U. Erez

Tel Aviv University, Tell Afif, Tel Aviv, Israel

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Publications (109)67.05 Total impact

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    ABSTRACT: The Gaussian parallel relay network, introduced by Schein and Gallager, consists of a concatenation of a Gaussian additive broadcast channel from a single encoder to a layer of relays followed by a Gaussian multiple-access channel from the relays to the final destination (decoder), where all noises are independent. This setup exhibits an inherent conflict between digital and analog relaying; while analog relaying [known as amplify-and-forward (A&F)] suffers from noise accumulation, digital relaying (known as decode-and-forward) looses the potential coherence gain in combining the relay noises at the decoder. For a large number of relays, the coherence gain is large, and thus analog relaying has better performance; however, it is limited to white channels of equal bandwidth. In this paper, we present a generalization of the analog approach to the case of bandwidth mismatch. Our strategy, coined rematch and forward (R&F), is based upon applying joint source-channel coding techniques that belong to a certain class of maximally analog schemes. Using such techniques, R&F converts the bandwidth of the broadcast section to that of the multiple-access section, creating an equivalent matched-bandwidth network over which A&F is applied. It is shown that this strategy exploits the full bandwidth of the individual channels, without sacrificing the coherence gain offered by A&F. Specifically, for given individual-link capacities, R&F remains within a constant gap from the network capacity for any number of relays and any bandwidth ratio between the sections. Finally, the approach is extended to the case of colored channels.
    IEEE Transactions on Information Theory 01/2014; 60(1):605-622. · 2.62 Impact Factor
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    Or Ordentlich, Uri Erez
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    ABSTRACT: Integer-Forcing (IF) is a new framework, based on compute-and-forward, for decoding multiple integer linear combinations from the output of a Gaussian multiple-input multiple-output channel. This work applies the IF approach to arrive at a new low-complexity scheme, IF source coding, for distributed lossy compression of correlated Gaussian sources under a minimum mean squared error distortion measure. All encoders use the same nested lattice codebook. Each encoder quantizes its observation using the fine lattice as a quantizer and reduces the result modulo the coarse lattice, which plays the role of binning. Rather than directly recovering the individual quantized signals, the decoder first recovers a full-rank set of judiciously chosen integer linear combinations of the quantized signals, and then inverts it. In general, the linear combinations have smaller average powers than the original signals. This allows to increase the density of the coarse lattice, which in turn translates to smaller compression rates. We also propose and analyze a one-shot version of IF source coding, that is simple enough to potentially lead to a new design principle for analog-to-digital converters that can exploit spatial correlations between the sampled signals.
    08/2013;
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    ABSTRACT: Lattice coding and decoding have been shown to achieve the capacity of the additive white Gaussian noise (AWGN) channel. This was accomplished using a minimum mean-square error scaling and randomization to transform the AWGN channel into a modulo-lattice additive noise channel of the same capacity. It has been further shown that when operating at rates below capacity but above the critical rate of the channel, there exists a rate-dependent scaling such that the associated modulo-lattice channel attains the error exponent of the AWGN channel. A geometric explanation for this result is developed. In particular, it is shown how the geometry of typical error events for the modulo-lattice channel coincides with that of a spherical code for the AWGN channel.
    08/2013;
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    ABSTRACT: Integer-forcing receivers generalize traditional linear receivers for the multiple-input multiple-output channel by decoding integer-linear combinations of the transmitted streams, rather then the streams themselves. Previous works have shown that the additional degree of freedom in choosing the integer coefficients enables this receiver to approach the performance of maximum-likelihood decoding in various scenarios. Nonetheless, even for the optimal choice of integer coefficients, the additive noise at the equalizer's output is still correlated. In this work we study a variant of integer-forcing, termed successive integer-forcing, that exploits these noise correlations to improve performance. This scheme is the integer-forcing counterpart of successive interference cancellation for traditional linear receivers. Similarly to the latter, we show that successive integer-forcing is capacity achieving when it is possible to optimize the rate allocation to the different streams. In comparison to standard successive interference cancellation receivers, the successive integer-forcing receiver offers more possibilities for capacity achieving rate tuples, and in particular, ones that are more balanced.
    07/2013;
