We introduce and study the problem of camera-radar fusion for 3-D depth reconstruction. This problem is motivated by autonomous driving applications, in which we can expect to have access to both front-facing camera and radar sensors. These two sensors are complementary in several respects: the camera is a passive sensor measuring azimuth and elevation; the radar is an active sensor measuring azimuth and range. Fusing their measurements is therefore beneficial. Our fusion solution uses a modified encoder-decoder deep convolutional neural network. We train and evaluate this network on over 100 000 samples collected in highway environments. Our results demonstrate an improvement in reconstruction accuracy and robustness from fusing the two sensors.

Motivated by requirements for future automotive radar, we study the problem of resolving target elevation from measurements by a one-dimensional horizontal radar antenna array. This is a challenging and ill-posed problem, since such measurements contain only indirect and highly ambiguous elevation cues. As a consequence, traditional model-based approaches fail. We instead propose to use a machine-learning-based approach that learns to exploit the subtle elevation cues and prior knowledge of the scene from the data. We design an encoder-decoder structured deep convolutional neural network that takes a radar return intensity image in the range-azimuth plane as input and produces a depth image in the elevation-azimuth plane as output. We train the network with over 200 000 radar frames collected in highway environments. Through experimental evaluations, we demonstrate the feasibility of resolving the highly ambiguous elevation information in such environments.

Communication in integrated circuits faces two major impediments: inter-wire capacitive coupling and noise. Coding can be used to address both these problems. So-called crosstalk-avoidance codes mitigate capacitive coupling, and traditional error-correction codes introduce resilience against channel errors. Unfortunately, crosstalk-avoidance and error-correction codes cannot be combined in a straightforward manner. On the one hand, crosstalk-avoidance encoding followed by error-correction encoding destroys the crosstalk-avoidance property. On the other hand, error-correction encoding followed by crosstalk-avoidance encoding causes the crosstalk-avoidance decoder to fail in the presence of errors. Existing approaches circumvent this difficulty by using additional bus wires to protect the parities generated from the output of the error-correction encoder, and are therefore inefficient. In this paper, we propose a novel joint crosstalk-avoidance and error-correction coding and decoding scheme that provides higher bus transmission rates compared with existing approaches. Our joint approach carefully embeds the parities such that the crosstalk-avoidance property is preserved. We analyze the rate and minimum distance of the proposed scheme. We also provide a density evolution analysis and predict iterative decoding thresholds for reliable communication under random bus erasures. This density evolution analysis is nonstandard, since the crosstalk-avoidance constraints are inherently nonlinear.

We study the role of caches in wireless interference networks. We focus on content caching and delivery across a Gaussian interference network, where both transmitters and receivers are equipped with caches. We provide a constant-factor approximation of the system's degrees of freedom (DoF), for arbitrary number of transmitters, number of receivers, content library size, receiver cache size, and transmitter cache size (as long as the transmitters combined can store the entire content library among them). We demonstrate approximate optimality with respect to information-theoretic bounds that do not impose any restrictions on the caching and delivery strategies. Our characterization reveals three key insights. First, the approximate DoF is achieved using a strategy that separates the physical and network layers. This separation architecture is thus approximately optimal. Second, we show that increasing transmitter cache memory beyond what is needed to exactly store the entire library between all transmitters does not provide more than a constant-factor benefit to the DoF. A consequence is that transmit zero-forcing is not needed for approximate optimality. Third, we derive an interesting tradeoff between the receiver memory and the number of transmitters needed for approximately maximal performance. In particular, if each receiver can store a constant fraction of the content library, then only a constant number of transmitters are needed. Our solution to the caching problem requires formulating and solving a new communication problem, the symmetric multiple multicast X-channel, for which we provide an exact DoF characterization.

Deduplication finds and removes long-range data duplicates. It is commonly used in cloud and enterprise server settings and has been successfully applied to primary, backup, and archival storage. Despite its practical importance as a source-coding technique, its analysis from the point of view of information theory is missing. This paper provides such an information-theoretic analysis of data deduplication. It introduces a new source model adapted to the deduplication setting. It formalizes both fixed and variable-length deduplication schemes, and it introduces a novel, multi-chunk deduplication scheme. It then provides an analysis of these three deduplication variants, emphasizing the importance of boundary synchronization between source blocks and deduplication chunks. The proposed multi-chunk deduplication scheme is shown to be order optimal under fairly mild assumptions.

