Home > Applications of DFT/FFT

Applications of DFT/FFT

Applications of DFT/FFT

    Spectral estimation. P. T. Gough, “A fast spectral estimation algorithm based on FFT,” IEEE Trans. SP, vol. 42, pp. 1317-1322, June 1994.
    Masking threshold for the psychoacoustic model is derived from an estimate of the power density spectrum obtained from a 512-point FFT. Used in MPEG Audio Coding. CD 11172-3. See the papers.
    D. Pan, “An overview of the MPEG/Audio Compression algorithm,”

IS&T/SPIE Symp., on Electronic Imaging: Science and Technology, vol.

2187, pp. 260-273, San Jose, CA, Feb. 1994.

    K. Brandengurg et al, “The ISO/MPEG audio codec: A generic standard for coding of high quality digital audio,” 92 AES convention, preprint 3336, Vienna, Austria, 1992. (www.aes.org)
    Interpolation using FFT. S.D. Stearn and R.A. David, “Signal processing algorithms,”

Chapter 10-Decimation and Interpolation Routines. Englewood Cliffs, NJ, Prentice-Hall, 1988. Convolution and correlation using FFT. Chapter 9-FFT convolution and correlation.

    Filtering LPF, BPF, HPF. Generalized cepstrum and homomorphic filtering. A.K.

Jain, “Fundamentals of digital image processing,” Englewood Cliffs, NJ, Prentice-

Hall, 1989. (Chapters 5 and 7)

    Multichannel carrier modulation (MCM) such as orthogonal frequency division

multiplexing (OFDM) for digital television terrestrial broadcasting,” IEEE Commun.

Magazine, vol. 32, pp. 46-52, May 1994.

    Several applications in spectral analysis, filtering, convolution, correlation etc. Refer

to following books among others.

a) S.D. Stearns and R.A. David, “Signal Processing Algorithms,” Englewood

Cliffs, NJ, Prentice Hall 1993.

b) S.D. Stearns and R.A. David, “Signal Processing Algorithms in Fortran and C,”

Englewood Cliffs, NJ, Prentice Hall 1993.

    Pulse compression (Radar systems-surveillance, tracking, target classification)-

Matched filter with a long impulse response-convolution via FFT. R. Cox “FFT-

based filter design boosts radar system’s process,” Electronic Design, pp. 81-84, March 31, 1988.

    Spectrum analysis. G. Dovel, “FFT analyzers make spectrum analysis a snap,” EDN,

pp. 149-155, Jan. 1989.

    Interpolation.
    R.W. Schafer and L.R. Rabiner, “A digital signal processing approach to

Interpolation,” Proc. IEEE, vol. 61, pp. 692-702, June 1973.

    M. Yeh, J.L. Melsa, and D.L. Cohn, “A direct FFT scheme for interpolation

decimation, and amplitude modulation,” in Proc. 16th Asilomar Conf. Circuits,

Syst., Comput., pp. 437-441, Nov. 1982.

    K.P. Prasad and P. Sathyanarayana, “Fast interpolation algorithm using FFT,”

Electron. Lett., vol. 22, pp. 185-187, Jan. 1986.

    J.W. Adams, “A subsequence approach to interpolation using the FFT,” IEEE

Trans. CAS, vol. 37, pp. 623-625, May 1987.

    P. Sathyanarayana, P.S. Reddy, and M.N.S. Swamy, “Interpolation of 2-D

Signals,” IEEE Trans. CAS, vol. 37, pp. 623-625, May 1990.

    T. Smith, M.R. Smith, and S.T. Nichols, “Efficient sinc function interpolation

techniques for center padded data,” IEEE Trans. ASSP, vol. 38, pp.1512-1517,

Sept. 1990.

    D. Fraser, “Interpolation by the FFT revisited-An experimental investigation,”

IEEE Trans. ASSP, vol. 37, pp.665-675, May, 1989.

    S.C. Chan, K.L. Ho and C.W. Kok, “Interpolation of 2-D signal by subsequence

FFT,” IEEE Trans. CAS-II: Analog and digital signal processing, vol. 40, pp. 115-118, Feb. 1993.

    Ghost cancellation. M.D. Kouam and J. Palicot, “Frequency domain ghost cancellation using small FFTs,” ICCE, pp. 138-139, Chicago, IL, June 1993.
    D.T.M. Slock and K. Maouche, “The fast subsampled updating recursive least

squares (FSURLS) algorithm for adaptive filtering based on displacement structure and the FFT,” SP, vol. 40, pp. 5-20, Oct. 1994.

    B. Farhang-Boroujeny and S. Gazor, “Generalized sliding FFT and its applications to

implementation of block LMS adaptive filters,” IEEE Trans. SP, vol. 42, pp.532-538, March 1994.

