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An alternative way of implementing this algorithm is to form an auxiliary function by copying the even elements of fj into the first N/2 locations, and the odd elements into the next N/2 elements in reverse order. | 12.4 FFT in Two or More Dimensions 521 An alternative way of implementing this algorithm is to form an auxiliary function by copying the even elements of fj into the first N 2 locations and the odd elements into the next N 2 elements in reverse order. However it is not easy to implement the alternative algorithm without a temporary storage array and we prefer the above in-place algorithm. Finally we mention that there exist fast cosine transforms for small N that do not rely on an auxiliary function or use an FFT routine. Instead they carry out the transform directly often coded in hardware for fixed N of small dimension 1 . CITED REFERENCES AND FURTHER READING Brigham E.O. 1974 The Fast Fourier Transform Englewood Cliffs NJ Prentice-Hall 10-10. Sorensen H.V. Jones D.L. Heideman M.T. and Burris C.S. 1987 IEEE Transactions on Acoustics Speech and Signal Processing vol. ASSP-35 pp. 849-863. Hou H.S. 1987 IEEE Transactions onAcoustics Speech andSignal Processing vol. ASSP-35 pp. 1455-1461 see for additional references . Hockney R.W. 1971 in Methods in Computational Physics vol. 9 New York Academic Press . Temperton C. 1980 Journal of Computational Physics vol. 34 pp. 314-329. Clarke R.J. 1985 Transform Coding of Images Reading MA Addison-Wesley . Gonzalez R.C. and Wintz P. 1987 Digital Image Processing Reading MA Addison-Wesley . Chen W. Smith C.H. and Fralick S.C. 1977 IEEE Transactions on Communications vol. COM-25 pp. 1004-1009. 1 12.4 FFT in Two or More Dimensions Given a complex function h k1 k2 defined over the two-dimensional grid 0 k1 N1 - 1 0 k2 N2 - 1 we can define its two-dimensional discrete Fourier transform as a complex function H n1 n2 defined over the same grid N2-1Ni-1 H ni n2 E E ex.p 2 ik2n2 N2 ex.p 2 ik1n1 N1 h k1 k2 2 0 fei 0 12.4.1 By pulling the subscripts 2 exponential outside of the sum over k1 or by reversing the order of summation and pulling the subscripts 1 outside of the sum over k2 we can see instantly that the two-dimensional FFT can be