Packet networks need to maintain the state in the form of forwarding tables at each switch. The cost of this state increases as networks support ever more sophisticated per-flow routing, traffic engineering, and service chaining. Per-flow or per-path state at the switches can be eliminated by encoding each packet's desired path in its header. A key component of such a method is an efficient encoding of paths through the network. We introduce a mathematical formulation of this optimal path-encoding problem. We prove that the problem is APX-hard, by showing that approximating it to within a factor less than 8/7 is NP-hard. Thus, at best, we can hope for a constant-factor approximation algorithm. We then present such an algorithm, approximating the optimal path-encoding problem to within a factor 2. Finally, we provide the empirical results illustrating the effectiveness of the proposed algorithm.

Recent work has studied the benefits of caching in the interference channel, particularly by placing caches at the transmitters. In this paper, we study the two-user Gaussian interference channel in which caches are placed at both the transmitters and the receivers. We propose a separation strategy that divides the physical and network layers. While a natural separation approach might be to abstract the physical layer into several independent bit pipes at the network layer, we argue that this is inefficient. Instead, the separation approach we propose exposes interacting bit pipes at the network layer, so that the receivers observe related (yet not identical) quantities. We find the optimal strategy within this layered architecture, and we compute the degrees-of-freedom it achieves. Finally, we show that separation is optimal in regimes where the receiver caches are large.

Caching of popular content during off-peak hours is a strategy to reduce network loads during peak hours. Recent work has shown significant benefits of designing such caching strategies not only to locally deliver the part of the content, but also to provide coded multicasting opportunities even among users with different demands. Exploiting both of these gains was shown to be approximately optimal for caching systems with a single layer of caches. Motivated by practical scenarios, we consider, in this paper, a hierarchical content delivery network with two layers of caches. We propose a new caching scheme that combines two basic approaches. The first approach provides coded multicasting opportunities within each layer; the second approach provides coded multicasting opportunities across multiple layers. By striking the right balance between these two approaches, we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap. We further show that there is no tension between the rates in each of the two layers up to the aforementioned gap. Thus, both the layers can simultaneously operate at approximately the minimum rate.

We consider a basic content distribution scenario consisting of a single origin server connected through a shared bottleneck link to a number of users each equipped with a cache of finite memory. The users issue a sequence of content requests from a set of popular files, and the goal is to operate the caches as well as the server such that these requests are satisfied with the minimum number of bits sent over the shared link. Assuming a basic Markov model for renewing the set of popular files, we characterize approximately the optimal long-term average rate of the shared link. We further prove that the optimal online scheme has approximately the same performance as the optimal offline scheme, in which the cache contents can be updated based on the entire set of popular files before each new request. To support these theoretical results, we propose an online coded caching scheme termed coded least-recently sent (LRS) and simulate it for a demand time series derived from the dataset made available by Netflix for the Netflix Prize. For this time series, we show that the proposed coded LRS algorithm significantly outperforms the popular least-recently used caching algorithm.

Coded caching is a recently proposed technique that achieves significant performance gains for cache networks compared to uncoded caching schemes. However, this substantial coding gain is attained at the cost of large delivery delay, which is not tolerable in delay-sensitive applications such as video streaming. In this paper, we identify and investigate the tradeoff between the performance gain of coded caching and the delivery delay. We propose a computationally efficient caching algorithm that provides the gains of coding and respects delay constraints. The proposed algorithm achieves the optimum performance for large delay, but still offers major gains for small delay. These gains are demonstrated in a practical setting with a video-streaming prototype.

Packet networks need to maintain state in the form of forwarding tables at each switch. The cost of this state increases as networks support ever more sophisticated per-flow routing, traffic engineering, and service chaining. Per-flow or per-path state at the switches can be eliminated by encoding each packet's desired path in its header. A key component of such a method is an efficient encoding of paths through the network. We introduce a mathematical formulation of this optimal path-encoding problem. We prove that the problem is APX-hard, by showing that approximating it to within a factor less than 8/7 is NP-hard. Thus, at best we can hope for a constant-factor approximation algorithm. We then present such an algorithm, approximating the optimal path-encoding problem to within a factor 2. Finally, we provide empirical results illustrating the effectiveness of the proposed algorithm.

Future wireless standards such as 5G envision dense wireless networks with large number of simultaneously connected devices. In this context, interference management becomes critical in achieving high spectral efficiency. Orthogonal signaling, which limits the number of users utilizing the resource simultaneously, gives a sum-rate that remains constant with increasing number of users. An alternative approach called interference alignment promises a throughput that scales linearly with the number of users. However, this approach requires very high SNR or long time duration for sufficient channel variation, and therefore may not be feasible in real wireless systems. We explore ways to manage interference in large networks with delay and power constraints. Specifically, we devise an interference phase alignment strategy that combines precoding and scheduling without using power control to exploit the diversity inherent in a system with large number of users. We show that this scheme achieves a sum-rate that scales almost logarithmically with the number of users. We also show upper bounds on the sum-rate within the restricted class of single-symbol phase alignment schemes. Specifically, we prove that no scheme in this class can achieve better than logarithmic scaling of the sum-rate.

