Discrete Fractional Fourier Transform

1
Discrete Fractional Fourier Transform

• Ideas for Today and Tomorrow

Shamim Nemati1.2 Supervisor Prof. Mark Yeary1
Supervisor Prof. Murad Ozaydin2 1School of
Electrical and Computer Engineering 2Department
of Mathematics
2
Contents

• Signal Representation
• Discrete Fourier Transform (DFT)
• Discrete Fractional Fourier Transform (DFrFT)
• Applications

3
Signal Representation

• Time Representation
• Imagine a data set of a thousand values
• X(t) 1, 5, 1, 5, 1, 5 (alternating 500 1s
  and 500 5s)
• Frequency Representation
• Let f0 1, 1, 1, 1 and f500 1, -1, 1, -1,
  . Then, it is obvious that the original data
  can be written as

Amplitude
Frequency
4
Discrete Fourier Transform

• Forward DFT denoted by

5
Integer Powers of DFT

• 0th power
• 1st power
• 2nd power
• 3rd power
• 4th power

6
Discrete Fractional Fourier Transform

• What does it mean?
• sth power of DFT ( ) is equivalent
• to a rotation by radians in
  the time-frequency plane!
• Example provides us with a 45o
  rotation in the time-frequency plane.

45o
7
Finding Chirp Rate
Chirp rate
Consider a chirp signal given by

• Calculate DFrFT for a discrete set of angles.
• Determine the angle ( ) at which the
  DFrFT has the highest magnitude.
• The chip rate is estimated via

8
References

• 1 B. Santhanam and J. McClellan, The Discrete
  Rotational Fourier Transform, IEEE Trans. on
  Signal Process., vol. 44, no. 4, APRIL 1996.
• 2 C. Candan, M. A. Kutay, and H. M. Ozaktas,
  The discrete fractional Fourier transform, IEEE
  Trans. Signal Process., vol. 48, no. 5, pp.
  1329-1337, May 2000.
• 3 Juan G. Vargas-Rubio and B. Santhanam, On
  the Multiangle Cenered Discrete Fractional
  Fourier Transform, IEEE Signal Process. Letters,
  vol. 12, no. 4, April 2005.
• 4 M. Ozaydin, S. Nemati, M. Yeary and V.
  DeBrunner, Orthogonal Projections and Discrete
  Fractional Fourier Transforms, IEEE DSP
  Workshop, September 2006, accepted.