Re: FFT algorithm for fixed number of samples



On Thu, 10 Jul 2008 01:23:57 -0700 (PDT), ARH <haghdoost@xxxxxxxxx>
wrote:

Hi

I am looking for a FFT algorithm for fixed number of sampling input.
usual FFT algorithms like Cooly-Tuky, radix-2, give dynamic number of
input samples an then divide and concoure it for calculation of the
output, but in my case study the number of input samples is static and
I can easily compute all of the twiddle factors before run time and
save them in a twiddle factor table.
Using twiddle factor table, FFT algorithm simplify to the matrix
multiplication algorithm which gave inputs them multiply proper
twiddle factor on it and generate output.
but this algorithm is similar to DFT not FFT, would you mind help me
to found an efficient algorithm for this purpose ?

Are there any material which described radix-2 FFT algorithm in
detail ? most of the textbook like "numerical receipts" just give the
implementation of algorithm and have pure description detail.

Regards
Alireza Haghdoost

Hi,
here are more FFT references than you probably
want but, who knows, maybe something here will
help you.

[-Rick-]
-----
[1] M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss
and the history of the FFT," IEEE Acoustics, Speech, and Signal
Processing Magazine, vol. 1, pp. 14-21, October 1984. also in IEEE
Press FFT Reprints, by P. Duhamel, 1995.
[2] H. V. Sorensen, C. S. Burrus, and M. T. Heideman, Fast Fourier
Transform Database. Boston: PWS Publishing, 1995. Update of
Technical Report TR-8402.
[3] L. R. Rabiner and C. M. Rader, eds., Digital Signal Processing,
selected reprints. New York: IEEE Press, 1972.
[4] DSP Committee, ed., Digital Signal Processing II, selected
reprints. New York: IEEE Press, 1979.
[5] DSP Committee, ed., Programs for Digital Signal Processing. New
York: IEEE Press, 1979.
[6] P. Duhamel, ed., Papers on the Fast Fourier Transform. New York:
IEEE Press, to appear in 1995.
[7] J. H. McClellan and C. M. Rader, Number Theory in Digital Signal
Processing. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1979.
[8] H. J. Nussbaumer, Fast Fourier Transform and Convolution
Algorithms. Heidelberg, Germany: Springer-Verlag, second ed., 1981,
1982.
[9] R. E. Blahut, Fast Algorithms for Digital Signal Processing.
Reading, MA: Addison-Wesley, Inc., 1984.
[10] M. T. Heideman, Multiplicative Complexity, Convolution, and the
DFT. Springer-Verlag, 1988.
[11] R. Tolimieri, M. An, and C. Lu, Algorithms for Discrete Fourier
Transform and Convolution. New York: Springer-Verlag, 1989.
[12] D. G. Myers, Digital Signal Processing, Efficient
Convolution and Fourier Transform Techniques. Sydney, Australia:
Prentice-Hall, 1990.
[13] R. E. Blahut, Algebraic Methods for Signal Processing and
Communications Coding. New York: Springer-Verlag, 1992.
[14] W. L. Briggs and V. E. Henson, The DFT: An Owner's Manual for
the Discrete Fourier Transform. Philadelphia: SIAM, 1995.
[15] W. W. Smith and J. M. Smith, Handbook of Real-Time Fast Fourier
Transforms. New York: IEEE Press, 1995.
[16] C. S. Burrus and T. W. Parks, DFT/FFT and Convolution
Algorithms. New York: John Wiley & Sons, 1985.
[17] J. S. Lim and A. V. Oppenheim, Advanced Topics in Signal
Processing, ch. 4. Prentice-Hall, 1988.
[18] J. W. Cooley, "How the FFT gained acceptance," IEEE Signal
Processing Magazine, vol. 9, pp. 10-13, January 1992. Also presented
at the ACM Conference on the History of Scientific and