Over the past decade, the bulk of wireless traffic has shifted from speech to content. This shift creates the opportunity to cache part of the content in memories closer to the end users, for example in base stations. Most of the prior literature focuses on the reduction of load in the backhaul and core networks due to caching, i.e., on the benefits caching offers for the wireline communication link between the origin server and the caches. In this paper, we are instead interested in the benefits caching can offer for the wireless communication link between the caches and the end users. To quantify the gains of caching for this wireless link, we consider an interference channel in which each transmitter is equipped with an isolated cache memory. Communication takes place in two phases, a content placement phase followed by a content delivery phase. The objective is to design both the placement and the delivery phases to maximize the rate in the delivery phase in response to any possible user demands. Focusing on the three-user case, we show that through careful joint design of these phases, we can reap three distinct benefits from caching: a load balancing gain, an interference cancellation gain, and an interference alignment gain. In our proposed scheme, load balancing is achieved through a specific file splitting and placement, creating a particular pattern of content overlap at the caches. This overlap allows to implement interference cancellation. Further, it allows us to construct several virtual transmitters, each responsible for a part of the requested content, which increases interference alignment possibilities.

We consider the problem of precise range estimation between pairs of moving vehicles using periodic broadcast messages. The vehicles are not time synchronized and one needs to explicitly account for the clock offset and clock drift in addition to the vehicle motion to obtain accurate range estimates. We develop a broadcast range estimation algorithm based on local polynomial smoothing of the vehicle motion and experimentally verify that the performance is close to that obtained using unicast round-trip time ranging. This broadcast approach is of interest particularly in the context of dedicated short-range communication (DSRC) wherein periodic broadcast safety messages are exchanged between vehicles. We propose to exploit these broadcast messages to perform ranging. Our scheme requires additional timestamp information to be transmitted as part of the DSRC messages, and we develop a novel timestamp compression algorithm to minimize the resulting overhead. We validate our proposed algorithm on experimental data and show that it is able to achieve sub-meter ranging accuracies in vehicular scenarios.

Communication systems are traditionally designed to have tight transmitter-receiver synchronization. This requirement has negligible overhead in the high-signal-to-noise ratio (SNR) regime. However, in many applications, such as wireless sensor networks, communication needs to happen primarily in the energy-efficient regime of low SNR, where requiring tight synchronization can be highly suboptimal. In this paper, we model the noisy channel with synchronization errors as a duplication/deletion/substitution channel. For this channel, we propose a new communication scheme that requires only loose transmitter-receiver synchronization. We show that the proposed scheme is asymptotically optimal for the Gaussian channel with synchronization errors in terms of energy efficiency as measured by the rate per unit energy. In the process, we also establish that the lack of synchronization causes negligible loss in energy efficiency. We further show that, for a general discrete memoryless channel with synchronization errors and a general input cost function admitting a zero-cost symbol, the rate per unit cost achieved by the proposed scheme is within a factor two of the information-theoretic optimum.

Replicating or caching popular content in memories distributed across the network is a technique to reduce peak network loads. Conventionally, the main performance gain of this caching was thought to result from making part of the requested data available closer to end-users. Instead, we recently showed that a much more significant gain can be achieved by using caches to create coded-multicasting opportunities, even for users with different demands, through coding across data streams. These coded-multicasting opportunities are enabled by careful content overlap at the various caches in the network, created by a central coordinating server. In many scenarios, such a central coordinating server may not be available, raising the question if this multicasting gain can still be achieved in a more decentralized setting. In this paper, we propose an efficient caching scheme, in which the content placement is performed in a decentralized manner. In other words, no coordination is required for the content placement. Despite this lack of coordination, the proposed scheme is nevertheless able to create coded-multicasting opportunities and achieves a rate close to the optimal centralized scheme.

We consider a basic content distribution scenario consisting of a single origin server connected through a shared bottleneck link to a number of users each equipped with a cache of finite memory. The users issue a sequence of content requests from a set of popular files, and the goal is to operate the caches as well as the server such that these requests are satisfied with the minimum number of bits sent over the shared link. Assuming a basic Markov model for renewing the set of popular files, we characterize approximately the optimal long-term average rate of the shared link. We further prove that the optimal online scheme has approximately the same performance as the optimal offline scheme, in which the cache contents can be updated based on the entire set of popular files before each new request. To support these theoretical results, we propose an online coded caching scheme termed coded least-recently sent (LRS) and simulate it for a demand time series derived from the dataset made available by Netflix for the Netflix Prize. For this time series, we show that the proposed coded LRS algorithm significantly outperforms the popular least-recently used (LRU) caching algorithm.