    M. Lodman et al, “A single chip stereo AC-3 audio decoder,” ICCE, pp. 234-235, Chicago, IL, June 1994. (FFT is used in time domain aliasing cancellation-TDAC)
    M. Perry, “Using 2-D FFTs for object recognition,” ICSPAT, pp. 1043-1048, DSP World Expo., Dallas, TX, Oct. 1994.
    J. Edwards, “Automatic bubble size detection using the zoom FFT,” ICSPAT, pp. 1511-1516, DSP World Expo., Dallas, TX, Oct. 1994.
    R.R. Holdrich, “Frequency analysis of non-stationary signals using time frequency

mapping of the DFT magnitudes,” ICSPAT, DSP World Expo, Dallas, TX, Oct. 1994. (www.icspat.com)

    B.A. Schnaufer and W.K. Jenkins, “A fault tolerant FIR adaptive filter based on the

FFT,” ICASSP-94, Vol. 3, pp. 393-396, Adelaide, Australia, April 1994.

    S.U. Zaman and W. Yates, “Use of the DFT for sychronization in packetized data

communications,” ICASSP-94, Vol. 3, pp. 261-264, Adelaide, Australia, April 1994.

    S. Sridharan, E. Dawson and B. Goldburg, “Fast Fourier transform based speech

encryption system,” IEE Proc., Part I Communication, Speech and Vision, Vol. 138,

pp. 215-223, Aug. 1993.

    B. Goldburg, S. Sridharan and E, Dawson, “Cryptanalysis of frequency domain

analog speech scramblers,” IEE Proc., Part I Communication, Speech and Vision,

Vol. 140, pp. 235-239, Aug. 1993.

    S. Sridharan, E. Dawson and B. Goldgurg, “Speech encryption using discrete

orthogonal transforms,” ICASSP-90, pp. 1647-1650, Albuquerque. NM, April 1990.

    Several applications of FFT in digital signal processing are illustrated in software/

hardware, books on using the software (Mathcad, MATLAB etc.)

    V. Murino and A. Trucco, “Underwater 3D imaging by FFT dynamic focusing

beamforming,” ICIP-94, pp. 890-894, Austin, TX, Nov. 1994.

    Y.P. Lin and P.P Vaidyanathan, “Application of DFT filter banks and cosine

modulated filter banks in filtering,” IEEE APCCAS, pp. 254-259, Taipei, Taiwan, Dec. 1994.

    M.F. Catedra et al, “The CG-FFT Method: Application of signal processing

techniques to electromagnetics,” (CG = Conjugate Gradient), Norwood, MA: Artech

House, 1994.

    “FFT-based RELP vocoder,” cited in A.S. Spanias, “Speech coding: A tutorial

review,” Proc. IEEE, Vol. 82, pp. 1542-1582, Oct. 1994.

    D. Petrinovic and H. Babic, “Window spectrum fitting for high accuracy harmonic

analysis,” ECCTD’95, Istanbul, Turkey, Aug. 1995.

    V.K. Jain, W.L. Collins and D.C. Davis, “High accuracy analog measurements via

interpolated FFT,” IEEE Trans. IAM, Vol. IM-28, pp. 113-122, June 1979.

    T. Grandke, “Interpolation algorithms for discrete Fourier transforms of weighted

signals,” IEEE Trans. IAM, Vol. 32, pp. 350-355, June 1983.

    G. Sperry, “Forensic applications utilizing FFT filters,” IS&T’s 48th Annual Conf.,
    , Washington, D.C., May 1995.
    W. Smith and J. Smith, “The handbook of real-time Fourier transforms,” IEEE Press, 1995.
    H. Ochi and N. Bershad, “A new frequency-domain LMS adaptive filter with reduced

sized FFTs,” IEEE ISCAS, pp. , Seattle, WA, April/May 1995

33. P. Estermann and A. Kaelin, “On the comparison of optimum least-squares and

computationally efficient DFT-based adaptive block filters,” ISCAS, pp. , Seattle,

WA, April/May 1995.

34. Detection of a few sinusoids in noise. Dual Tone Multi-Frequency signalling

(DTMF). Use pruned FFT.

35. A. Niederlinski and J. Figwer, “Using the DFT to synthesize bivariate orthogonal

white noise series,” IEEE Trans. SP, Vol. 43, pp. 749-758, March 1995.

36. (8x8) 2D FFT spatially on video frame/field followed by temporal DPCM for a DS-3

NTSC TV codec built by Grass Valley, P. O. Box 1114, Grass Valley, CA 95945.

37. D.I. Laurenson and G.J.R. Povey, “The application of a generalized sliding FFT

algorithm to prediction for a RAKE receiver system operating over the mobile

channels,” ICC-95, pp. , Seattle, WA, June 1995.

38. R.N. McDonough and A.D. Whalen, “Detection of signals in noise,” Orlando, Fl:

Academic Press, 1995. (Detection processing based on the FFT)

39. Dolby AC-2 and AC-3 audio coders.

Time Domain Aliasing Cancellation (TDAC) transform involves MDCT (modified

DCT) and MDST (modified DST). Both MDCT and MDST and their inverses can

be implemented via FFT. Several papers from Dolby Labs.

A.G. Elder and S.G. Turner, “A real-time PC based implementation of AC-2 digital

audio compression,” 95th AES convention, paper #3773, New York, NY, Oct. 1995.

