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This paper considers truncated type-II hybrid automatic repeat-request (ARQ) schemes with noisy feedback over block fading channels. With these ARQ techniques, the number of retransmissions is limited, and, similar to forward error correction (FEC), error-free delivery of data packets cannot be guaranteed. Bounds on the average number of transmissions, the average coding rate as well as the reliability of the schemes are derived using random coding techniques, and the performance is compared with FEC. The random coding bounds reveal the achievable performance with block codes and maximum-likelihood soft-decision decoding. Union upper bounds and simulation results show that over block fading channels, these bounds can be closely approached with simple terminated convolutional codes and soft-decision Viterbi decoding. Truncated type-II hybrid ARQ and the corresponding FEC schemes have the same probability of packet erasure; however, the truncated ARQ schemes offer a trade-off between the average coding rate and the probability of undetected error. Truncated ARQ schemes have significantly higher average coding rates than FEC at high and medium signal-to-noise ratio even with noisy feedback. Truncated ARQ can be viewed as adaptive FEC that adapts to the instantaneous channel conditions. Index Terms-Block fading channel, error control, noisy feedback, random coding bound, truncated ARQ, union bound.

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The demand for mobile data is likely to grow at a pace more than envisaged in the coming years. Further, as applications such as the Internet of Things (IoT) come to fruition, there will be increased diversity in the types of devices demanding Internet connectivity and their requirements. Significant increase in data rate requirements is also expected due to services such as Ultra High Definition (UHD) video streaming and cloud computing. To meet all these demands, physical layer waveform candidates for future generations of communications need to be robust and inherently capable of extending into multiple domains (space, time, frequency, users, transmission media, code etc.) to ensure efficient utilization of resources. Multiple domains can be innately integrated into the design process of modulation schemes by using tensors, which are multi-way arrays. This paper introduces a unified tensor framework, providing a foundation for multi-domain communication systems that can be used to represent, design and analyse schemes that span several domains. Transmitted signals are represented by Nth order time function tensors which are coupled, using a system tensor of order N + M, with the received signals which are represented by another tensor of order M through the contracted convolution. We begin with the continuous time representation of the tensor system model and present both the strict multi-domain generalization of the Nyquist criterion for zero interference (inter-tensor and intra-tensor interference) as well as a relaxation. We present an equivalent discrete time system model, and as an example of using the tensor framework we derive tensor based linear equalization methods to combat multi-domain interference. An application to multi-user MIMO-GFDM illustrates the utility of this novel framework for derivation of joint domain signal processing techniques.

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