Abstract

Direct calculation of fractional Fourier transforms from the expressions derived for their optical implementation is laborious. An extension of the discrete Fourier transform would have only O(N2) computational complexity. We define such a system, offer a general way to compute the fractional discrete Fourier transform matrix, and numerically validate the algorithm.

© 1996 Optical Society of America

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References

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  1. A. W. Lohmann, B. H. Soffer, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
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1994 (1)

1993 (2)

A. W. Lohmann, B. H. Soffer, J. Opt. Soc. Am. A 10, 2181 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, Opt. Commun. 101, 163 (1993).
[CrossRef]

1982 (1)

1980 (1)

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Equations (31)

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H ( f n ) = h ( t ) exp ( 2 π i f n t ) d t k = 0 N 1 h k exp ( 2 π i f n t k ) Δ t Δ t k = 0 N 1 h k exp ( 2 π i k n / N ) ,
i = 1 ,             f n = n N Δ t , n = N 2 , , N 2 ,             t k = k Δ t .
H n = H N n .
W exp ( 2 π i / N ) ,
H = M N × N h ,
M n k = W n × k exp ( 2 π i n k / N ) .
N = 2 m .
H n = k = 0 N 1 h k W n × k .
n = ( n m 1 n m 2 n 1 n 0 ) = n m 1 × 2 m 1 + n m 2 × 2 m 2 + + n 1 × 2 + n 0 , k = ( k m 1 k m 2 k 1 k 0 ) = k m 1 × 2 m 1 + k m 2 × 2 m 2 + + k 1 × 2 + k 0 , n i , k i = 0 or 1 , i = 0 , 1 , 2 , m 1 ,
H n k 0 k 1 k m 2 k m 1 h ( k m 1 k m 2 k 1 k 0 ) × W ( n m 1 n m 2 n 1 n 0 ) ( k m 1 k m 2 k 1 k 0 ) .
W N = W 2 m = W j × 2 m 1 , H n k 0 { k 1 k m 2 [ k m 1 h ( k m 1 k m 2 k 1 k 1 ) W k m 1 ( 00 0 n 0 ) ] × W k m 2 ( 00 n 1 n 0 ) W k 1 ( 0 n m 2 n 1 n 0 ) } W k 0 ( n m 1 n m 2 n 1 n 0 ) .
F 0 [ f ( x ) ] = f ( x ) ,
F 1 [ f ( x ) ] = F ( u ) , ( the conventional Fourier transform ) ,
F α { F β [ f ( x ) ] } = F α + β [ f ( x ) ] , 0 α , β 1 .
F α { F β [ f ( x ) ] } = F β { F α [ f ( x ) ] } .
H = M N × N α h , 0 α 1 ,
M 0 = I ( the identity matrix ) ,
M 1 = M ( the DFT matrix ) ,
M α M β = M α + β ,
M n k α = M k n α ( symmetric condition ) ,
M α M β = M β M α .
H = ( M N × N 1 / P ) P h , P = 1 , 2 , 3 , ,
( M 1 / P ) n k P = exp ( 2 π i n k / N ) .
M M H = M H M .
M = U Λ U H ,
M 1 / P = U Λ 1 / P U H .
M 1 / P = [ 2 1 4 2 2 2 + 1 4 + 2 2 1 4 2 2 1 4 + 2 2 ] × [ 2 1 / P 0 0 ( 2 ) 1 / P ] [ 2 1 4 2 2 1 4 2 2 2 + 1 4 + 2 2 1 4 + 2 2 ] .
M 2 × 2 1 / 2 = [ 0.1742 + 1.0151 i 0.4204 0.4204 i 0.4204 0.4204 i 1.0150 + 0.1742 i ] , M 2 × 2 1 / 3 = [ 0.6434 + 0.8297 i 0.1984 0.3437 i 0.1984 0.3437 i 1.0403 + 0.1424 i ] , M 2 × 2 1 / 4 = [ 0.8179 + 0.6582 i 0.1129 0.2726 i 0.1129 0.2726 i 1.0437 + 0.1129 i ] .
M 1 / P = [ 1 2 1 2 1 2 0 1 2 1 2 0 1 2 1 2 1 2 1 2 0 1 2 1 2 0 1 2 ] × [ 2 1 / P ( 2 ) 1 / P ( 2 i ) 1 / P 2 1 / P ] × [ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 1 2 0 0 1 2 0 1 2 ] .
( M N × N 1 / P ) P = ( A N × N + i B N × N ) P .
M 2 × 2 1 / 2 = [ 0.1742 + 1.0151 i 0.4204 0.4204 i 0.4204 0.4204 i 1.0150 + 0.1742 i ] , M 2 × 2 1 / 3 = [ 0.6434 + 0.8297 i 0.1984 0.3437 i 0.1984 0.3437 i 1.0403 + 0.1424 i ] , M 2 × 2 1 / 4 = [ 0.4985 + 0.6582 i 0.6582 0.2726 i 0.6582 0.2726 i 1.8179 + 0.1129 i ] .

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