Uri Erez

Tel Aviv University, Tell Afif, Tel Aviv, Israel

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Publications (118)73.77 Total impact

  • [Show abstract] [Hide abstract]
    ABSTRACT: A framework is developed for decode-and-forward based relaying using standard coding and decoding that are good for the single-input single-output (SISO) additive white Gaussian noise channel. The framework is applicable to various scenarios and demonstrated for several important cases. Each of these scenarios is transformed into an equivalent Gaussian multiple-input multiple-output (MIMO) common-message broadcast problem, which proves useful even when all links are SISO ones. Over the effective MIMO broadcast channel, a recently developed Gaussian MIMO common-message broadcast scheme is applied. This scheme transforms the MIMO links into a set of parallel SISO channels with no loss of mutual information, using linear pre- and post-processing combined with successive decoding. Over these resulting SISO channels, “off-the-shelf” scalar codes may be used.
    No preview · Article · Dec 2015 · IEEE Transactions on Information Theory
  • Anatoly Khina · Yuval Kochman · Uri Erez
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    ABSTRACT: In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an i.i.d. ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented, which was shown to be optimal in the limit of high signal-to-noise ratios, where the sum-capacity is dictated by the minimal ("bottleneck") channel gain. In the present work, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions, which transform the channel into effective parallel scalar dirty multiple-access channels. This approach however suffers from a "bottleneck" effect for each effective scalar channel and therefore the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high signal-to-noise ratios, under an equal rank assumption. This approach is then extended to any number of users. Finally, an application to physical-layer network coding for the MIMO two-way relay channel is presented.
    No preview · Article · Oct 2015
  • O. Ordentlich · U. Erez · B. Nazer
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    ABSTRACT: We consider a Gaussian multiple-access channel where each user's message is identified with a vector of elements from a finite field, and the receiver's goal is to decode a linear combination of these finite field vectors. It is further assumed that each transmitter can causally observe the channel's output through a clean feedback link. We propose a novel coding scheme for this setup, which can be seen as an extension of the Cover-Leung scheme for the computation problem. This scheme is shown to achieve computation rates higher than the best known computation rates for the same scenario without feedback. In particular, for the symmetric two-user Gaussian multiple-access channel, the proposed scheme attains a symmetric computation rate greater than 1/2 log(3/4 + SNR).
    No preview · Article · Jun 2015
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    Or Ordentlich · Uri Erez
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    ABSTRACT: Sampling above the Nyquist-rate is at the heart of sigma-delta modulation, where the increase in sampling rate is translated to a reduction in the overall (minimum mean-squared-error) reconstruction distortion. This is attained by using a feedback filter at the encoder, in conjunction with a low-pass filter at the decoder. The goal of this work is to characterize the optimal trade-off between the per-sample quantization rate and the resulting mean-squared-error distortion, under various restrictions on the feedback filter. To this end, we establish a duality relation between the performance of sigma-delta modulation, and that of differential pulse-code modulation when applied to (discrete-time) band-limited inputs. As the optimal trade-off for the latter scheme is fully understood, the full characterization for sigma-delta modulation, as well as the optimal feedback filters, immediately follow.
    Preview · Article · Jan 2015
  • A. Khina · Y. Kochman · U. Erez
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    ABSTRACT: The Gaussian multiple-input multiple-output two-way relay channel is considered. By applying linear pre- and post-processing, the channel matrices are transformed into triangular form having equal diagonals. Over the obtained triangular channels, dirty-paper coding is applied, yielding parallel symmetric scalar two-way relay channels; thus, reducing the coding task to that of coding over the scalar symmetric two-way relay channel. Any existing coding technique can then be readily applied over these resulting channels. This technique allows to obtain new achievable rates in the symmetric case.
    No preview · Article · Dec 2014
  • E. Domanovitz · U. Erez
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    ABSTRACT: The integer-forcing receiver architecture has recently been proposed as a high-performance, yet low-complexity, equalization scheme, that is applicable when all data streams are encoded with the same linear code. It was further shown in [1], that this receiver architecture, when coupled with space-time linear precoding is able to achieve the capacity of the open-loop multiple-input multiple-output channel, up to a constant gap that depends only on the number of transmit antennas. The gap, however, is quite large and thus provides performance guarantees that are useful only for high values of capacity. In this work, we consider the problem of multicast over multiple-input multiple-output channels to a modest number of users, and with space-only linear precoding. It is assumed that channel state information is available to the transmitter, allowing it to optimize the precoding matrix so as to maximize the achievable transmission rate. It is numerically demonstrated that this architecture allows to very closely approach the multicast capacity at all transmission rates regimes.
    No preview · Article · Dec 2014
  • Oded Fischler · Uri Erez
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    ABSTRACT: Recently, an open-loop transmission scheme for multiple-input multiple-output Gaussian channels based on precoded integer-forcing was proposed. The transmitter encodes the data into independent streams, all taken from the same linear code. The coded streams are then linearly precoded using a unitary matrix. At the receiver side, integer-forcing equalization is applied, followed by single-stream decoding. It was shown that this communication architecture achieves capacity up to a finite gap. In the present work we consider precoded integer-forcing for parallel Gaussian channels. We derive tighter bounds for this class of channels, which are related to the minimum product distance figure of merit. We further suggest a practical scheme that is applicable for all transmission rates, where the precoding matrix is capacity-dependent, chosen so as to maximize the achievable rate for a given value of capacity. For example, it is shown that for the case of two and three parallel channels, the scheme universally (for any value of capacity) achieves 94% and 82% of capacity, respectively.
    No preview · Conference Paper · Jun 2014
  • E. Haim · Y. Kochman · U. Erez
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    ABSTRACT: We consider the finite-blocklength performance of singular channels (e.g., the binary erasure channel). At least for symmetric singular channels, it is known that the next correction term after the channel dispersion is the "constant" term. We show that for such channels, the asymptotic significance of tie breaking (i.e., making a fair decision in the case of multiple codewords of equal likelihood) is greater than for non-singular channels. Specifically, for an ensemble of codebooks, where the codewords are independent with the same marginal distribution, the constant correction term is increased by exactly one nat.
    No preview · Article · Jan 2014
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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.
    Preview · Article · Jan 2014 · IEEE Transactions on Information Theory
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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.
    Preview · Article · Aug 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.
    Preview · Article · Aug 2013
  • Or Ordentlich · Uri Erez · Bobak Nazer
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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.
    No preview · Article · Jul 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.
    Preview · Article · Jun 2013 · IEEE Transactions on Information Theory
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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).
    Preview · Article · May 2013
  • Or Ordentlich · Uri 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.
    No preview · Article · May 2013 · IEEE Transactions on Information Theory
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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.
    Preview · Article · Feb 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.
    Preview · Article · Jan 2013 · IEEE Transactions on Information Theory
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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.
    Preview · Article · Sep 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.
    No preview · Conference Paper · Sep 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.
    Preview · Article · Jul 2012 · IEEE Transactions on Information Theory

Publication Stats

3k Citations
73.77 Total Impact Points

Institutions

  • 1998-2013
    • Tel Aviv University
      • Department of Electrical Engineering - Systems
      Tell Afif, Tel Aviv, Israel
  • 2004-2005
    • Massachusetts Institute of Technology
      • Department of Electrical Engineering and Computer Science
      Cambridge, Massachusetts, United States