A Note on the Deletion Channel Capacity

Memoryless channels with deletion errors as defined by a stochastic channel matrix allowing for bit drop outs are considered in which transmitted bits are either independently deleted with probability $d$ or unchanged with probability $1-d$. Such channels are information stable, hence their Shannon capacity exists. However, computation of the channel capacity is formidable, and only some upper and lower bounds on the capacity exist. In this paper, we first show a simple result that the parallel concatenation of two different independent deletion channels with deletion probabilities $d_1$ and $d_2$, in which every input bit is either transmitted over the first channel with probability of $\lambda$ or over the second one with probability of $1-\lambda$, is nothing but another deletion channel with deletion probability of $d=\lambda d_1+(1-\lambda)d_2$. We then provide an upper bound on the concatenated deletion channel capacity $C(d)$ in terms of the weighted average of $C(d_1)$, $C(d_2)$ and the parameters of the three channels. An interesting consequence of this bound is that $C(\lambda d_1+(1-\lambda))\leq \lambda C(d_1)$ which enables us to provide an improved upper bound on the capacity of the i.i.d. deletion channels, i.e., $C(d)\leq 0.4143(1-d)$ for $d\geq 0.65$. This generalizes the asymptotic result by Dalai as it remains valid for all $d\geq 0.65$. Using the same approach we are also able to improve upon existing upper bounds on the capacity of the deletion/substitution channel.Comment: Submitted to the IEEE Transactions on Information Theor

An Upper Bound on the Capacity of non-Binary Deletion Channels

We derive an upper bound on the capacity of non-binary deletion channels. Although binary deletion channels have received significant attention over the years, and many upper and lower bounds on their capacity have been derived, such studies for the non-binary case are largely missing. The state of the art is the following: as a trivial upper bound, capacity of an erasure channel with the same input alphabet as the deletion channel can be used, and as a lower bound the results by Diggavi and Grossglauser are available. In this paper, we derive the first non-trivial non-binary deletion channel capacity upper bound and reduce the gap with the existing achievable rates. To derive the results we first prove an inequality between the capacity of a 2K-ary deletion channel with deletion probability $d$, denoted by $C_{2K}(d)$, and the capacity of the binary deletion channel with the same deletion probability, $C_2(d)$, that is, $C_{2K}(d)\leq C_2(d)+(1-d)\log(K)$. Then by employing some existing upper bounds on the capacity of the binary deletion channel, we obtain upper bounds on the capacity of the 2K-ary deletion channel. We illustrate via examples the use of the new bounds and discuss their asymptotic behavior as $d \rightarrow 0$.Comment: accepted for presentation in ISIT 201

Upper bounds on the capacity of deletion channels using channel fragmentation

Cataloged from PDF version of article.We study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop-outs or symbol insertions with particular emphasis on the binary and non-binary deletion channels. We offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels. The resulting output symbols are combined appropriately to come up with an equivalent input–output representation of the original channel which allows for derivation of new upper bounds on the channel capacity. We consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels. With two specific applications of this idea, a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel, we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels. The resulting inequalities prove useful in deriving tighter capacity upper bounds for: 1) independent identically distributed (i.i.d.) deletion channels when the deletion probability exceeds 0.65 and 2) nonbinary deletion channels. Some extensions of these results, for instance, to the case of deletion/substitution channels are also explored

Spectrally Effiecient Alamouti Code Structure in Asynchronous Cooperative Systems

Cataloged from PDF version of article.A relay communication system with two amplify and forward (AF) relays under flat fading channel conditions is considered where the signals received from the relays are not necessarily time aligned. We propose a new time-reversal (TR)-based scheme providing an Alamouti code structure which needs a smaller overhead in transmitting every pair of data blocks in comparison with the existing schemes and, as a result, increases the transmission rate significantly (as much as 20%) in exchange for a small performance loss. The scheme is particularly useful when the delay between the two relay signals is large, e.g., in typical underwater acoustic (UWA) channels

Achieving Delay Diversity in Asynchronous Underwater Acoustic (UWA) Cooperative Communication Systems

Cataloged from PDF version of article.In cooperative UWA systems, due to the low speed of sound, a node can experience significant time delays among the signals received from geographically separated nodes. One way to combat the asynchronism issues is to employ orthogonal frequency division multiplexing (OFDM)-based transmissions at the source node by preceding every OFDM block with an extremely long cyclic prefix (CP) which reduces the transmission rates dramatically. One may increase the OFDM block length accordingly to compensate for the rate loss which also degrades the performance due to the significantly time-varying nature of UWA channels. In this paper, we develop a new OFDM-based scheme to combat the asynchronism problem in cooperative UWA systems without adding a long CP (in the order of the long relative delays) at the transmitter. By adding a much more manageable (short) CP at the source, we obtain a delay diversity structure at the destination for effective processing and exploitation of spatial diversity by utilizing a low complexity Viterbi decoder at the destination, e.g., for a binary phase shift keying (BPSK) modulated system, we need a two-state Viterbi decoder. We provide pairwise error probability (PEP) analysis of the system for both time-invariant and block fading channels showing that the system achieves full spatial diversity. We find through extensive simulations that the proposed scheme offers a significantly improved error rate performance for time-varying channels (typical in UWA communications) compared to the existing approaches

Achievable Rates for Noisy Channels with Synchronization Errors

Cataloged from PDF version of article.We develop several lower bounds on the capacity of binary input symmetric output channels with synchronization errors, which also suffer from other types of impairments such as substitutions, erasures, additive white Gaussian noise (AWGN), etc. More precisely, we show that if a channel suffering from synchronization errors as well as other type of impairments can be decomposed into a cascade of two component channels where the first one is another channel with synchronization errors and the second one is a memoryless channel (with no synchronization errors), a lower bound on the capacity of the original channel in terms of the capacity of the component synchronization error channel can be derived. A primary application of our results is that we can employ any lower bound derived on the capacity of the component synchronization error channel to find lower bounds on the capacity of the (original) noisy channel with synchronization errors. We apply the general ideas to several specific classes of channels such as synchronization error channels with erasures and substitutions, with symmetric q-ary outputs and with AWGN explicitly, and obtain easy-to-compute bounds. We illustrate that, with our approach, it is possible to derive tighter capacity lower bounds compared to the currently available bounds in the literature for certain classes of channels, e.g., deletion/substitution channels and deletion/AWGN channels (for certain signal-to-noise ratio (SNR) ranges). © 2014 IEEE