It has recently been demonstrated that for single-layer cache networks, jointly designing caching and delivery can enable significant benefits over conventional caching. This was based on strategically designing the cached content to induce coded multicasting opportunities even among users with different demands and without foreknowledge of the user demands. In this work, we extend this coded caching approach to a multi-hop hierarchical content delivery network with two layers of caches. We propose a new caching scheme that combines two basic approaches. The first approach provides coded multicasting opportunities within each layer (through decoding and forwarding); the second approach provides coded multicasting opportunities across multiple layers (through strategic forwarding without decoding). By striking the right balance between these two approaches, we show that the proposed scheme achieves the optimal communication rates to within a constant multiplicative and additive gap. We further show that there is no tension between the rates in each of the two layers up to the aforementioned gap. Thus, both layers can simultaneously operate at approximately the minimum rate.

Communication networks are often designed and analyzed assuming tight synchronization among nodes. However, in applications that require communication in the energy-efficient regime of low signal-to-noise ratios, establishing tight synchronization among nodes in the network can result in a significant energy overhead. Motivated by a recent result showing that near-optimal energy efficiency can be achieved over the point-to-point AWGN channel without requiring tight synchronization, we consider the question of whether the potential gains of cooperative communication can be achieved in the absence of synchronization. We focus on the symmetric Gaussian diamond network and establish that cooperative-communication gains are indeed feasible even with unsynchronized nodes. More precisely, we show that the capacity per unit energy of the unsynchronized symmetric Gaussian diamond network is within a constant factor of the capacity per unit energy of the corresponding synchronized network. To this end, we propose a distributed relaying scheme that does not require tight synchronization but nevertheless achieves most of the energy gains of coherent combining.

Communication networks are often designed and analyzed assuming tight synchronization among nodes. However, in applications that require communication in the energy-efficient regime of low signal-to-noise ratios, establishing tight synchronization among nodes in the network can result in a significant energy overhead. Motivated by a recent result showing that near-optimal energy efficiency can be achieved over the additive white Gaussian noise channel without requiring tight synchronization, we consider the question of whether the potential gains of cooperative communication can be achieved in the absence of synchronization. We focus on the symmetric Gaussian diamond network and establish that cooperative-communication gains are indeed feasible even with unsynchronized nodes. More precisely, we show that the capacity per unit energy of the unsynchronized symmetric Gaussian diamond network is within a constant factor of the capacity per unit energy of the corresponding synchronized network. To this end, we propose a distributed relaying scheme that does not require tight synchronization but nevertheless achieves most of the energy gains of coherent combining.

Caching is a technique to reduce peak traffic rates by prefetching popular content into memories at the end users. Conventionally, these memories are used to deliver requested content in part from a locally cached copy rather than through the network. The gain offered by this approach, which we term local caching gain, depends on the local cache size (i.e., the memory available at each individual user). In this paper, we introduce and exploit a second, global, caching gain not utilized by conventional caching schemes. This gain depends on the aggregate global cache size (i.e., the cumulative memory available at all users), even though there is no cooperation among the users. To evaluate and isolate these two gains, we introduce an information-theoretic formulation of the caching problem focusing on its basic structure. For this setting, we propose a novel coded caching scheme that exploits both local and global caching gains, leading to a multiplicative improvement in the peak rate compared with previously known schemes. In particular, the improvement can be on the order of the number of users in the network. In addition, we argue that the performance of the proposed scheme is within a constant factor of the information-theoretic optimum for all values of the problem parameters.

Communication systems are traditionally designed to have tight transmitter-receiver synchronization. This requirement has negligible overhead in the high-SNR regime. However, in many applications, such as wireless sensor networks, communication needs to happen primarily in the energy-efficient regime of low SNR, where requiring tight synchronization can be highly suboptimal. In this paper, we model the noisy channel with synchronization errors as an insertion/deletion/substitution channel. For this channel, we propose a new communication scheme that requires only loose transmitter-receiver synchronization. We show that the proposed scheme is asymptotically optimal for the Gaussian channel with synchronization errors in terms of energy efficiency as measured by the rate per unit energy. In the process, we also establish that the lack of synchronization causes negligible loss in energy efficiency. We further show that, for a general discrete memoryless channel with synchronization errors and a general cost function (with a zero-cost symbol) on the input, the rate per unit cost achieved by the proposed scheme is within a factor two of the information-theoretic optimum.

