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Fractional Fourier domain analysis of decimation and interpolation

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Abstract

The sampling rate conversion is always used in order to decrease computational amount and storage load in a system. The fractional Fourier transform (FRFT) is a powerful tool for the analysis of nonstationary signals, especially, chirp-like signal. Thus, it has become an active area in the signal processing community, with many applications of radar, communication, electronic warfare, and information security. Therefore, it is necessary for us to generalize the theorem for Fourier domain analysis of decimation and interpolation. Firstly, this paper defines the digital frequency in the fractional Fourier domain (FRFD) through the sampling theorems with FRFT. Secondly, FRFD analysis of decimation and interpolation is proposed in this paper with digital frequency in FRFD followed by the studies of interpolation filter and decimation filter in FRFD. Using these results, FRFD analysis of the sampling rate conversion by a rational factor is illustrated. The noble identities of decimation and interpolation in FRFD are then deduced using previous results and the fractional convolution theorem. The proposed theorems in this study are the bases for the generalizations of the multirate signal processing in FRFD, which can advance the filter banks theorems in FRFD. Finally, the theorems introduced in this paper are validated by simulations.

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Abbreviations

FT:

Fourier transform

FD:

Fourier domain

FRFT:

Fractional Fourier transform

FRFD:

Fractional Fourier domain

DTFRT:

Discrete-time FRFT

F p [.]:

Notation for FRFT with order p

X p (u):

Transform result of x(t) for FRFT with order p

\(\tilde F_p [.]\) :

Notation for DTFRT with order p

\(\tilde X_p [.]\) :

Transform result of x(t) for D TFRT with order p

L,M:

The integer factor

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Authors and Affiliations

  1. Department of Electronic Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Meng XiangYi, Tao Ran & Wang Yue

Authors
  1. Meng XiangYi
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  2. Tao Ran
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  3. Wang Yue
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Corresponding author

Correspondence to Tao Ran.

Additional information

Supported partially by the National Natural Science Foundation of China (Grant Nos. 60232010 and 60572094), and the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 60625104)

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Meng, X., Tao, R. & Wang, Y. Fractional Fourier domain analysis of decimation and interpolation. SCI CHINA SER F 50, 521–538 (2007). https://doi.org/10.1007/s11432-007-0040-7

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  • DOI: https://doi.org/10.1007/s11432-007-0040-7

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