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    ABSTRACT: Multicast is the general method of conveying the same information to multiple users over a broadcast channel. In this work, the Gaussian multiple-input multiple-output broadcast channel is considered, with multiple receive nodes, each equipped with an arbitrary number of antennas. A "closed loop" scenario is assumed, for which a practical multicast scheme is constructed which approaches capacity, by applying judiciously chosen unitary operations at the transmit and receives nodes that triangularize the channel matrices such that the resulting matrices have equal diagonals. This, along with the utilization of successive interference cancellation, reduces the coding and decoding tasks to those of coding and decoding over the single-antenna additive white Gaussian noise channel. Over the resulting effective channel, any "off-the-shelf" code may be employed. For the two-user case, it was recently shown that such joint unitary triangularization is always possible. In this work it is shown that for more users, joint triangularization of the time extensions of the channel matrices is necessary in general, which corresponds to carrying out the unitary processing over multiple channel uses. It is further shown that exact triangularization, where all resulting diagonals are equal, is not always possible, and appropriate conditions for the existence of such are established for certain cases. When exact triangularization is not possible, an approximate construction is proposed, that achieves the desired equal diagonals up to constant-length prefix and suffix. By enlarging the number of channel uses processed together, the loss in rate due to the prefix and the suffix can be made arbitrarily small.
    06/2013;
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    Elad Domanovitz, Uri Erez
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    ABSTRACT: The performance limits of scalar coding for multiple-input single-output channels are revisited in this work. By employing randomized beamforming, Narula et al. demonstrated that the loss of scalar coding is universally bounded by ~ 2.51 dB (or 0.833 bits/symbol) for any number of antennas and channel gains. In this work, by using randomized beamforming in conjunction with space-time codes, it is shown that the bound can be tightened to ~ 1.1 dB (or 0.39 bits/symbol).
    05/2013;
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    Eli Haim, Yuval Kochman, Uri Erez
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    ABSTRACT: We consider upper bounds on the error probability in channel coding. We derive an improved maximum-likelihood union bound, which takes into account events where the likelihood of the correct codeword is tied with that of some competitors. We compare this bound to various previous results, both qualitatively and quantitatively. With respect to maximal error probability of linear codes, we observe that when the channel is additive, the derivation of bounds, as well as the assumptions on the admissible encoder and decoder, simplify considerably.
    02/2013;
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    Or Ordentlich, Uri Erez
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    ABSTRACT: An open-loop single-user multiple-input multiple-output communication scheme is considered where a transmitter, equipped with multiple antennas, encodes the data into independent streams all taken from the same linear code. The coded streams are then linearly precoded using the encoding matrix of a perfect linear dispersion space-time code. At the receiver side, integer-forcing equalization is applied, followed by standard single-stream decoding. It is shown that this communication architecture achieves the capacity of any Gaussian multiple-input multiple-output channel up to a gap that depends only on the number of transmit antennas.
    01/2013;
  • O. Ordentlich, U. Erez
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    ABSTRACT: A static (constant channel gains) real K-user interference channel is considered, where all interference (cross) channel gains are integers. For such channels, previous results demonstrate that the number of degrees of freedom is very sensitive to slight variations in the direct channel gains. In this paper, we derive an achievable rate region for such channels that is valid for finite SNR. At moderate values of SNR, the derived rate region is robust to slight variations in the direct channel gains. At asymptotic high SNR conditions, known results on the degrees of freedom are recovered. The new rate region is based on lattice interference alignment. The result is established via a new coding theorem for the two-user Gaussian multiple-access channel where both users use a single linear code.
    IEEE Transactions on Information Theory 01/2013; 59(5):2735-2759. · 2.62 Impact Factor
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    Or Ordentlich, Uri Erez
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    ABSTRACT: This paper gives a simplified proof for the existence of nested lattice codebooks that allow to achieve the capacity of the additive white Gaussian noise channel. The proof is self-contained and relies only on basic probabilistic and geometrical arguments. An ensemble of nested lattices which is different than the one used in previous proofs is introduced. This ensemble, in addition to giving rise to a simple proof, can be easily generalized to an ensemble of nested lattices chains. As a result, the proof technique given here easily extends to showing the existence of "good" chains of nested lattices.
    09/2012;
  • Or Ordentlich, Uri Erez
