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Realization of Second-Order Structure of Recursive Algorithm for Discrete Cosine Transform

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Abstract

A computational and hardware-efficient second-order infinite impulse response filter structure is proposed in this paper. It can compute discrete cosine transform (DCT) with improved processing speed and is valid for N = 2r, where N is the length of the input sequence and r > 1. A new algorithm is also proposed in this paper which is an improvement over previously reported algorithms in the literature. The proposed algorithm reduces the total number of real multiplications and additions in comparison with the existing algorithms. Using the suggested algorithm, computational cycles required to compute a DCT coefficient are less which further reduces the truncation error while processing a long length of input data.

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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

Authors
  1. Pragati Dahiya
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  2. Priyanka Jain
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Corresponding author

Correspondence to Pragati Dahiya.

Additional information

The authors wish to acknowledge research fellowship from University Grants of Commission, Government of India.

Appendix

Appendix

Computation of DCT coefficients for an input sequence x(n) using MATLAB 8.3 version.

Given a sequence of N = 8 samples

$$ x\left( n \right) = \left[ {10 9 8 4 6 7 9 5} \right],\quad n = 0, 1, 2, \ldots ,7 $$
(A1)

Computation of X(k) using Eq. (1a)

$$ \begin{aligned}&X\left( k \right) = \left[ { 20.5061\;\;2.5347\;\;2.8837\;\;2.1439\;\; - 2.8284\;\;0.6550\;\; - 0.4291\;\;1.8842} \right],\\ &\qquad k = 0, 1, 2, \ldots ,7\end{aligned} $$
(A2)

From Eq. (11b), we get:

$$ w_{0} \left( n \right) = \left[ { 5 0 1 - 2} \right],\quad n = 0,1,2,3 $$
(A3)
$$ w_{1} \left( n \right) = \left[ {5 3} \right],\quad n = 0, 1 $$
(A4)
$$ w_{2} \left( n \right) = \left[ { - 8} \right],\quad n = 0 $$
(A5)
$$ x^{3} \left( 0 \right) = \left[ {58} \right] $$
(A6)

Using Eq. (17) to obtain \( G_{ji}^{p} \left( k \right) \)

$$ G_{3i}^{p} \left( k \right) = \left[ {5.0693 4.2877 1.3100 3.7685} \right],\quad i = 0, 1, 2, 3 $$
(A7)
$$ G_{1i}^{p} \left( k \right) = \left[ {5.7674 - 0.8582} \right],\quad i = 0, 1, 2 $$
(A8)
$$ G_{0i}^{p} \left( k \right) = \left[ { - 5.6569 } \right],\quad i = 0 $$
(A9)

By multiplying the above sequence \( G_{ji}^{p} \left( k \right) \) and Eq. (A6) by constant \( \sqrt {\frac{2}{N} } A_{k} \), we get X(k):

$$ X\left( k \right) = \left[ { 20.5061 2.5347 2.8837 2.1439 - 2.8284 0.6550 - 0.4291 1.8842} \right], \quad k = 0, 1, 2, \ldots ,7 $$
(A10)

It can be seen from Eq. (A10), and we obtained the DCT coefficients as given in (A2).

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Dahiya, P., Jain, P. Realization of Second-Order Structure of Recursive Algorithm for Discrete Cosine Transform. Circuits Syst Signal Process 38, 791–804 (2019). https://doi.org/10.1007/s00034-018-0885-6

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  • DOI: https://doi.org/10.1007/s00034-018-0885-6

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