Numerical Computation, Princeton, NJ, May 1987 and published in: A
History of Scientific Computing, edited by S. G. Nash, Addison-Wesley,
1990, pp. 133-140.
[19] P. Duhamel and M. Vetterli, "Fast Fourier transforms: a tutorial
review and a state of the art," Signal Processing, vol. 19, pp.
259-299, April 1990.
[20] J. W. Cooley, "The structure of FFT algorithms," April 1990.
Notes for a Tutorial at IEEE ICASSP-90.
[21] S. Winograd, "On computing the discrete Fourier transform,"
Mathematics of Computation, vol. 32, pp. 175-199, January 1978.
[22] S. Winograd, "On the multiplicative complexity of the
discrete Fourier transform," Advances in Mathematics, vol. 32, pp.
83-117, May 1979. also in[7].
[23] S. Winograd, Arithmetic Complexity of Computation. SIAM CBMS-NSF
Series, No. 33, Philadelphia: SIAM, 1980.
[24] J. W. Cooley and J. W. Tukey, "An algorithm for the machine
calculation of complex Fourier series," Math. Computat., vol. 19, pp.
297-301, 1965.
[25] R. Yavne, "An economical method for calculating the discrete
Fourier transform," in Proceedings of the Fall Joint Computer
Conference, pp. 115-125, 1968.
[26] P. Duhamel and H. Hollmann, "Split radix FFT algorithm,"
Electronic Letters, vol. 20, pp. 14-16, January 5 1984.
[27] P. Duhamel, "Implementation of `split-radix' FFT algorithms for
complex, real, and real-symmetric data," IEEE Trans. on ASSP, vol. 34,
pp. 285-295, April 1986. A shorter version appeared in the ICASSP-85
Proceedings, p. 20.6, March 1985.
[28] J. B. Martens, "Recursive cyclotomic factorization - a
new algorithm for calculating the discrete Fourier transform,"
IEEE Trans. on ASSP, vol. 32, pp. 750-762, August 1984.
[29] M. Vetterli and H. J. Nussbaumer, "Simple FFT and DCT
algorithms with reduced number of operations," Signal Processing,
vol. 6, pp. 267-278, August 1984.
[30] M. Vetterli and P. Duhamel, "Split-radix algorithms for length -
pm DFT's," IEEE Trans. on ASSP, vol. 37, pp. 57-64, January 1989.
Also, ICASSP-88 Proceedings, pp. 1415-1418, April 1988.
[31] R. Stasinski, "The techniques of the generalized fast Fourier
transform algorithm," IEEE Transactions on Signal Processing, vol. 39,
pp. 1058-1069, May 1991.
[32] H. V. Sorensen, M. T. Heideman, and C. S. Burrus, "On
calculating the split-radix FFT," IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 34, pp. 152-156, February 1986.
[33] A. Saidi, "Decimation-in-time-frequency FFT algorithm," in
Proceedings of the IEEE International Conference on Acoustics,
Speech, and Signal Processing, (IEEE ICASSP-94, Adelaide,
Australia), pp. III:453-456, April 19-22 1994.
[34] M. T. Heideman and C. S. Burrus, "On the number of
multiplications necessary to compute a length-2n DFT," IEEE
Transactions on Acoustics, Speech, and Signal Processing, vol. 34, pp.
91-95, February 1986.
[35] J. K. Beard, "An in-place, self-reordering FFT," in Proceedings
of the ICASSP-78, (Tulsa), pp. 632- 633, April 1978.
[36] H. W. Johnson and C. S. Burrus, "An in-place, in-order radix-2
FFT," in ICASSP-84 Proceedings, p. 28A.2, March 1984.
[37] C. Temperton, "Self-sorting in-place fast Fourier transforms,"
SIAM Journal of Sci. Statist. Comput., vol. 12, no. 4, pp. 808-823,
1991.
[38] C. S. Burrus, "Unscrambling for fast DFT algorithms," IEEE
Transactions on Acoustics, Speech, and Signal Processing, vol. 36, pp.
1086-1087, July 1988.
[39] P. R"osel, "Timing of some bit reversal algorithms," Signal
Processing, vol. 18, pp. 425-433, December 1989.
