Kapil Bhattad

Minimal network coding for multicast

We give an information flow interpretation for multicasting using network coding. This generalizes the fluid model used to represent flows to a single receiver. Using the generalized model, we present a decentralized algorithm to minimize the number of packets that undergo network coding. We also propose a decentralized algorithm to construct capacity achieving multicast codes when the processing at some nodes is restricted to routing. The proposed algorithms can be coupled with existing decentralized schemes to achieve minimum cost multicast

Evolution of reference signals for LTE-advanced systems

3GPP LTE Release 10 standards (also known as LTE-Advanced) adopted some of the state-of-the-art radio access technologies that include carrier aggregation, eight-layer downlink spatial multiplexing, and four-layer uplink spatial multiplexing. For facilitating these enhancements, reference signals have significantly evolved in LTE-Advanced. This article examines underlying design principles of the LTE-Advanced reference signals. Specifically, newly introduced dedicated demodulation reference signals and channel state information reference signals for downlink and improvements of demodulation reference signals and sounding reference signals in uplink are discussed.

On the Distortion SNR Exponent of Some Layered Transmission Schemes

We consider the problem of joint source-channel coding for transmitting K samples of a complex Gaussian source overT bK uses of a block-fading multiple-input multiple-output (MIMO) channel with M transmit and N receive antennas. We consider the case when we are allowed to code over L blocks. The channel gain is assumed to be constant over a block and channel gains for different blocks are assumed to be independent. The performance measure of interest is the rate of decay of the expected mean-squared error with the signal-to-noise ratio (SNR), called the distortion SNR exponent. We first show that using a broadcast strategy similar to that of Gunduz and Erkip, but with a different power and rate allocation policy, the optimal distortion SNR exponent can be achieved for 0 les b les (|N - M| + 1)/ min(M,N) and for b > MNL2. This is the first time the optimal exponent is characterized for 1/min(M, N) < b < (|N - M| + 1)/min(M, N). Then, we propose a digital layered transmission scheme that uses both time layering and superposition. The new scheme is at least as good as currently known schemes for the entire range of bandwidth expansion factors b, whereas at least for some M, N, and b, it is strictly better than the currently known schemes.

On the Distortion Exponent of Some Layered Transmission Schemes

We consider the problem of joint source-channel coding for transmitting K samples of a complex Gaussian source over T = bK users of a quasi-static multiple input multiple output (MIMO) channel with M transmit and N receive antennas. The performance measure of interest is the rate of decay of the expected mean squared error with the signal-to-noise ratio (SNR), called the distortion exponent. We first show that using a broadcast strategy as in [D. Gunduz and E. Erkip, 2006], but with a different power and rate allocation policy, the optimal distortion exponent can be achieved for bandwidth efficiencies, 0 les b < (|N - M| + 1)/min(M, N). This is the first time the optimal exponent for 1/min(M, N) < b < (|N - M| + l)/min(M, N) is characterized. We then propose a digital layered transmission scheme that uses both time layering and superposition. This includes many known schemes in [D. Gunduz and E. Erkip, 2006; G. Caire and K. Narayanan, 2005] as special cases. The proposed scheme is at least as good as the currently best known schemes for the entire range of bandwidth efficiencies, whereas at least for some M , N, and b, it is strictly better than the currently best known schemes.

An MSE-Based Transfer Chart for Analyzing Iterative Decoding Schemes Using a Gaussian Approximation

An alternative to extrinsic information transfer (EXIT) charts called mean-square error (MSE) charts that use a measure related to the MSE instead of mutual information is proposed. Using the relationship between mutual information and minimum mean-square error (MMSE) for the additive white Gaussian noise (AWGN) channel, a relationship between the rate of any code and the area under a plot of MMSE versus signal-to-noise ratio (SNR) is obtained, when the a priori log-likelihood ratio (LLR) is from a binary input Gaussian channel. Using this result, a justification is provided for designing concatenated codes by matching the EXIT curves of the inner and outer decoder, when the LLRs are assumed to be Gaussian which is also the typical assumption used for code design using EXIT charts. Even though the Gaussian assumption is almost never true, the results presented in this paper represent a step toward the analysis of iterative decoding schemes using a single parameter. Finally, for the special case of AWGN channel it is shown that any capacity-achieving code has an EXIT curve that is a step function

