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Extrapolation of signals using the method of alternating projections in fractional Fourier domains

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Abstract

The need for extrapolation of signals in time domain or frequency domain often arises in many applications in the area of signal and image processing. One of the approaches used for the extrapolation of the signals is the method of alternating projections (MAP) in conventional Fourier domains (CFD). Here we propose an extension of this approach using the fractional Fourier transform (FRFT) called here as the method of alternating projections in the FRFT domains (MAPFD). It is shown through the simulation results that the mean square error (MSE) between the true signal and the extrapolated signal obtained from the given signal is a function of the angle parameter of the FRFT, and the MAPFD gives lower MSE than the MAP in the CFD for the class of signals bandlimited in the FRFT domains, e.g., chirp signals. Moreover, the performance of the extrapolation using the MAPFD is shown to be shift-variant along the time axis.

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Abbreviations

α :

FRFT angle parameter

F α (u):

FRFT of the signal f (t)

K α (t,u):

The kernel of the FRFT

\(\varsigma\) :

The time-limiting operation

\({\Im}\) :

The identity operator

\({\wp}\) :

The bandlimiting operator in the CFD

\({\wp}_{\alpha}\) :

The bandlimiting operator in the FRFT domain with angle \(\alpha \ne \pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2\)

Ω α :

Bandwidth of the signal f (t) in the FRFT domain α

rect(t/T):

Rectangular window of width T around origin of the time axis

g n (t):

The extrapolated signal after nth iteration

f e (t):

The error signal between the extrapolated signal g n (t) and the true signal f (t)

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Malaviya National Institute of Technology, Jaipur, India

    K. K. Sharma

  2. Department of Electrical Engineering, Indian Institute of Technology, New Delhi, India

    S. D. Joshi

Authors
  1. K. K. Sharma
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  2. S. D. Joshi
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Correspondence to K. K. Sharma.

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Sharma, K.K., Joshi, S.D. Extrapolation of signals using the method of alternating projections in fractional Fourier domains. SIViP 2, 177–182 (2008). https://doi.org/10.1007/s11760-007-0047-y

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  • DOI: https://doi.org/10.1007/s11760-007-0047-y

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