[40] J. Jeong and W. J. Williams, "A fast recursive bit-reversal
algorithm," in Proceedings of the ICASSP- 90, (Albuquerque, NM), pp.
1511-1514, April 1990.
[41] D. M. W. Evans, "A second improved digit-reversal permutation
algorithm for fast transforms," IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 37, pp. 1288-1291, August 1989.
[42] J. J. Rodriguez, "An improved FFT digit-reversal
algorithm," IEEE Transactions on Acoustics, Speech, and Signal
Processing, vol. 37, pp. 1298-1300, August 1989.
[43] J. S. Walker, "A new bit reversal algorithm," IEEE Transactions
on Acoustics, Speech, and Signal Processing, vol. 38, pp. 1472-1473,
August 1990.
[44] A. A. Yong, "A better FFT bit-reversal algorithm without
tables," IEEE Transactions on Signal Processing, vol. 39, pp.
2365-2367, October 1991.
[45] D. Sundararajan, M. O. Ahamad, and M. N. S. Swamy, "A fast FFT
bit-reversal algorithm," IEEE Transactions on Circuits and Systems,
II, vol. 41, pp. 701-703, October 1994.
[46] J. M. Rius and R. De Porrata-Doria, "New FFT bit-reversal
algorithm," IEEE Transactions on Signal Processing, vol. 43, pp.
991-994, April 1995.
[47] L. R. Morris, Digital Signal Processing Software. Toronto,
Canada: DSPSW, Inc., 1982, 1983.
[48] R. Meyer and K. Schwarz, "FFT implementation on DSP-chips,"
Sept. 18 1990. preprint.
[49] D. P. Kolba and T. W. Parks, "A prime factor FFT algorithm using
high speed convolution," IEEE Trans. on ASSP, vol. 25, pp. 281-294,
August 1977. also in[7].
[50] C. S. Burrus and P. W. Eschenbacher, "An in-place, in-order
prime factor FFT algorithm," IEEE Transactions on Acoustics, Speech,
and Signal Processing, vol. 29, pp. 806-817, August 1981. Reprinted in
it DSP Software, by L.R. Morris, 1983 and IEEE Press FFT Reprints, by
P. Duhamel, 1995.
[51] C. Temperton, "A self-sorting in-place prime factor
real/half-complex FFT algorithm," Journal of Computational
Physics, vol. 75, pp. 199-216, 1988.
[52] C. Temperton, "Nesting strategies for prime factor FFT
algorithms," Journal of Computational Physics, vol. 82, pp.
247-268, June 1989.
[53] C. Temperton, "A generalized prime factor FFT algorithm for any
n = 2p 3q5r," SIAM Journal of Sci. Stat. Comp., 1992. to appear.
[54] R. Stasinski, "Prime factor DFT algorithms for new small-N DFT
modules," IEE Proceedings, Part G, vol. 134, no. 3, pp. 117-126, 1987.
[55] H. W. Johnson and C. S. Burrus, "The design of optimal
DFT algorithms using dynamic programming," IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. 31, pp. 378-387, April
1983.
[56] H. W. Johnson and C. S. Burrus, "On the structure of efficient
DFT algorithms," IEEE Transactions on Acoustics, Speech, and Signal
Processing, vol. 33, pp. 248-254, February 1985.
[57] H. W. Johnson and C. S. Burrus, "Large DFT modules: N = 11, 13,
17, 19, and 25," Tech. Rep. 8105, Department of Electrical
Engineering, Rice University, Houston, TX 77251-1892, 1981.
[58] C. Temperton, "A new set of minimum-add small-n rotated DFT
modules," Journal of Computational Physics, vol. 75, pp. 190-198,
1988.
[59] C. Lu, J. W. Cooley, and R. Tolimieri, "FFT algorithms
for prime transform sizes and their implementations on VAX,
IBM3090VF, and IBM RS/6000," IEEE Transactions on Signal Processing,
vol. 41, pp. 638-648, February 1993.
[60] S. Chu and C. S. Burrus, "A prime factor FFT algorithm
using distributed arithmetic," IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. 30, pp. 217-227, April 1982.
[61] K. T. Chen, "A new record: the largest known prime number," IEEE