G. Davison, L. Fielder and M. Antill, “Low-complexity transform coder for satellite

link applications,” 89th AES convention, Los Angeles, CA, Sept, 1990. (www.aes.org)

“Dolby AC-3 multi-channel digital audio compression system,” Algorithm

Description, Revision 1.12, Dolby Labs. 22, Feb. 1994.

40. J.W. Woods and S.D. O’Neil, “Subband coding of images,” IEEE Trans. ASSP, vol.

ASSP-34, pp. 1278-1288, Oct. 1986.

(Implementation of the FIR filters with the QMF banks by FFT)

41. T. Q. Nguyen, “Partial reconstruction filter banks-theory, design and

implementation,” Technical Report 991, Lincoln Lab, MIT, Lexington, MA, 22 June,

1995. (DFT filter banks)

42. S. Holm, “FFT pruning applied to time domain interpolation and peak localization,”

IEEE Trans. ASSP, vol. ASSP-35, pp. 1776-1778, Dec. 1987.

43. G. Brunn, “Z-Transform DFT filters and FFTs,” IEEE Trans. ASSP, vol. ASSP-26,

pp. 56-63. Feb. 1978.

44. F.J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier

transform,” Proc. IEEE, vol. 66, pp. 51-83, Jan. 1978.

45. H. Murakami, “Sampling rate conversion systems using a new generalized form of

the discrete Fourier transform,” IEEE Trans. SP, vol. 43, pp. 2095-2102, Sept. 1995.

46. R. Gluth, “Regular FFT-related transform kernels for DCT/DST based polyphase

filter banks,” ICASSP-91, pp. 2205-2208, Toronto, Canada, May 1991.

47. R.M. Gray and J.W. Goodman, “Fourier transforms: An introduction for engineers,”

Norwell, MA: Kluwer Academic Publishers, 1995.

48. N. Bean and M. Stewart, “A note on the use of fast Fourier transforms in Buzen’s

algorithm,” Australian Telecom., Network & Applns., Sydney, Australia, Dec. 1995.

49. P.C. Sapino and J.D. Martin, “Maximum likelihood PSK classification using the DFT

of phase histogram,” IEEE 1995 GLOBECOM, Singapore, Nov. 1995.

50. A.G. Exposito, J.A.R. Macias and J.L.R. Macias “Discrete Fourier transform

computation for digital relaying,” Electric Power and Energy Systems, vol. 16, pp.

229-233, 1994.

51. J. Smith and W. Smith, “Handbook of real-time fast Fourier transforms,” IEEE Press

book, Dial-A-Book’s Web address, http://dab.psi.net/Dial A Book/index.html

52. A.D. Poularikas, (Ed.) “ The transforms and applications handbook,” Boca Raton, FL:

CRC Press, 1996.

53. A. Wannasarnmatha, S Hara and N. Morinaga, “A novel FSK demodulation method

using short time DFT analysis for LEO satellite communication systems,” IEEE 1995

GLOBECOM, pp. 549-553, Singapore, Nov. 1995.

54. B.G. Haskell, “Frame-to-frame coding of television pictures using two-dimensional

Fourier transforms,” IEEE Trans. IT, vol. IT-20, pp. 119-120, Jan. 1974.

55. C.D. Kuglin and D.C. Hines, “The phase correlation image alignment method,” In

Proc. IEEE Int. Conf. Cybern. Soc., San Francisco, CA, pp. 163-165, Sept. 1975.

    M. Ziegler, “Hierarchical motion estimation using the phase correlation method in

140 Mbit/s HDTV coding,” In Signal Processing of HDTV, ed. L. Chiariglione.

Amsterdam, The Netherland: Elsevier, pp. 131-137, 1990.

    H.M. Hang, Y.-M. Chou, and T.-H.S. Chao, “Motion estimation using frequency

components,” In Proc. SPIE VCIP, Boston, MA, vol.1818, pp. 74-85, Nov. 1992.

    K.O. Egiazarian et al, “Nonlinear filters based on ordering by FFT structure,”

Photonics West, IS&T/SPIE Symp. on Electronic Imaging: Science & Technology,

Vol. 2662, pp. , San Jose, CA, Feb. 1996.

    N. Kuroyanagi, L. Guo and N. Suehiro, “Proposal of a novel signal separation

principle based on DFT with extended frame buffer analysis,” IEEE GLOBECOM,

pp. , Singapore, Nov.1995.

    R.M. Piedra, “Efficient FFT implementation on reduced-memory DSPs,” ICSPAT,

Boston, MA, Oct. 1995.

    K. Hue, “A 256 fast Fourier transform processor,” ICSPAT, Boston, MA, Oct. 1995.

(www.icspat.com)

    S. Nakamura and A. Sasou, “A pitch extraction algorithm using combined wavelet and Fourier transforms,” ICSPAT, Boston, MA, Oct. 1995.
    J.J Shynk, “Frequency domain and multirate adaptive filtering,” IEEE SP Magazine,

vol.9, pp. 14-37, Jan. 1992.