Degree Optimization and Stability Condition for the Min-Sum Decoder

The min-sum (MS) algorithm is arguably the second most fundamental algorithm in the realm of message passing due to its optimality (for a tree code) with respect to the block error probability [1]. There also seems to be a fundamental relationship of MS decoding with the linear programming decoder [2]. Despite its importance, its fundamental properties have not nearly been studied as well as those of the sum-product (also known as BP) algorithm. We address two questions related to the MS rule. First, we characterize the stability condition under MS decoding. It turns out to be essentially the same condition as under BP decoding. Second, we perform a degree distribution optimization. Contrary to the case of BP decoding, under MS decoding the thresholds of the best degree distributions for standard irregular LDPC ensembles are significantly bounded away from the Shannon threshold. More precisely, on the AWGN channel, for the best codes that we find, the gap to capacity is 1 dB for a rate 0.3 code and it is 0.4 dB when the rate is 0.9 (the gap decreases monotonically as we increase the rate). We also used the optimization procedure to design codes for modified MS algorithm where the output of the check node is scaled by a constant 1/alpha. For alpha = 1.25, we observed that the gap to capacity was lesser for the modified MS algorithm when compared with the MS algorithm. However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates between 0.3 and 0.9. We conclude by posing what we consider to be the most important open questions related to the MS algorithm.

A decision feedback based scheme for Slepian-Wolf coding of sources with hidden Markov correlation

We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a hidden Markov model (HMM). We propose a decision feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian-Wolf limit.

A Note on the Rate of Decay of Mean-Squared Error With SNR for the AWGN Channel

The problem of transmitting a Gaussian source over an additive white Gaussian noise (AWGN) channel when the channel signal-to-noise ratio (SNR) is unknown at the transmitter and is known at the receiver is considered. The performance metric used is distortion SNR exponent which is defined as the rate of decay of mean-squared error (MSE) distortion with SNR. The optimal exponent is shown to be equal to the ratio of the number of channel uses to number of source samples. A superposition-based scheme is proposed that can achieve an exponent arbitrarily close to the optimal value.

Tradeoff Between Diversity and Decodable-Rate: Diversity Multiplexing Tradeoff for Fixed Encoding Schemes

In (Zheng, 2003), Zheng and Tse formulated the problem of determining the diversity multiplexing tradeoff for MIMO systems. In their formulation they considered a family of coding schemes {C(rho)} for an M times N MIMO channel where C(rho) denotes the coding scheme corresponding to an SNR of rho. For any coding scheme family with a multiplexing rate of r, i.e., whose rate grows as r log(1 + rho), they characterized the diversity order, dDMT*(r), the largest possible rate of decay for the probability of outage (Zheng, 2003). Here we consider a problem where the encoding scheme C at the transmitter is fixed (independent of SNR) and the receiver attempts to recover information bits at a rate of r log(1 + rho) from the received signal, where rho is the receive SNR. r is termed the decodable multiplexing rate (DMR). We define diversity of a scheme as the rate of decay of the probability that a receiver with receive SNR rho decodes a rate less than r log(1 + rho). We ask the question, what is the best possible diversity for a given DMR? We present a superposition scheme and show that for the MISO/SIMO channel, i.e. when min(M, N) = 1, the superposition scheme has a diversity order equal to max(M, N)(1 - r) which is also the best possible decay rate in this case. For the MIMO channel we show that the superposition scheme achieves a diversity of MN(1 - r) for r < 1. For the block fading SISO channel with coding over L blocks, the scheme achieves a diversity of L(1 - Lr) for r < 1/L.