Caching is a technique to reduce peak traffic rates by prefetching popular content in memories at the end users. This paper proposes a novel caching approach that can achieve a significantly larger reduction in peak rate compared to previously known caching schemes. In particular, the improvement can be on the order of the number of end users in the network. Conventionally, cache memories are exploited by delivering requested contents in part locally rather than through the network. The gain offered by this approach, which we term local caching gain, depends on the local cache size (i.e., the cache available at each individual user). In this paper, we introduce and exploit a second, global, caching gain, which is not utilized by conventional caching schemes. This gain depends on the aggregate global cache size (i.e., the cumulative cache available at all users), even though there is no cooperation among the caches. To evaluate and isolate these two gains, we introduce a new, information-theoretic formulation of the caching problem focusing on its basic structure. For this setting, the proposed scheme exploits both local and global caching gains, leading to a multiplicative improvement in the peak rate compared to previously known schemes. Moreover, we argue that the performance of the proposed scheme is within a constant factor from the information-theoretic optimum for all values of the problem parameters.

We consider the problem of distributed computation of a target function over a two-user deterministic multiple-access channel. If the target and channel functions are matched (i.e., compute the same function), significant performance gains can be obtained by jointly designing the communication and computation tasks. However, in most situations there is mismatch between these two functions. In this work, we analyze the impact of this mismatch on the performance gains achievable with joint communication and computation designs over separation-based designs. We show that for most pairs of target and channel functions there is no such gain, and separation of communication and computation is optimal.

We consider the Gaussian “diamond” or parallel relay network, in which a source node transmits a message to a destination node with the help ofNrelays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of orderNbits and up to a multiplicative gap of orderN2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric GaussianN-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relaysNin the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric GaussianN-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factorO(log4(N)) of capacity with preconstant in the order notation independent of the channel gains.

Interference alignment is a key technique for communication scenarios with multiple interfering links. In several such scenarios, interference alignment was used to characterize the degrees of freedom of the channel. However, these degree-of-freedom capacity approximations are often too weak to make accurate predictions about the behavior of channel capacity at finite signal-to-noise ratios (SNRs). The aim of this paper is to significantly strengthen these results by showing that interference alignment can be used to characterize capacity to within a constant gap. We focus on real, time-invariant, frequency-flat X-channels. The only known solutions achieving the degrees of freedom of this channel are either based on real interference alignment or on layer-selection schemes. Neither of these solutions seems sufficient for a constant-gap capacity approximation. In this paper, we propose a new communication scheme and show that it achieves the capacity of the Gaussian X-channel to within a constant gap. To aid in this process, we develop a novel deterministic channel model. This deterministic model depends on the 1/2 log (SNR) most-significant bits of the channel coefficients rather than only the single most-significant bit used in conventional deterministic models. The proposed deterministic model admits a wider range of achievable schemes that can be translated to the Gaussian channel. For this deterministic model, we find an approximately optimal communication scheme. We then translate this scheme for the deterministic channel to the original Gaussian X-channel and show that it achieves capacity to within a constant gap. This is the first constant-gap result for a general, fully-connected network requiring interference alignment.

Consider several source nodes communicating across a wireless network to a destination node with the help of several layers of relay nodes. Recent work by Avestimehr has approximated the capacity of this network up to an additive gap. The communication scheme achieving this capacity approximation is based on compress-and-forward, resulting in noise accumulation as the messages traverse the network. As a consequence, the approximation gap increases linearly with the network depth. This paper develops a computation alignment strategy that can approach the capacity of a class of layered, time-varying wireless relay networks up to an approximation gap that is independent of the network depth. This strategy is based on the compute-and-forward framework, which enables relays to decode deterministic functions of the transmitted messages. Alone, compute-and-forward is insufficient to approach the capacity as it incurs a penalty for approximating the wireless channel with complex-valued coefficients by a channel with integer coefficients. Here, this penalty is circumvented by carefully matching channel realizations across time slots to create integer-valued effective channels that are well suited to compute-and-forward. Unlike prior constant gap results, the approximation gap obtained in this paper also depends closely on the fading statistics, which are assumed to be i.i.d. Rayleigh.

Interference alignment is a key technique for communication scenarios with multiple interfering links. In several such scenarios, interference alignment was used to characterize the degrees-of-freedom of the channel. However, these degrees-of-freedom capacity approximations are often too weak to make accurate predictions about the behavior of channel capacity at finite signal-to-noise ratios. The aim of this paper is to significantly strengthen these results by showing that interference alignment can be used to characterize capacity to within a constant gap. We focus on real time-invariant frequency-flat X-channels, for which only the degrees-of-freedom are known. We propose a new communication scheme and show that it achieves the capacity of the Gaussian X-channel to within a constant gap. To aid in this process, we develop a novel deterministic channel model, admitting a wider range of achievable schemes that can be translated to the Gaussian channel. For this deterministic model, we find an approximately optimal communication scheme. We then translate this scheme for the deterministic channel to the original Gaussian X-channel and show that it achieves capacity to within a constant gap. This is the first constant-gap result for a fully-connected network requiring interference alignment.