    A.K. Jain and J-Jasiulek, “A class of FFT based algorithms for linear estimation and

boundary value problems,” IEEE Trans. ASSP, vol. ASSP-31, pp.1435-1446, Dec. 1983.

    Pacific Cyber / Metrix, Inc., 6693 Sierra Lane, Dublin, CA 94568, Ph: 510-829-8700, Fax: 510-829-9796 (VSP-91 Vector processor, IK complex FFT in 8 sec, 64K complex FFT in 8.2 msec).
    Viper-5, FFT (IM CFFT in 21 msec),Texas Memory Systems, Inc. 11200

Westheimer , #1000, Houston, TX 77042, Ph:713-266-3200, Fax: 713-266-0332,

Website: http://www.texmemsys.com

    K.C. Lo and A. Purvis, “Reconstructing randomly sampled signals by the FFT,”

ISCAS-96, vol. 2, pp.124-127, Atlanta, GA, May 1996.

    J.C. Schatzman, “Index mappings for the fast Fourier transform,” IEEE Trans. SP, vol. 44, pp.717-719, March 1996.
    D. Sundararajan and M.O. Ahmad, “Vector split-radix algorithm for DFT

computation,” ISCAS-96, vol.2, pp. 532-535, Atlanta, GA, May 1996.

    B.S. Reddy and B.N. Chatterji, “An FFT based technique for translation, rotation and scale-invariant image registration,” IEEE Trans. IP, vol. 5, pp. 1266-1271, Aug.1996.
    M. Webster and R. Roberts, “Adaptive channel truncation for FFT detection in DMT

systems-error component partitioning,” 30th Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, CA, Nov. 1996.

    Y. Dezhong, “Fast interpolation of n-dimensional signal by subsequence FFT,”

IEEE Trans. CAS II, Analog and digital signal processing, vol.43, pp. 675-676, Sept.

1996.

    H. Murakami, “Perfect reconstruction condition on the DFT domain for the block

maximally decimated filter bank,” IEEE ICCS/ISPACS 96, pp. 6.3.1-6.3.3, Singapore, Nov. 1996.

    K.C. Teh, K.H. Li and C. Kot, “Rejection of partial baud interference in FFT spread

spectrum systems using FFT based self normalizing receivers,” IEEE ICCS/ISPACS 96, pp. 1.4.1-1.4.4, Singapore, Nov. 1996.

    K.O. Eglazarian et al, “Adaptive LMS FFT-ordered L-filters,” IS & T/SPIE’s 9th

Annual Symp., Electronic Imaging, vol. 3026, San Jose, CA, Feb. 1997.

    A. Celentano and Y. Di Lecce, “FFT-based technique for image signature

generation,” IS & T/SPIE’s 9th Annual Symp., Electronic Imaging, vol. 3022, San

Jose, CA, Feb. 1997.

    S. Yamasaki, “A reconstruction method of damaged two-dimensional signal blocks

using error correction coding based on DFT,” IEEE APCCAS, pp. 215-219, Seoul,

Korea, Nov. 1996.

    G.M. Dillard, “Recursive computation of the discrete Fourier transform with

applications to an FSK communication receiver,” IEEE NTC Record, pp.263-265,

1974.

    G.M. Dillard, “Recursive computation of the discrete Fourier transform with

applications to a pulse radar system,” Comput. and Elec. Engrg., vol.1, pp. 143-152,

1973.

    P.E. Pang and D. Hatzinakos, “An efficient implementation of affine transformation

using one-dimensional FFTs,” ICASSP-97, vol.4, pp. 2885-2888, Munich, Germany,

April 1997.

    F. Clavean, M. Poirier and D. Gingras, “FFT-based cross-covariance processing of

optical signals for speed and length measurement,” ICASSP-97, vol. 5, pp.4097-4100,

Munich, Germany, April 1997.

    A. Buttar el al, “FFT and OFDM receiver ICs for DVB-T decoders,” IEEE ICCE Chicago, IL, June 1997.
    Y. Jhung and S. Park, “Architecture of dual mode audio filter for AC-3 and

MPEG,” (Reconstruction filter based on FFT structure), IEEE ICCE, Chicago, IL,

June 1997.

    G. Zhou and X-G. Xia, “Multiple frequency detection in under sampled complexed-

valued waveforms with close multiple frequencies,” Electron. Lett., (Accepted)

(multiple frequency detection by multiple DFTs).

    K. Brandenburg et al, “Transform coding of high quality digital audio at low bit

rates-algorithms and implementations,” IEEE ICC 1990, pp.332.2.1 thru-332.2.5,

1990.

    G. Davidson, L. Fielder and M. Antill, “Low-complexity transform coder for satellite link applications,” 89th AES Convention, Preprint 2966, Los Angeles, CA, Sept. 1990.(www.aes.org)
    Digital audio compression (AC-3) ATSC standard, 20 Dec. 1995.(www.atsc.org)
    L.D. Fielder and G.A. Davidson, “AC-2: A family of low complexity transform-based music coders,” AES 10th Int. Conf., London, England, Sept. 1991.
    Dolby AC-3, Multi-channel digital audio compression system algorithm description, Dolby Labs. Inc., Revision 1.12, 22 Feb. 1994.
    C.C. Todd et al, “AC-3: Flexible perceptual coding for audio transmission and

storage,” 96th convention of AES, Preprint 3796, Amsterdam, Feb./March 1994.

