Ahmed Badr

Streaming codes for channels with burst and isolated erasures

We study low-delay error correction codes for streaming recovery over a class of packet-erasure channels that introduce both burst-erasures and isolated erasures. We propose a simple, yet effective class of codes whose parameters can be tuned to obtain a tradeoff between the capability to correct burst and isolated erasures. Our construction generalizes previously proposed low-delay codes which are effective only against burst erasures. We establish an information theoretic upper bound on the capability of any code to simultaneously correct burst and isolated erasures and show that our proposed constructions meet the upper bound in some special cases. We discuss the operational significance of column-distance and column-span metrics and establish that the rate 1/2 codes discovered by Martinian and Sundberg [IT Trans. 2004] through a computer search indeed attain the optimal column-distance and column-span tradeoff. Numerical simulations over a Gilbert-Elliott channel model and a Fritchman model show significant performance gains over previously proposed low-delay codes and random linear codes for certain range of channel parameters.

Streaming Codes for Multicast Over Burst Erasure Channels

We study low-delay erasure correction codes in a real-time streaming setup. The encoder observes a stream of source packets and outputs the channel packets in a causal fashion, which are broadcast to two receivers over burst-erasure channels. Each receiver must decode the source packets sequentially with a deadline of T_i, while its channel can introduce an erasure burst of maximum length B_i, where i ∈ {1,2} and w.l.o.g. B₂ > B1. We study the associated capacity as a function of the burst lengths and decoding deadlines. We observe that the operation of the system can be divided into two main regimes. The so-called large-delay regime corresponds to the case when either T₁ ≥ B₂ or T₂ ≥ B₁ + B₂. We show that for these parameters, the optimal code is obtained through simple modifications of previously proposed single-user codes by Martinian et al. and the diversity embedded streaming codes proposed by Badr et al. When both T₁ <; B₂ and T₂ <; B₁ + B₂, the system is said to be in the low-delay regime. We propose a new code construction and establish its optimality when T₂ ≥ T₁ + B₁. In the case when T₂ <; T₁ + B₁, we establish upper and lower bounds on the capacity and characterize the exact capacity when either T₁ = B₁ or T₂ = B₂. Our upper bounds in the low-delay regime are based on novel information theoretic arguments that capture the tension between the decoding constraints at the two receivers.

Streaming Codes With Partial Recovery Over Channels With Burst and Isolated Erasures

We study forward error correction codes for low-delay, real-time streaming communication over packet erasure channels. Our encoder operates on a stream of source packets in a sequential fashion, and the decoder must output each packet in the source stream within a fixed delay. We consider a class of practical channel models with correlated erasures and introduce new “streaming codes” for efficient error correction over these channels. For our analysis, we propose a simplified class of erasure channels that introduce both burst and isolated erasures within the same decoding window. We demonstrate that the previously proposed streaming codes can lead to significant number of packet losses over such channels. Our proposed constructions involve a layered coding approach, where a burst-erasure code is first constructed, and additional layers of parity-checks are concatenated to recover from the isolated erasure patterns. We also introduce another construction that requires a significantly smaller field-size and decoding complexity, but incurs some performance loss. Numerical simulations over the Gilbert-Elliott and Fritchman channel models indicate that by addressing patterns involving both burst and isolated erasures within the same window, our proposed codes achieve significant gains over previously proposed streaming codes.

Convolutional Codes With Maximum Column Sum Rank for Network Streaming

The column Hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels. We introduce a metric known as the column sum rank that parallels the column Hamming distance when streaming over a network with link failures. We prove the rank analogues of several known column Hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory. Our construction involves finding a class of super-regular matrices that preserve this property after multiplication with non-singular block diagonal matrices in the ground field.

