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters $R$, $N$, $P_e$, and the quality of the channel $W$ quantified by its capacity $I(W)$ and its Bhattacharyya parameter $Z(W)$. In previous work, two main regimes were studied. In the error Exponent regime, the channel $W$ and the rate $R
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Scaling Exponent of list decoders with applications to polar codes
Information Theory Workshop, 2013Co-Authors: Marco Mondelli, Hamed S Hassani, Rudiger UrbankeAbstract:Motivated by the significant performance gains which polar codes experience when they are decoded with successive cancellation list decoders, we study how the Scaling Exponent changes as a function of the list size L. In particular, we fix the block error probability Pe and we analyze the tradeoff between the blocklength N and the back-off from capacity C-R using Scaling laws. By means of a Divide and Intersect procedure, we provide a lower bound on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the blocklength grows large. We show that, although list decoding can significantly improve the involved constants, the Scaling Exponent itself, i.e., the speed at which capacity is approached, stays unaffected. This result applies in particular to polar codes, since their minimum distance tends to infinity as N increases. Some considerations are also pointed out for the genie-aided successive cancellation decoder when transmission takes place over the binary erasure channel.