(www.aes.org)

    Y. Kim, M. Shin and H. Cho, “The performance analysis and the simulation of MC-

CDMA system using IFFT/FFT,” ICSPAT 97, San Diego, CA, Sept. 1997.

    C. Pateros, “Coarse frequency acquisition using multiple FFT windows,” ICSPAT

97, San Diego, CA, Sept. 1997.

    M. Zhao, “Channel separation and combination using fast Fourier transform,”

ICSPAT 97, San Diego, CA, Sept. 1997.

    D. Ridge, Y. Hu and T.J. Ding, “PLD based FFTs,” ICSPAT 97, San Diego, CA,

Sept. 1997.

    Pyschoacoustic model for NBC (nonbackward compatible)-AAC (Advanced audio

coder) audio coder, IS for MPEG-2, Also adopted by MPEG-4 in T/F coder, IS13818-7, April 1997.

    FFT in ‘CELP’ coder MPEG-4, ISO/IEC 14496-3, 1997.
    “A harmonic method for active power filters using recursive DFT,” EECON 20,

Bangkok, Thailand, Nov. 1997.

    B. Widrow et al, “Fundamental relations between the LMS algorithm and the DFT,”

IEEE Trans. CAS, vol. CAS-34, pp. 814-820, July 1987.

    A. C. Kot, S. Li and K.C.Teh, “FFT-based clipper receiver for fast frequency

hopping spread spectrum system,” IEEE ISCAS’98, Monterey, CA, June 1998.

    S.K. Stevens and B. Suter, “Low power, high performance FFT design,” IEEE

ISCAS’98, Monterey, CA, June 1998.

    A. Salsano et al, “16-point high speed (I) FFT for OFDM modulation,” IEEE

ISCAS’98, Monterey, CA, June 1998.

    A.E. Cetin, O.N. Gerek, and Y.Yardimci, “Equiripple FIR filter design by the FFT

algorithm,” IEEE SP Magazine, vol.14, pp.60-64, March 1997.

    P. Kraniauskar, “A plain man’s guide to the FFT,” IEEE SP Magazine, vol.11, pp. 24-35, April 1994.
    Y. T. Han, D.K. Kang and J.S. Koh, “An ASIC design of the MPEG-2 audio

encoder,” ICSPAT’97, San Diego, CA, Sept. 1997. (www.icspat.com).

    H. Murakami, K. Nakamura and Y. Takuno, “High-harmonics analysis by recursive

DFT algorithm,” ICSPAT 98, Toronto, Canada, Sept. 1998.

    H.K. Garg, “Digital signal processing algorithms: number theory, fast Fourier

transforms and convolutions,” Boca Raton, FL: CRC Press, 1998.

    S. Richardson, “DFT strategies for mixed signal and analog designs–from layout-

system,” IEEE ICECS 98, Lisbon, Portugal, Sept. 1998.

    A. Celantano and V.D. Lecce, "A FFT based technique for image signature

generation," Proc. of SPIE: storage and retrieval for image and video databases V, vol.

3022, pp. 457-466, Feb. 1997.

    M. Morhac and V. Matousek, "Fast adaptive Fourier-based transform and its use in

multidimensional data compression," Signal Processing, vol. 68, pp. 141-153, July 1998.

    S. J. Sangwine, "The problem of defining the Fourier transform of a colour image,"

IEEE ICIP, pp. , Chicago, IL, Oct. 1998.

    C. Tellambura,” A reduced-complexity coding technique for limiting peak- to-average power ratio in OFDM,” IEEE ISPACS’98,pp.447-450, Melbourne, Australia, Nov. 1998.
    A. Zaknich, and Y. Attikiouzel,’A comparison of template matching with neural network-approaches in the recognition of numeric characters hand-stamped in aluminum,” IEEE ISPACS’98, pp. 98-102, Melbourne, Australia, Nov. 1998.
    H. Choi, H. Kim, and T. Kim,”Robust watermarks for images in subband domain,”IEEE ISPACS’98,pp. 168-172,Melbourne, Australia, Nov. 1998.
    T.M. Peters and J.C. Williams, “The Fourier transform in biomedical engineering,” Boston, MA :Birkhauser, 1998.
    Y.P. Lin and S.M. Phoong, “Asymptotical optimality of DFT based DMT transceivers,” ISCAS , pp. , Orlando, FL, May-June 1999.
    M. Joho, H. Mathis and G.S. Moschytz,” An FFT-based algorithm for multichannel blind deconvolution,” ISCAS , pp. , Orlando, FL, May-June 1999.
    F. Deguillaume et al, “Robust 3D DFT video watermarking,” SPIE Photonics West, vol. 3657, pp.113-124, San Jose, CA, Jan. 1999.
    J. Marciano and T. B. Vu, “Implementation of a broadband frequency-invariant (FI) array beamformer using the two-dimensional discrete Fourier transform. (2D-DFT), “ IEEE ISPACS 99.
    W. Philips, “On computing the FFT of digital images in qradtree format”, IEEE Trans. IP, vol. 47, pp. 2059-2060, July 1999.
    R. W. Cox and R. Tong, “Two-and three-dimensional image rotation using the FFT”, IEEE Trans. IP, vol. 8, pp. 1297-1299, Sept. 1999.
    H. Murakami, “Generalized DIT and DIF algorithms for signals of composite length,” IEEE ISPACS 99, Pukhet, Thailand, Dec. 1999.
    Altera Application Note 84, “Implementing FFT with on-chip RAM in FLEX 10K devices,” Feb. 1998
    C. Jing and H.-M. Tai, “Implementation of modulated lapped transform using programmable logic,” ICCE 99, pp.20-21, Los Angeles, CA, June 1999.X. Kang et al, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE Trans. CSVT, vol.13, pp. 776-786, Aug. 2003.