We consider the problem of delivering content cached in a wireless network of n nodes randomly located on a square of area n. The network performance is described by the 2n× n-dimensional caching capacity region of the wireless network. We provide an inner bound on this caching capacity region, and, in the high path-loss regime, a matching (in the scaling sense) outer bound. For large path-loss exponent, this provides an information-theoretic scaling characterization of the entire caching capacity region. The proposed communication scheme achieving the inner bound shows that the problems of cache selection and channel coding can be solved separately without loss of order-optimality. On the other hand, our results show that the common architecture of nearest-neighbor cache selection can be arbitrarily bad, implying that cache selection and load balancing need to be performed jointly.

We analyze the asymptotic behavior of compute-and-forward relay networks in the regime of high signal-to-noise ratios. We consider a section of such a network consisting of K transmitters and K relays. The aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters. An upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays, transforming the network into a K × K multiple-input multiple-output (MIMO) channel. The number of degrees of freedom of compute-and-forward is hence at most K. In this paper, we analyze the degrees of freedom achieved by the lattice coding implementation of compute-and-forward proposed recently by Nazer and Gastpar. We show that this lattice implementation achieves at most 2/(1+1/K) ≤ 2 degrees of freedom, thus exhibiting a very different asymptotic behavior than the MIMO upper bound. This raises the question if this gap of the lattice implementation to the MIMO upper bound is inherent to compute-and-forward in general. We answer this question in the negative by proposing a novel compute-and-forward implementation achieving K degrees of freedom.

We study the scaling of the capacity per unit energy of a wireless network as a function of the number of nodes and the deployment area. We show that in a network of n nodes located randomly in a region of area scaling linearly with n and communicating over Gaussian fading channels with power pathloss exponent α, the per-node capacity per unit energy scales essentially as Θ(n1-α/2) in the low path-loss regime (2 ≤ α ≤ 3) and essentially as Θ(n-1/2) in the high path-loss regime (α ≥ 3); while if the area is held constant, it scales essentially as Θ(n) and Θ(n(α-1)/2) in the low and high path-loss regimes, respectively. We propose a novel communication scheme, phase-aligned amplify-and-forward, which is shown to be order-optimal in the low path-loss regime-no other known scheme achieves the same scaling. We show that the well-known multi-hop scheme is order-optimal in the high path-loss regime.

We consider the Gaussian “diamond” or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains, and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound.

We analyze the asymptotic behavior of compute-and-forward relay networks in the regime of high signal-to-noise ratios. We consider a section of such a network consisting of K transmitters and K relays. The aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters. An upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays, transforming the network into a K × K multiple-input multiple-output (MIMO) channel. The number of degrees of freedom of compute-and-forward is hence at most K. In this paper, we analyze the degrees of freedom achieved by the lattice coding implementation of compute-and-forward proposed recently by Nazer and Gastpar. We show that this lattice implementation achieves at most 2=(1+1=K) ≤ 2 degrees of freedom, thus exhibiting a very different asymptotic behavior than the MIMO upper bound. This raises the question if this gap of the lattice implementation to the MIMO upper bound is inherent to compute-and-forward in general. We answer this question to the negative by proposing a novel compute-and-forward implementation achieving K degrees of freedom.

We consider arbitrary dense wireless networks, in which n nodes are placed in an arbitrary (deterministic) manner on a square region of unit area and communicate with each other over Gaussian fading channels. We provide inner and outer bounds for the n × n-dimensional unicast and the n × 2n-dimensional multicast capacity regions of such a wireless network. These inner and outer bounds differ only by a factor O(log(n)), yielding a fairly tight scaling characterization of the entire regions. The communication schemes achieving the inner bounds use interference alignment as a central technique and are, at least conceptually, surprisingly simple.

We consider arbitrary traffic patterns in arbitrarily placed extended wireless networks. We provide sufficient conditions for the approximate optimality of multi-hop communication over such networks. For exponential power decay, we show that these sufficient conditions are always satisfied, resulting in a scaling characterization of the entire capacity region for any node placement.

We consider the question of determining the scaling of then2-dimensional balanced unicast and then2n-dimensional balanced multicast capacity regions of a wireless network withnnodes placed uniformly at random in a square region of areanand communicating over Gaussian fading channels. We identify this scaling of both the balanced unicast and multicast capacity regions in terms of¿(n) , out of2ntotal possible, cuts. These cuts only depend on the geometry of the locations of the source nodes and their destination nodes and the traffic demands between them, and thus can be readily evaluated. Our results are constructive and provide optimal (in the scaling sense) communication schemes.