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    ABSTRACT: This paper gives a simplified proof for the existence of nested lattice codebooks that allow to achieve the capacity of the additive white Gaussian noise channel. The proof is self-contained and relies only on basic probabilistic and geometrical arguments. An ensemble of nested lattices which is different than the one used in previous proofs is introduced. This ensemble, in addition to giving rise to a simple proof, can be easily generalized to an ensemble of nested lattices chains. As a result, the proof technique given here easily extends to showing the existence of "good" chains of nested lattices.
    09/2012;
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    Eli Haim, Yuval Kochman, Uri Erez
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    ABSTRACT: In this work we show how an improved lower bound to the error exponent of the memoryless multiple-access (MAC) channel is attained via the use of linear codes, thus demonstrating that structure can be beneficial even in cases where there is no capacity gain. We show that if the MAC channel is modulo-additive, then any error probability, and hence any error exponent, achievable by a linear code for the corresponding single-user channel, is also achievable for the MAC channel. Specifically, for an alphabet of prime cardinality, where linear codes achieve the best known exponents in the single-user setting and the optimal exponent above the critical rate, this performance carries over to the MAC setting. At least at low rates, where expurgation is needed, our approach strictly improves performance over previous results, where expurgation was used at most for one of the users. Even when the MAC channel is not additive, it may be transformed into such a channel. While the transformation is lossy, we show that the distributed structure gain in some "nearly additive" cases outweighs the loss, and thus the error exponent can improve upon the best known error exponent for these cases as well. Finally we apply a similar approach to the Gaussian MAC channel. We obtain an improvement over the best known achievable exponent, given by Gallager, for certain rate pairs, using lattice codes which satisfy a nesting condition.
    07/2012;
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    ABSTRACT: Interference alignment has emerged as a powerful tool in the analysis of multi-user networks. Despite considerable recent progress, the capacity region of the Gaussian K-user interference channel is still unknown in general, in part due to the challenges associated with alignment on the signal scale using lattice codes. This paper develops a new framework for lattice interference alignment, based on the compute-and-forward approach. Within this framework, each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer-valued coefficients and then solving these equations for its desired codeword. For the special case of symmetric channel gains, this framework is used to derive the approximate sum capacity of the Gaussian interference channel, up to an explicitly defined outage set of the channel gains. The key contributions are the capacity lower bounds for the weak through strong interference regimes, where each receiver should jointly decode its own codeword along with part of the interfering codewords. As part of the analysis, it is shown that decoding K linear combinations of the codewords can approach the sum capacity of the K-user Gaussian multiple-access channel up to a gap of no more than K log(K)/2 bits.
    06/2012;
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    ABSTRACT: Multicasting is the general method of conveying the same information to multiple users over a broadcast channel. In this work, the Gaussian MIMO broadcast channel is considered, with multiple users and any number of antennas at each node. A "closed loop" scenario is assumed, for which a practical capacity-achieving multicast scheme is constructed. In the proposed scheme, linear modulation is carried over time and space together, which allows to transform the problem into that of transmission over parallel scalar sub-channels, the gains of which are equal, except for a fraction of sub-channels that vanishes with the number of time slots used. Over these sub-channels, off-the-shelf fixed-rate AWGN codes can be used to approach capacity.
    04/2012;
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    ABSTRACT: This short note gives an overview of the concept of integer-forcing architectures for MIMO systems. Such architectures employ a linear receiver structure, i.e., a static linear transform followed by separate single-stream decoders, thus enjoying a significant complexity advantage over joint decoding of all streams. In standard linear receivers, the component single-stream decoders each decode one of the original data streams. In the integer-forcing architecture, each component decoder decodes an integer linear combination of several streams. Finally, these integer linear combinations are inverted to recover the original messages. Integer-forcing architectures can significantly outperform other linear MIMO receivers, including receivers that employ successive interference cancellation.
    01/2012;
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    E. Haim, Y. Kochman, U. Erez