FFT Software/Hardware

    “FFT tools” Software Package. Adds FFT capability to Lotus 1-2-3 & enhances FFT

capability of Microsoft Excel. 1024- point FFT under a second on a 486DX/33 PC.

Up to 8192-point FFT with choice of windows, Blackman, Hamming, Hanning, Parzen, tapered retangular & triangular taper. DH Systems Inc., 1940 Cotner Ave.,

Los Angeles, CA 90025. Phone: 800-747-4755, Fax: 310-478-4770.

    ADSP-21060 Benchmarks (@ 40MHz)

1024-point complex FFT (Radix 4 with digit reverse) 0.46ms (18221 cycles).

Analog Devices, Inc., 1 Technology Way, P.O. Box 9106, Norwood, MA 02062,

Phone: 617-461-3771, Fax: 617-461-4447.

    ZR34161 16 bit VSP. High performance programmable 16-bit DSP. 1-D and 2-D

FFTs, several DSP operations, 1024 point radix-2 complex FFT in 2178 sec. ZP34325 32 bit VSP. 1-D and 2-D FFTs, several DSP operations ZR38000 and ZR38001 can execute 1024 point radix-2 complex FFT in 0.88msec. Zoran

Corporation, 1705 Wyatt Drive, Santa Clara, CA 95054. Phone: 408-986-1314, Fax:

408-986-1240.

    FT 200 series Multiprocessors 1 K complex FFT < 550 sec, 1K x 1K Real to complex FFT 782 msec.

Alacron, 71 Spitbrook Road, Suite 204, Nashua, NH 03060

Phone: 603-891-2750, Fax: 603-891-2745.

    E. Bidet, C. Joanblanq and P. Senn, (CNET, Grenoble, France), “A fast single chip

implementation of 8192 complex points FFT,” CICC94, pp. , San Diego, CA,

May 1994.

    “Image processing toolbox” (2-D transforms) MATLAB, Mathworks, 24 Prime

Park Way, Natick, MA 01760, E-mail : info@mathworks.com, FAX: 508-653-6284. Signal processing toolbox (FFT, DCT, Hilbert, Filter design) (www.mathworks.com)

(FTP server ftp.mathworks.com)

    IMSA 100: programmable DSP. Implement FFT, convolution, correlation etc.

SGS-Thomson Microelectronics, 1000 East Bell Road, Phoenix, AZ 85022-2699.

(www.st.com)

8. Windows DLL version of the prime factor FFT sub-routine library, Alligator

Technologies, 17150 Newhope Street # 114, P.O. Box 9706, Fountain Valley, CA

92728-9706, Phone: 714-850-9984, FAX: 714-850-9987.

9. CRP1M40 PC/ISA-bus float point DSP board can process DFT, FFTs, DCTs, FCTs,

adaptive filtering etc., 1K complex FFT in 82 sec at 40MHz. Can upgrade up to 1

Megapoint FFT. Catalina Research Inc., 985 Space Center Dr., Suite 105, Colorado

Springs, CO 80915, Phone: 719-637-0880, FAX: 719-637-3839.

10. Ultra DSP-1 board, 1K complex FFT in 90 sec, Valley Technologies Inc. RD #4,

Route 309, Tamaqua, PA 18252, Phone: 717-668-3737, FAX: 717-668-6360.

11. SIGLAB Software, FFT, correlation etc., Monarch, DSP software, The Athena

Group, Inc. 3424 NW 31st Street, Gainesville, FL 32605, Phone :904-371-2567,

FAX: 904-373-5182.

12. Signal ++ DSP Library (C++), Several transforms including CZT, wavelet, cosine,

sine, Hilbert, FFT and various DSP operations. Sigsoft, 15856 Lofty Trail Drive, San

Diego, CA 92127, Phone:619-673-0745.

