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Efficient MDCT Recursive Structure for VLSI Implementation

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Abstract

Modified discrete cosine transform (MDCT) is a computationally intensive fundamental processing block for mapping time into frequency in various versions of MPEG audio coding. An efficient algorithm for MDCT computation is a basic requirement to realize low-cost audio codec. This paper introduces a new recursive algorithm and architecture for MDCT. The proposed architecture requires minimum hardware and execution time. Folding of the input sequence is performed multiple times in the subsequent groups which results in a significant reduction in real multiplication and real addition operations. Also, fewer number of execution cycles are required to generate the output through this algorithm. The proposed structure is suitable for parallel VLSI implementation as it is simple, regular and modular.

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Acknowledgements

The author would like to acknowledge the anonymous reviewers for their valuable comments. This study was supported by University Grant of Commissions, Government of India [Grant No. 3579(NET-DEC.2013)].

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Authors and Affiliations

  1. Department of Electronic and Communication Engineering, Delhi Technological University, Bawana Road, Delhi, India

    Pragati Dahiya & Priyanka Jain

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  1. Pragati Dahiya
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  2. Priyanka Jain
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Correspondence to Priyanka Jain.

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Appendix

Appendix

1.1 Computation of MDCT Coefficients for an Input Sequence \( x\left( n \right) \) Using MATLAB 8.5 Version

Given a sequence of \( N = 16 \) samples

$$ x\left( n \right) = \left[ {9, 5, 10, 4, 8, 12, 6, 7, 23, 17, 11, 32, 15, 8, 16, 17 } \right],\quad n = 0, 1, 2, \ldots , 15 $$
(34)

Computation of \( X\left( k \right) \) using Eq. (1)

$$ X\left( k \right) = \left[ { - 125.1546, - 39.4699,15.3834,6.2849, - 46.5619, - 35.1487,1.7654,15.2230 } \right], \quad k = 0, 1, 2, \ldots , 7 $$
(35)

Changing the order of input sequence by applying Eqs. (4) and (5) in Eq. (34)

$$ g\left( i \right) = \left[ { - 15, - 8, - 16, - 17,9,5,10,4,8,12,6,7,23,17,11,32} \right], \quad i = 0, 1, 2, \ldots , 15 $$
(36)

From Eq. (23b), we get

$$ w_{0} \left( n \right) = \left[ { - 3.2091, - 6.4826, - 17.6485, - 27.4492 } \right],\quad n = 0, 1, 2,3 $$
(37)
$$ w_{1} \left( n \right) = \left[ { - 19.7217, 6.3975} \right],\quad n = 0,1 $$
(38)
$$ w_{2} \left( n \right) = \left[ { - 35.1727} \right],\quad n = 0 $$
(39)
$$ x^{3} \left( 0 \right) = \left[ { - 101.4571 } \right] $$
(40)

Using Eq. (32) to obtain \( H_{ji}^{p} \left( k \right) \)

$$ H_{3i}^{p} \left( k \right) = \left[ { - 23.6976, 31.1557, - 21.6911,15.2230} \right],\quad i = 0, 1, 2, 3 $$
(41)
$$ H_{1i}^{p} \left( k \right) = \left[ { - 15.7723, - 13.4577 } \right],\quad i = 0, 1, 2 $$
(42)
$$ H_{0i}^{p} \left( k \right) = \left[ { - 24.8708} \right],\quad i = 0 $$
(43)

By Eq. (33), we get MDCT coefficients:

$$ X\left( k \right) = \left[ { - 125.1546, - 39.4699,15.3834,6.2849, - 46.561, - 35.1487,1.7654,15.2230} \right],\quad k = 0,1,2, \ldots ,7 $$
(44)

From Eq. (44), we obtained the MDCT coefficients as given in (35).

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Dahiya, P., Jain, P. Efficient MDCT Recursive Structure for VLSI Implementation. Circuits Syst Signal Process 39, 1372–1386 (2020). https://doi.org/10.1007/s00034-019-01195-x

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