Multicast streaming codes (Mu-SCo) for burst erasure channels

Streaming codes sequentially encode the source stream and reproduce each source packet with a fixed delay. We study multicast streaming codes (MU-SCo) that simultaneously serve two users : one user, whose channel introduces a burst-erasure length of B1 and tolerates a delay of T1 and a second user, whose channel introduces a burst-erasure length of B2 and tolerates a delay of T2. We show that the streaming capacity intricately depends on the burst-delay parameters and provide explicit constructions that attain the capacity for a wide range of parameters. In particular we identify regimes where (a) avoiding interference between the parity checks of the two users is optimal (b) the weaker user treats parity checks of the stronger user as side information to decode part of the source stream and (c) the capacity is achieved by a single user code. Our results shed new insights into the role of delay on user ordering in broadcast channels.

Robust streaming erasure codes based on deterministic channel approximations

We study near optimal error correction codes for real-time communication. In our setup the encoder must operate on an incoming source stream in a sequential manner, and the decoder must reconstruct each source packet within a fixed playback deadline of T packets. The underlying channel is a packet erasure channel that can introduce both burst and isolated losses. We first consider a class of channels that in any window of length T +1 introduce either a single erasure burst of a given maximum length B, or a certain maximum number N of isolated erasures. We demonstrate that for a fixed rate and delay, there exists a tradeoff between the achievable values of B and N, and propose a family of codes that is near optimal with respect to this tradeoff. We also consider another class of channels that introduce both a burst and an isolated loss in each window of interest and develop the associated streaming codes. All our constructions are based on a layered design and provide significant improvements over baseline codes in simulations over the Gilbert-Elliott channel.

Diversity Embedded Streaming Erasure Codes (DE-SCo): Constructions and Optimality

Streaming erasure codes encode a source stream to guarantee that each source packet is recovered within a fixed delay at the receiver over a burst-erasure channel. This paper introduces diversity embedded streaming erasure codes (DE-SCo), that provide a flexible trade-off between the channel quality and receiver delay. When the channel conditions are good, the source stream is recovered with a low delay, whereas when the channel conditions are poor the source stream is still recovered, albeit with a larger delay. Information theoretic analysis of the underlying burst-erasure broadcast channel reveals that DE-SCo achieve the minimum possible delay for the weaker user, without sacrificing the performance of the stronger user. A larger class of multicast streaming erasure codes (MU-SCo) that achieve optimal tradeoff between rate, delay and erasure-burst length is also constructed.

Streaming codes for a double-link burst erasure channel

A sender and receiver are connected by two links, which both pass through a burst erasure channel. The channel induces an erasure burst of length B onto both links, but the bursts are separated by d time units. Source packets arrive at the sender, and are encoded with a streaming code such that the receiver can decode with a delay T. If source packet s[t] arrives at the sender at time t, then the receiver must be able to decode s[t] by time t+T from its received packets. Given the parameters B, T and d, we find the upper bound for the rate of the streaming code, and also discover codes that can operate at capacity for certain parameter values. The code constructions also internally make use of SCo codes. Finally, we find that by exploiting the dependence of the burst erasure locations on either link, we can achieve a higher rate than if we simply used single-link SCo codes on each link.

FEC for VoIP using dual-delay streaming codes

We introduce a new class of forward error correction (FEC) codes for VoIP communications which support different recovery delay depending on the channel conditions. Specifically, our proposed class of Dual-Delay (DD) codes can recover from challenging long bursts of losses with close to theoretical minimum delay so as to meet playback deadlines for recovered packets. They further improve conversational interactivity by achieving lower recovery delay during periods of random isolated losses. These DD codes are shown to achieve lower residual loss rates when compared to existing codes over a wide range of parameters of the Gilbert-Elliott channel. Experiments over real world packet traces further show performance gains of DD codes in terms of perceptually motivated ITU-T G.107 E-model.

Convolutional codes with maximum column sum rank for network streaming

The column Hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels. We introduce a metric known as the column sum rank that parallels the column Hamming distance when streaming over a network with link failures. We prove the rank analogues of several known column Hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory. Our construction involves finding a class of super-regular matrices that preserve this property after multiplication with non-singular block diagonal matrices in the ground field.