    DSP works-real time windows-based signal processing software. FFT, Convolution,

Filtering etc. (includes multirate digital filters, QMF bank). Complete bundled hardware and software packages, DSP operations. Momentum Data Systems, 1520

Nutmeg Place #108, Costa Mesa, CA 92626, Phone: 714-557-6884, Fax: 714-557-6969.

    Version 1.1 ProtoSim, PC based software, FFT, Bode plots, convolution, filtering

etc. Systems Engineering Associates Inc., Box 3417, RR#3, Montpelier, VT 05602,

Phone: 802-223-6194, Fax: 802-223-6195.

    FFT, filtering hitogram techniques, image manipulation etc. Software in ‘C’ on a

diskette. H.R. Myler and A.R. Weeks, “Computer imaging recipes in C,” Englewood

Cliffs, NJ: Prentice Hall, 1993.

    C.D. Burrus et al, “Computer based exercises for signal processing using MATLAB,” Englewood Cliffs, NJ: Prentice Hall, 1994.
    O. Alkin, “DSP-A laboratory approach using PC-DSP,” Englewood Cliffs, NJ: Prentice Hall, 1994.
    1024 point complex FFT in 82 sec. DSP MAX-P40 board, Butterfly DSP, Inc.

1614 S.E. 120th Ave., Vancouver, WA 98684, Ph: 206-892-5597, Fax: 206-254-2524.

    DSP board: DSP Lab one. Various DSP software. Real-time signal capture, analysis,

and generation plus high-level graphics. Standing Applications Lab, 1201 Kirkland

Ave., Kirkland, WA 98033, Ph: 206-453-7855, Fax: 206-453-7870.

    Sig XTM, A general purpose signal processing package, Technisoft, P.O. Box 2525,

Livemore, CA 94551, ph: 510-443-7213, FAX: 510-743-1145.

    DSP Designer, Design environment for DSP, Zola Technologies, Inc. 6195 Heards Creek Dr., N.W., Suite 201, Atlanta, GA 30328, Ph: 404-843-2973, Fax: 404-843-0116.
    S.K. Mitra and J.F. Kaiser, “Handbook for digital signal processing,” Filter design and implementation, FFT implementation on various DSPs, New York, NY: John Wiley, 1993.
    F.J. Taylor, “Principles of signals and systems,” (includes a data disk of MATLAB

and MONARCH example files), New York, NY: McGraw-Hill, 1994.

    Standard filter design software, DGS Associates, Inc. Ph: 415-325-4373, Fax:415-

325-7278.

    Digital Alpha AXP parallel systems and TMS320C40. Parallel DSP& Image

Processing Systems. Traquair Data Systems, Inc., Tower Bldg., 112 Prospect St.,

Ithaca, NY 14850, Ph: 607-272-4417, Fax: 607-272-6211.

    DT VEE and VB-EZ for windows. Software for Microsoft windows. Filters, FFTs

etc., Data Translation, 100 Locke Drive, Marlboro, MA 01752-1192,

Ph: 508-481-3700 or 800-525-8528.

    Mahtematica, (includes FFT, Bessel Functions), Wolfram Research, Inc., 800-441-

MATH, Ph: 217-398-0700, Fax: 217-398-0747, E-mail: info@wri.com.

    Mathcad 5.0, Mathsoft Inc., P.O. Box 1018, Cambridge, MA 02142-1519,

Ph: 800-967-5075, Ph: 217-398-0700, Fax: 217-398-0747.

    A41102 FFT processor, Lake DSP Pty. Ltd. Suite 4/166 Maroubra Road, Maroubra

2035, Australia, Ph: 61-2-314-2104, Fax: 61-2-314-2187.

    DSP/Veclib. Vast library of DSP functions for TI’s TMS 320C40 architecture,

Spectroanalysis, 24 Murray Road, West Newton, MA 02165, Ph: 617-894-8296,

Fax: 617-894-8297.

    Matrix-based interactive language: Signal Processing FFTs, -matrix, objective numerical analysis, Harmonic Software Inc. Ph:206-367-8742, Fax:206-367-1067.

32. A. Hiregange, R. Subramaniyan and N. Srinivasa, “1-D FFT and 2-D DCT routines

for the Motorola DSP 56100 family,” ICSPAT, Dallas, TX, Oct. 1994.

    W. Chen and S. King, “Implementation of real input valued FFT on Motorola DSPs,”

ICSPAT, Dallas, TX, Oct. 1994.

    FFT, Hilbert transform, Acolade +. Enhanced software for communication system,

CAE, Amber technologies Inc., 47 Junction Square Dr., Concord, MA 01742-9879,

Ph: 508-369-0515, Fax: 508-371-9642.

    C.S. Burrus, “Teaching the FFT using Matlab,” ICASSP-92, Vol. 4, pp. 93-96, San

Francisco, CA, March 1992.

    E. Bernard et al, “A pipeline architecture for modified higher radix FFT,” ICASSP-

92, Vol. 5, pp. 617-620, San Francisco, CA, March 1992.

    J.I. Guo, C.M. Liu and C.W. Jen, “A memory-based approach to design and

implement systolic arrays for DFT and DCT,” ICASSP-92, Vol. 5, pp. 621-624, San

Francisco, CA, March 1992.