We study the problem of determining the multicast capacity region of a wireless network of n nodes randomly located in an extended area and communicating with each other over Gaussian fading channels. We obtain an explicit information- theoretic characterization of the scaling of the multicast capacity region for n nodes in terms of 2n weighted cuts. These cuts only depend on the geometry of the locations of the source nodes and their destination nodes and the traffic demands between them, and thus can be readily evaluated. The results are constructive and provide a two-layer architecture for achieving nearly the entire multicast capacity region in the scaling sense: The top layer routes traffic from each of the source nodes to its set of destination nodes, and the bottom layer physically distributes/concentrates traffic among appropriate nodes through one of the two cooperative communication schemes - hierarchical relaying and multi-hopping - depending on the wireless-channel characteristics.

We consider the problem of delivering content cached in a wireless network of n nodes randomly located on a square of area n. In the most general form, this can be analyzed by considering the 2ntimesn-dimensional caching capacity region of the wireless network. We propose a communication scheme for transmission of messages cached in the network. This provides an inner bound to the caching capacity region.

In recent work, Ozgur, Leveque, and Tse (2007) obtained a complete scaling characterization of throughput scaling for random extended wireless networks (i.e.,nnodes are placed uniformly at random in a square region of arean). They showed that for small path-loss exponentsalphaisin(2,3], cooperative communication is order optimal, and for large path-loss exponentsalpha>3, multihop communication is order optimal. However, their results (both the communication scheme and the proof technique) are strongly dependent on the regularity induced with high probability by the random node placement. In this paper, we consider the problem of characterizing the throughput scaling in extended wireless networks with arbitrary node placement. As a main result, we propose a more general novel cooperative communication scheme that works for arbitrarily placed nodes. For small path-loss exponentsalphaisin(2,3], we show that our scheme is order optimal for all node placements, and achieves exactly the same throughput scaling as in Ozgur. This shows that the regularity of the node placement does not affect the scaling of the achievable rates foralphaisin(2,3]. The situation is, however, markedly different for large path-loss exponentsalpha>3. We show that in this regime the scaling of the achievable per-node rates depends crucially on the regularity of the node placement. We then present a family of schemes that smoothly ldquointerpolaterdquo between multihop and cooperative communication, depending upon the level of regularity in the node placement. We establish order optimality of these schemes under adversarial node placement foralpha>3.

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application in many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm are known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. More precisely, we consider the impact of having a slowly time-varying domain over which the minimization takes place. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps somewhat surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.

A novel quickest detection setting is proposed, generalizing the well-known Bayesian change-point detection model. Suppose{(Xi,Yi)}iges 1 is a sequence of pairs of random variables, and thatSis a stopping time with respect to{Xi}iges 1. The problem is to find a stopping timeTwith respect to{Yi}iges 1 that optimally tracksS, in the sense thatTminimizes the expectedreactiondelay\BBE(T-S)+, while keeping thefalse-alarmprobabilityP(Talphaisin[0,1]. This problem formulation applies in several areas, such as in communication, detection, forecasting, and quality control.

We consider the problem of characterizing per node throughput scaling in arbitrary extended wireless networks. Recently, Özgür, Lévêque, and Tse (2007) obtained a complete characterization of throughput scaling for random extended networks (i.e., nodes are placed in a square region uniformly at random) under a fast fading channel model. They proposed a hierarchical cooperative communication scheme to establish this result. However, their results (both the communication scheme and the proof technique) are strongly dependent on the “regularity” induced with high probability by the random node placement. As a main result of this paper, we propose a more general (and very different) hierarchical cooperative communication scheme that works for arbitrarily placed nodes (with a minimum-separation requirement). Under our scheme, we obtain exactly the same per node throughput scaling as in Özgür et. al., showing that much less regularity is necessary for successful hierarchical cooperation. Our result holds under both fast and slow fading channel model. For small path-loss exponents α ∈ (2, 3], we show that our scheme is order optimal for all node placements with minimum-separation requirement. We also show that for certain node placements, our scheme is order optimal for all α ≫ 3 as well.

We consider the problem of characterizing per node throughput scaling in arbitrary extended wireless networks. For extended networks with random node placement, the following threshold phenomenon exists: for path loss exponent alpha les 3, hierarchical cooperative communication achieves the optimal throughput scaling; for alpha > 3, multi-hop communication achieves the optimal throughput scaling. We establish that for arbitrary node placement, due to the lack of ldquoregularityrdquo, such a threshold phenomenon does not exist. More precisely, while hierarchical cooperative communication is still order optimal for alpha les 3, there are node placements such that multi-hop communication is not order optimal for alpha > 3. We then present a family of schemes that smoothly ldquointerpolatesrdquo between multi-hop and hierarchical cooperative communication, depending upon the ldquolevel of regularityrdquo of the node placement. We establish optimality of these schemes under adversarial node placement for alpha > 3.

