We consider the transmission of correlated sources over a multiple access channel(MAC). Multiple access channels are important building blocks in many practical communication systems, e.g., local area networks(LAN), cellular systems, wireless multi-hop networks. Thus this topic has been studied for last several decades. One recent motivation is estimating a random field via wireless sensor networks. Often the sensor nodes are densely deployed resulting in correlated observations. These sensor nodes need to transmit their correlated observations to a fusion center which uses this data to estimate the sensed random field. Sensor nodes have limited computational and storage capabilities and very limited energy. Since transmission is very energy intensive, it is important to minimize it. This motivates our problem of energy efficient transmission of correlated sources over a sensor network. Sensor networks are often arranged in a hierarchical fashion. Neighboring nodes can first transmit their data to a cluster head which can further compress information before transmission to the fusion center. The transmission of data from sensor nodes to their cluster-head is usually through a MAC. At the fusion center the underlying physical process is estimated. The main trade-off possible is between the rates at which the sensors send their observations and the distortion incurred in estimation at the fusion center. The availability of side information at the encoders and/or the decoder can reduce the rate of transmission. In this thesis, the above scenario is modeled as an information theoretic problem. Efficient joint source-channel codes are discussed under various assumptions on side information and distortion criteria. Sufficient conditions for transmission of discrete/continuous alphabet sources with a given distortion over a discrete/continuous alphabet MAC are given. We recover various previous results as special cases from our results. Furthermore, we study the practically important case of the Gaussian MAC(GMAC) in detail and propose new joint source-channel coding schemes for discrete and continuous sources. Optimal schemes are identified in different scenarios. The protocols like TDMA, FDMA and CDMA are widely used across systems and standards. When these protocols are used the MAC becomes a system of orthogonal channels. Our general conditions can be specialized to obtain sufficient conditions for lossy transmission over this system. Using this conditions, we identify an optimal scheme for transmission of Gaussian sources over orthogonal Gaussian channels and show that the Amplify and Forward(AF) scheme performs close to the optimal scheme even at high SNR. Next we investigate transmission of correlated sources over a fast fading MAC with perfect or partial channel state information available at both the encoders and the decoder. We provide sufficient conditions for transmission with given distortions. We also provide power allocation policies for efficient transmission. Next, we use MAC with side information as a building block of a hierarchical sensor network. For Gaussian sources over Gaussian MACs, we show that AF performs well in such sensor network scenarios where the battery power is at a premium. We then extend this result to the hierarchical network scenario and show that it can perform favourably to the Slepian-Wolf based source coding and independent channel coding scheme. In a hierarchical sensor network the cluster heads often need to send only a function of the sensor observations to the fusion center. In such a setup the sensor nodes can compress the data sent to the cluster head exploiting the correlation in the data and also the structure of the function to be computed at the cluster head. Depending upon the function, exploiting the structure of the function can substantially reduce the data rate for transmission. We provide efficient joint source-channel codes for transmitting a general class of functions of the sources over the MAC.

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