Spectrum, vol. 27, p. 47, February 1990.
[62] H. V. Sorensen, D. L. Jones, C. S. Burrus, and M. T. Heideman,
"On computing the discrete Hartley transform," IEEE Transactions on
Acoustics, Speech, and Signal Processing, vol. 33, pp. 1231-1238,
October 1985.
[63] R. N. Bracewell, The Hartley Transform. Oxford Press, 1986.
[64] F. M. Wang and P. Yip, "Fast prime factor decomposition
algorithms for a family of discrete trigonometric transforms,"
Circuits, Systems, and Signal Processing, vol. 8, no. 4, pp. 401-419,
1989.
[65] K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms,
Advantages, Applications. San Diego, CA: Academic Press, 1990.
[66] H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus,
"Real valued fast Fourier transform algorithms," IEEE Transactions
on Acoustics, Speech, and Signal Processing, vol. 35, pp.
849-863, June 1987. also in IEEE Press FFT Reprints, by P. Duhamel,
1995.
[67] P. Duhamel and M. Vetterli, "Improved Fourier and Hartley
transfrom algorithms, application to cyclic convolution of real data,"
IEEE Trans. on ASSP, vol. 35, pp. 818-824, June 1987.
[68] M. Popovic and D. Sevic, "A new look at the comparison of the
fast Hartley and Fourier transforms," IEEE Transactions on Signal
Processing, vol. 42, pp. 2178-2182, August 1994.
[69] P. R. Uniyal, "Transforming real-valued sequences: fast
Fourier versus fast Hartley transform algorithms," IEEE
Transactions on Signal Processing, vol. 42, pp. 3249-3254, November
1994.
[70] G. Bruun, "Z-transform DFT filters and FFTs," IEEE
Transactions on ASSP, vol. 26, pp. 56-63, February 1978.
[71] R. Storn, "On the Bruun algorithm and its inverse," Frequenz,
vol. 46, pp. 110-116, 1992. a German journal.
[72] C. M. Rader and N. M. Brenner, "A new principle for fast Fourier
transformation," IEEE Transactions on Acoustics, Speech, and Signal
Processing, vol. ASSP-24, pp. 264-266, June 1976.
[73] K. M. Cho and G. C. Temes, "Real-factor FFT algorithms,"
in Proceedings of IEEE ICASSP-78, (Tulsa, OK), pp. 634-637, April
1978.
[74] P. Duhamel, B. Piron, and J. M. Etcheto, "On computing the
inverse DFT," IEEE Transactions on ASSP, vol. 36, pp. 285-286,
February 1978.
[75] R. Singleton, "An algorithm for computing the mixed radix fast
Fourier transform," IEEE Transactions on Audio and Electroacoustics,
vol. AU-17, pp. 93-103, June 1969.
[76] J. A. Glassman, "A generalization of the fast Fourier
transform," IEEE Transactions on Computers, vol. C-19, pp. 105-116,
Feburary 1970.
[77] W. E. Ferguson, Jr., "A simple derivation of Glassman general-n
fast Fourier transform," Comput. and Math. with Appls., vol. 8, no. 6,
pp. 401-411, 1982. Also, in Report AD-A083811, NTIS, Dec. 1979.
[78] L. Rabiner, R. Schafer, and C. Rader, "The chirp z-transform
algorithm," IEEE Transactions on Audio Electroacoustics, vol. AU-17,
pp. 86-92, June 1969.
[79] G. A. Sitton, "The QFT algorithm," unpublished paper, 1985.
[80] H. Guo, G. A. Sitton, and C. S. Burrus, "The quick discrete
Fourier transform," in Proceedings of the IEEE International
Conference on Acoustics, Speech, and Signal Processing, (IEEE
ICASSP-94, Adelaide, Australia), pp. III:445-448, April 19-22 1994.
[81] H. Guo, G. A. Sitton, and C. S. Burrus, "The quick Fourier
transform, an FFT based on symmetries," IEEE Transactions on Signal
Processing, submitted Oct. 1994.
[82] I. W. Selesnick and C. S. Burrus, "Automating the design
of prime length FFT programs," in Proceedings of the IEEE
