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Design of Nonuniform Transmultiplexers with Block Samplers and Single-Input Single-Output Linear Time-Invariant Filters Based on Perfect Reconstruction Condition

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Abstract

This paper proposes a design of a nonuniform transmultiplexer with block samplers and single-input single-output linear time-invariant filters. First, the perfect reconstruction condition of the nonuniform transmultiplexer is derived. Then, the design problem is formulated as an optimization problem. In particular, the perfect reconstruction error is minimized in the \(L_{1}\) norm sense subject to the frequency selectivities of the filters. Computer numerical simulation results show that the designed nonuniform transmultiplexer with block samplers is robust to the channel noise and achieves a small reconstruction error.

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Acknowledgments

This paper was supported partly by the National Nature Science Foundation of China (No. 61372173), the Guangdong Higher Education Engineering Technology Research Center for Big Data on Manufacturing Knowledge Patent (No. 501130144), the Hundred People Plan from the Guangdong University of Technology and the Young Thousand People Plan from the Ministry of Education of China.

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Authors and Affiliations

  1. School of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

    Qing Liu, Bingo Wing-Kuen Ling & Zhijing Yang

  2. Faculty of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

    Qingyun Dai

  3. Australian Joint Research Centre for Building Information Modelling, School of Built Environment, Curtin University, Perth, WA, 6845, Australia

    Changzhi Wu & Xiangyu Wang

Authors
  1. Qing Liu
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  2. Bingo Wing-Kuen Ling
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  3. Qingyun Dai
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  4. Zhijing Yang
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  5. Changzhi Wu
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Corresponding author

Correspondence to Bingo Wing-Kuen Ling.

Appendix: The Proof of Theorem 1

Appendix: The Proof of Theorem 1

Proof

Since

$$\begin{aligned} G_{j,i} (z )= & {} H_{j}(z )F_{i}(z )=\left( {\sum \limits _{k=0}^{M-1} {z^{-k}H_{j,k} ({z^{M}} )}} \right) \left( {\sum \limits _{k=0}^{M-1} {z^{-k}F_{i,k} ({z^{M}} )}} \right) \\= & {} \left( {H_{j,0} ({z^{M}} )+z^{-1}H_{j,1} ({z^{M}} )+\cdots +z^{-({M-1} )}H_{j,M-1} ({z^{M}} )} \right) \left( F_{i,0} ({z^{M}} )\right. \\&+\,z^{-1}F_{i,1} ({z^{M}} )+\cdots +z^{-({M-1} )}F_{i,M-1} ({z^{M}} )) \\= & {} H_{j,0} ({z^{M}} )F_{i,0} ({z^{M}} )+z^{-1}H_{j,0} ({z^{M}} )F_{i,1} ({z^{M}} )\\&+\cdots +z^{-({M-1} )}H_{j,0} ({z^{M}} )F_{i,M-1} ({z^{M}} ) \\&+\,z^{-1}H_{j,1} ({z^{M}} )F_{i,0} ({z^{M}} )+z^{-2}H_{j,1} ({z^{M}} )F_{i,1} ({z^{M}} )\\&+\cdots +z^{-M}H_{j,1} ({z^{M}} )F_{i,M-1} ({z^{M}} ) \\&+\cdots \\&+\,z^{-({M-1} )}H_{j,M-1} ({z^{M}} )F_{i,0} ({z^{M}} )+z^{-M}H_{j,M-1} ({z^{M}} )F_{i,1} ({z^{M}} )\\&+\cdots +z^{-2({M-1} )}H_{j,M-1} ({z^{M}} )F_{i,M-1} ({z^{M}} ) \\= & {} H_{j,0} ({z^{M}} )F_{i,0} ({z^{M}} )+\,z^{-1}H_{j,0} ({z^{M}} )F_{i,1} ({z^{M}} )\\&+\cdots +z^{-({M-1} )}H_{j,0} ({z^{M}} )F_{i,M-1} ({z^{M}} ) \\&+\,z^{-M}H_{j,1} ({z^{M}} )F_{i,M-1} ({z^{M}} )+z^{-1}H_{j,1} ({z^{M}} )F_{i,0} ({z^{M}} )\\&+\,z^{-2}H_{j,1} ({z^{M}} )F_{i,1} ({z^{M}} )\\&+\cdots +z^{-({M-1} )}H_{j,1} ({z^{M}} )F_{i,M-2} ({z^{M}} ) \\&+\cdots \\&+\,z^{-M}H_{j,M-1} ({z^{M}} )F_{i,1} ({z^{M}} )+z^{-({M+1} )}H_{j,M-1} ({z^{M}} )F_{i,2} ({z^{M}} )\\&+\cdots +z^{-({2M-2} )}H_{j,M-1} ({z^{M}} )F_{i,M-1} ({z^{M}} )\\&+\,z^{-({M-1} )}H_{j,M-1} ({z^{M}} )F_{i,0} ({z^{M}} ) \\= & {} \sum \limits _{k=0}^{M-1} {\sum \limits _{l=0}^{M-1} {z^{-\left( {\hbox {mod}({M+k-l,M} )+l} \right) }H_{j,\hbox {mod}({M+k-l,M} )} ({z^{M}} )F_{i,l} ({z^{M}} )}} \\= & {} \sum \limits _{k=0}^{M-1} {\sum \limits _{l=0}^{M-1} {z^{-({k+\frac{M}{2}\left( {1+\hbox {sgn}_{1}({l-k} )} \right) } )}H_{j,\hbox {mod}({M+k-l,M} )} ({z^{M}} )F_{i,l} ({z^{M}} )}} \\= & {} \sum \limits _{k=0}^{M-1} {z^{-k}G_{j,i,k} ({z^{M}} )}, \end{aligned}$$

we have \(G_{j,i,k} ({z^{M}} )=\sum \limits _{l=0}^{M-1} {z^{-\frac{M}{2}\left( {1+\hbox {sgn}_{1}({l-k} )} \right) }H_{j,\hbox {mod}({M+k-l,M} )} ({z^{M}} )F_{i,l} ({z^{M}} )} \) for \(i=0,1,\ldots ,N-1\), for \(j=0,1,\ldots ,N-1\) and for \(k=0,1,\ldots ,M-1\). This completes the proof. \(\square \)

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Liu, Q., Ling, B.WK., Dai, Q. et al. Design of Nonuniform Transmultiplexers with Block Samplers and Single-Input Single-Output Linear Time-Invariant Filters Based on Perfect Reconstruction Condition. Circuits Syst Signal Process 35, 4081–4098 (2016). https://doi.org/10.1007/s00034-015-0238-7

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