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Implementations of Sorted-QR Decomposition for MIMO Receivers: Complexity, Reusability and Efficiency Analysis

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Matrix decomposition of the channel matrix in the form of QR decomposition (QRD) is needed for advanced multiple input and multiple output (MIMO) demapping algorithms like sphere decoder. Due to the computation-intensive nature of the QRD, its implementation has to be highly efficient. Flexibility in several forms, e.g. support for different algorithms, reusability of wireless implementations, portability, etc. is highly sought in wireless devices. The contradictory nature of flexibility and efficiency requires tradeoffs to be made between them in system development. In this paper, we have analyzed such tradeoffs by implementing two minimum mean squared error-sorted QRD algorithms. The algorithms have been implemented in four different methods with varying degree of reusability and in five different forms of portability. The performance of the implementations is evaluated by using the real-time constraints from the LTE standard. For all the implementations, modular equations for accurately estimating the execution time are derived.

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  1. 1.

    In this context, the term waveform is used to refer a complete wireless standard like UMTS, LTE, etc.

  2. 2.

    Mapping denotes the process of spatially and temporally distributing the functionality of a waveform in an efficient way onto the resources of a hardware such that the constraints of waveform and requirements of system designer are met.

  3. 3.

    Unitary matrix is the complex equivalent of an orthogonal matrix. A n ×n dimensional matrix U of complex entries is said to be unitary when \(\mathbf{U}^{H}\mathbf{U}=\mathbf{U}\mathbf{U}^{H}= \mathbf{I_n}\). A n ×n dimensional matrix Q of real entries is said to be orthogonal when Q T = Q  − 1 implying QQ T = QQ T = I.

  4. 4.

    Q format represents the fixed-point number format where the number of fractional bits and integer bits is specified. For example, Q2.13 represents a 16-bit number with a sign bit, 2 integer bits and 13 fractional bits.


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This research project was performed in the Ultra High-Speed Mobile Information and Communication (UMIC) research centre under the support of the Technical Center for Information Technology and Electronics (WTD-81), Germany.

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Correspondence to Venkatesh Ramakrishnan.

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Ramakrishnan, V., Veerkamp, T., Ascheid, G. et al. Implementations of Sorted-QR Decomposition for MIMO Receivers: Complexity, Reusability and Efficiency Analysis. J Sign Process Syst 69, 41–53 (2012). https://doi.org/10.1007/s11265-011-0649-z

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  • MIMO
  • Sorted QRD
  • Modified Gram Schmidt
  • Givens rotation
  • Software defined radio
  • Mapping exploration
  • Flexibility
  • Portability
  • Reusability
  • Efficiency