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    ABSTRACT: Recently there has been significant interest in the analysis of finite-blocklength performance in different settings. Specifically, there is an effort to extend the performance bounds, as well as the Gaussian approximation (dispersion) beyond point-to-point settings. This proves to be a difficult task, as the performance may be governed by multiple dependent constraints. In this work we shed light on these difficulties, using the multiple-access channel as a test case. We show that a local notion of dispersion is more informative than that of dispersion regions sought after thus far. On the positive side, we show that for channels possessing certain symmetry, the dispersion problem reduces to the single-user one. Furthermore, for such channels, linear codes enable to translate single-user achievability bounds to the multiple-access channel.
    Electrical & Electronics Engineers in Israel (IEEEI), 2012 IEEE 27th Convention of; 01/2012
  • E. Domanovitz, U. Erez
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    ABSTRACT: In a recent work by Zhan et al., a new equalization technique for multiple-input multiple-output channels, applicable when linear codes are used for transmission, was proposed. In this technique, coined integer-forcing equalization, the channel matrix is equalized at the receiver to one consisting of only integer entries. It was demonstrated that for quasi-static independent flat Rayleigh fading, and when independent streams are sent over each transmit antenna, this equalization technique allows to approach quite closely maximum-likelihood performance, using standard coding and decoding of off-the-shelf linear codes designed for transmission over a scalar white Gaussian channel. In particular, the technique is optimal in the diversity-multiplexing tradeoff sense. In the present work, we describe how this receiver structure may be seamlessly combined with linear transmit diversity methods. We show that for quasi-static independent flat Rayleigh fading, this combination allows to approach closely the outage capacity of the channel using linear pre- and post-processing and standard scalar coding and decoding.
    Electrical & Electronics Engineers in Israel (IEEEI), 2012 IEEE 27th Convention of; 01/2012
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    ABSTRACT: This work considers practical implementation of the decode-and-forward relaying protocol for the full-duplex Gaussian relay channel. Unlike previous works which developed coding techniques tailored to this protocol, it is shown that standard codes which are good for the Gaussian scalar channel of fixed signal 5a8 -to-noise ratio suffice to approach the theoretical performance promised by this protocol. The proposed technique employs only linear operations and successive interference cancelation in conjunction with fixed signal-to-noise ratio base codes, and the achievable rate is solely dictated by the performance of these base codes. The same approach and results carry over to the multiple-antenna case as well.
    Information Theory Workshop (ITW), 2012 IEEE; 01/2012
  • E. Haim, Y. Kochman, U. Erez
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    ABSTRACT: We consider the error exponent of the memoryless multiple-access (MAC) channel. We show that if the MAC channel is modulo-additive, then any error probability, and hence any error exponent, achievable by a linear code for the corresponding single-user channel, is also achievable for the MAC channel. Specifically, for an alphabet of prime cardinality, where linear codes achieve the best known exponents in the single-user setting (and the optimal exponent above the critical rate), this performance carries over to the MAC setting. At least at low rates, where expurgation is needed, our approach strictly improves performance over previous results, where expurgation was used at most for one of the users. Even when the MAC channel is not additive, it may be transformed into such a channel. While the transformation is lossy, we show that the distributed structure gain in some “nearly additive” cases outweighs the loss, and thus we can improve upon the best known exponent for these cases as well. This approach is related to that previously proposed for the Gaussian MAC channel, and is based on “distributed structure”.
    Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on; 01/2012
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    ABSTRACT: We derive an achievable rate region for the Gaussian K-user multiple-access channel (MAC) where all users transmit codewords from a chain of nested lattices. For any set of channel coefficients, this rate region contains points within a constant gap from the sum capacity boundary of the MAC. The main tool used is the recently proposed compute-and-forward framework. A new transformation of a MAC to a modulo-lattice multiple-input multiple-output (MIMO) channel is introduced based on this framework. Specifically, from one noisy linear combination of the transmitted signals the receiver attempts to decode K linearly independent equations with integer-valued coefficients. While the individual rates at which these equations can be decoded are highly sensitive to the exact channel gains, their sum is always within a constant gap from the sum capacity boundary of the MAC. The transformation is then utilized for establishing the desired rate region.
    Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on; 01/2012

Publication Stats

2k Citations
67.05 Total Impact Points

Institutions

  • 1998–2014
    • Tel Aviv University
      • School of Electrical Engineering
      Tell Afif, Tel Aviv, Israel
  • 2004–2011
    • Massachusetts Institute of Technology
      • Department of Electrical Engineering and Computer Science
      Cambridge, Massachusetts, United States
  • 2010
    • University of California, Berkeley
      • Department of Electrical Engineering and Computer Sciences
      Berkeley, California, United States