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables or equivalently of finding a point in the intersection of two sets. The iterative nature and simplicity of the algorithm has led to its application to many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm is quite well-known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. Our result is a generalization of the work by Csiszar and Tusnady on alternating minimization procedures. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.

We present a generalization of the well-known Bayesian change-point detection problem. Specifically, let {(Xi,Yi)}iges1be a sequence of pairs of random variables, and let S be a stopping time with respect to {Xi}iges1. We assume that the (Xi, Yi)'s take values in the same finite alphabet X times Y. For a fixed kappa ges 1, we consider the problem of finding a stopping time Ti}iges1that optimally tracks S, in the sense that T minimizes the average reaction time E(T - S)+, while it keeps the false-alarm probability P(Tkappa), and constructs the associated optimal stopping times T. In this paper, we provide a sufficient condition on {(Xi,Yi)}iges1and S under which the algorithm running time is polynomial in kappa, and we illustrate this condition with two examples: a Bayesian change-point problem and a pure tracking stopping time problem.

We consider communication through a cascade of discrete memoryless channels (DMCs). The source and destination node of this cascade are allowed to use coding schemes of arbitrary complexity, but the intermediate relay nodes are restricted to process only blocks of a fixed length. We investigate how the processing at the relays must be chosen in order to maximize the capacity of the cascade, that is, the maximum achievable end-to-end rate between the source and the destination. For infinite cascades with fixed intermediate processing length at the relays, we prove that this intermediate processing can be chosen to be identical without loss of optimality, and that the capacity of the cascade coincides with the rate of the best zero-error code of length equal to the block length of the intermediate processing. We further show that for fixed and identical intermediate processing at all relays, convergence of capacity as the length of the cascade goes to infinity is exponentially fast. Finally, we characterize how the block length of the intermediate processing must scale with the length of the cascade to guarantee a constant end-to-end rate. We prove that it is sufficient that the block length scales logarithmically with the network length in order to achieve any rate above the zero-error capacity. We show that in many cases of interest logarithmic growth is also necessary.

We consider communication over the Gaussian multiple access channel (MAC) with unknown set of active users. The proposed multiple access strategy is distributed and achieves a maximum sum rate point on the boundary of the capacity region for this channel for any set of active users S simultaneously, as if S were known at the transmitters. The proposed coding scheme splits each user into a set of virtual users, each of which can be decoded using a single-user decoder at the receiver instead of having to decode all users jointly. We also present a generalization of this scheme to the case where the channel gains differ between users and each user only knows its own channel gain.

We consider communication through a cascade of L identical discrete memoryless channels (DMCs). The source and destination node are allowed to use coding schemes of arbitrary complexity, but the intermediate relay nodes are restricted to process only blocks of N symbols. It is well known that for any L and N rarr infin the relays can use a capacity achieving code and communicate reliably as long as the rate of this code is below the capacity of the underlying DMC. The capacity of the cascade is hence equal to the network min-cut capacity. For finite N and L rarr infin, we showed in previous work that the optimal intermediate processing is the highest rate zero-error code of length N for the underlying DMC. The capacity of the cascade coincides with the rate of this zero-error code, and is always below the zero-error capacity. In this work, we characterize how N must scale with L in order to achieve rates in between the zero-error and the min-cut capacity. In particular, we have observed that N = thetas (log L) is sufficient to achieve any rate below the min-cut capacity. Here, we develop a novel upper bound on the capacity of cascades with optimal intermediate processing that applies for any (N, L) pairs and use it to show that N = thetas (log L) is necessary to achieve certain rates above the zero-error capacity. Furthermore, we propose a method to evaluate our upper bound by establishing a connection with the set-cover problem in algorithms

A class of structured LDPC codes-turbo-structured LDPC (TS-LDPC) codes-composed of two subtrees connected by an interleaver is introduced in this paper. TS-LDPC codes with good girth properties are easy to design: careful design of the interleaver component prevents short cycles in its Tanner graph. A methodology to design TS-LDPC codes with arbitrary column weight j/spl ges/2 and arbitrary girth is also presented. In addition, a complexity reduced decoding algorithm is described. Simulation results demonstrate the good performance of TS-LDPC codes when compared to random LDPC codes of the similar size and rate.

We introduce a method to design structured LDPC codes with large girth and flexible code rates. The method is simple to explain: we divide the nodes in the Tanner graph into groups and connect nodes in these groups according to a set of parameters called shifts. We derive a general theorem on the shifts to prevent small cycles. Simulations show that these codes, GS-LDPC codes, outperform random LDPC codes.