International Symposium on Circuits and Systems, (ISCAS-92, San Diego,
CA), pp. 133-136, May 1992.
[83] I. W. Selesnick and C. S. Burrus, "Multidimensional
mapping techniques for convolution," in Proceedings of the IEEE
International Conference on Signal Processing, (IEEE ICASSP-93,
Minneapolis), pp. III-288-291, April 1993.
[84] I. W. Selesnick and C. S. Burrus, "Automatic generation
of prime length FFT programs," IEEE Transactions on Signal
Processing, to appear. 1995.
[85] I. W. Selesnick and C. S. Burrus, "Extending Winograd's
small convolution algorithm to longer lengths," in Proceedings of
the IEEE International Symposium on Circuits and Systems, (IEEE ISCAS-
94, London), pp. 2.449-452, June 30 1994.
[86] W. K. Hocking, "Performing Fourier transforms on extremely
long data streams," Computers in Physics, vol. 3, pp. 59-65,
January 1989.
[87] R. Stasinski, "Extending sizes of effective convolution
algorithms," Electronics Letters, vol. 26, no. 19, pp. 1602-1604,
1990.
[88] R. C. Agarwal and J. W. Cooley, "New algorithms for digital
convolution," IEEE Trans. on ASSP, vol. 25, pp. 392-410, October 1977.
[89] I. Pitas and C. S. Burrus, "Time and error analysis of digital
convolution by rectangular transforms," Signal Processing, vol. 5, pp.
153-162, March 1983.
[90] R. Meyer, R. Reng, and K. Schwarz, "Convolution algorithms on
DSP processors," in Proceedings of the ICASSP-91, (Toronto, Canada),
pp. 2193-2196, May 1991.
[91] R. C. Agarwal and C. S. Burrus, "Number theoretic transforms to
implement fast digital convolution," Proceedings of the IEEE, vol. 63,
pp. 550-560, April 1975. also in IEEE Press DSP Reprints II, 1979.
[92] H. V. Sorensen and C. S. Burrus, "Efficient computation of the
short-time FFT," in Proceedings of the IEEE International Conference
on Acoustics, Speech, and Signal Processing, (New York), pp. 1894-
1897, April 1988.
[93] H. V. Sorensen, C. S. Burrus, and D. L. Jones, "A new
efficient algorithm for computing a few DFT points," in
Proceedings of the IEEE International Symposium on Circuits and
Systems, (Espoo, Finland), pp. 1915-1918, June 1988.
[94] C. Roche, "A split-radix partial input/output fast Fourier
transform algorithm," IEEE Transactions on Signal Processing, vol. 40,
pp. 1273-1276, May 1992.
[95] H. V. Sorensen and C. S. Burrus, "Efficient computation of the
DFT with only a subset of input or output points," IEEE Transactions
on Signal Processing, vol. 41, pp. 1184-1200, March 1993.
[96] R. Meyer and K. Schwarz, "FFT implementation on DSP-chips _
theory and practice," in Proceedings of the ICASSP-90, (Albuquerque,
NM), pp. 1503-1506, April 1990.
[97] H. V. Sorensen, C. A. Katz, and C. S. Burrus, "Efficient FFT
algorithms for DSP processors using tensor product decomposition," in
Proceedings of the IEEE International Conference on Acoustics,
Speech, and Signal Processing, (Albuquerque, NM), pp. 1507-1510, April
1990.
[98] J. Johnson, R. W. Johnson, D. Rodriguez, and R. Tolimieri, "A
methodology for designing, modifying, and implementing Fourier
transform algorithms on various architectures," Circuits, Systems and
Signal Processing, vol. 9, no. 4, pp. 449-500, 1990.
[99] C. Van Loan, Matrix Frameworks for the Fast Fourier Transform.
Philadelphia, PA: SIAM, 1992.
[100] R. Tolimieri, M. An, and C. Lu, Mathematics of Multidimensional
Fourier Transform Algorithms. New York: Springer-Verlag, 1993.


.



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