    D. Rodriguez, “Tensor product algebra as a tool for VLSI implementation of the

discrete Fourier transform,” ICASSP-91, pp.1025-1028, Toronto, Canada, May 1991.

    R. Bhatia, M. Furuta and J. Ponce, “A quasi radix-16 FFT VLSI processor,”

ICASSP-91, pp. 1085-1088, Toronto, Canada, May 1991.

    H. Miyanaga, H. Yamaguchi and K. Matsuda, “A real-time 256x256 point two-

dimensional FFT single chip processor,” ICASSP-91, pp. 1193-1196, Toronto, Canada, May 1991.

    F. Kocsis, “A fully pipelined high speed DFT architecture,” ICASSP-91, pp. 1569-

1572, Toronto, Canada, May 1991.

    S.R. Malladi et al, “A high speed pipelined FFT processor,” ICASSP-91, pp. 1609-

1612, Toronto, Canada, May 1991.

    C. Lu, J.W. Cooley and R. Tolimieri, “Variants of the Winograd mutiplicative FFT

algorithms and their implementation on IBM RS/6000,” ICASSP-91, pp. 2185-2188,

Toronto, Canada, May 1991.

    D. Rodriguez, “A new FFT algorithm and its implementation on the DSP 96002,”

ICASSP-91, pp. 2189-2192, Toronto, Canada, May 1991.

    High-order Spectral Analysis (ISA-PC32) Software. Integral Signal Processing, Inc.,

P.O. Box 27661, Austin, TX 78755-2661, Voice: 512-346-1451, Fax: 512-346-8290.

    Toshiba IP 9506 Image Processor. High speed image processing on a single chip.

Quickest FFT process. Toshiba, I.E. OEM Division, 9740 Irvine Blvd., CA 92718,

Phone: 714-583-3180.

    E. Bidet, C. Joanblanq and P. Senn, “A fast 8K FFT VLSI chip for large OFDM

single frequency network,” 7th Intl. Workshop on HDTV, pp. , Torino, Italy, Oct. 1994.

    FFT-523. A dedicated FFT accelerator for HP’s 68000-based series 200 workstations.

Ariel Corp., 433 River Road, Highland Park, NJ 8904. Phone and Fax: 908-249-2900, E-mail: ariel@ariel.com.

    T.J. Tobias, “In-line split radix FFT for the 80386 family of microprocessors,”

ICSPAT, Santa Clara, CA, Sept. 28-Oct. 1, 1993. (128 point FFT in 700 msec. On a

386, 40MHz PC).

    C. Lu et al, “Efficient multidimensional FFT module implementation on the Intel I860 processor,” ICSPAT, pp. 473-477, Santa Clara, CA, Sept. 28- Oct. 1, 1993.
    Y. Solowiejczyk, “2-D FFTs on a distributed memory multiprocessing DSP based

architectures,” ICSPAT, pp. , Santa Clara. CA, Sept.28-Oct. 1, 1993.

    W. Chen and S. King, “Implementation of real input valued FFT on Motorola

DSPs,” ICSPAT, pp. , Santa Clara, CA, Sept. 28- Oct. 1, 1993.

    ORIGIN for windows, voice spectrum, statistics, FFT , power spectrum, phase

unwrap, Microcal software, One Roundhouse Plaza, Northampton, MA 01060,

Phone: 413-586-2013, Fax: 413-585-0126.(www.microcal.com)

    Sharp Electronics, 5700 NW Pacific Rim Blvd., Camas, WA 98607, Phone: 206-834-8908, Fax: 206-834-8903, (Real time DSP chip set: LH 9124 DSP and LH 9320 Address Generator) 1024 point complex FFT in 80 sec. Real and complex radix-2, radix-4 and radix-16 FFTs.
    TMC 2310 FFT processor, complex FFT (forward or inverse) of up to 1024 points

(514sec). Raytheon Semiconductor, 300 Ellis St, Mountain View, CA 94043-7016,

800-722-7074, Fax: 415-966-7742.

    TMS 320C80 Multimedia Video Processor (MVP), 64-point and 256-point complex

radix-2 FFT, TI, Market Communications Manager, MS736, P.O. Box 1443,

Houston, TX 77251-1443, (Phone: 1-800-477-8924).

    Visilog, Image Processing & Analysis Software: FFTs and various processing

operations, Noesis, 6800 Cote de Liesse, Suite 200. St. Laurent, Quebec, H4T2A7,

Canada, Phone: 514-345-1400, Fax: 514-345-1575, Internet: noesis@cam.org

    Valley Technologies Inc., Rd. 4, Route 309, Tamaqua, PA 18252, Phone: 717-668-

3737, Fax: 717-668-6360, E-mail: valtech@cris.com.

    Stanford Graphics 3.0, Visual Numerics, 9990 Richmand Avenue, Suite 400,

Houston, TX 77042, Phone: 713-954-6424, Fax: 713-781-9260.

    J.I. Guo, C.M. Lui and C.W. Jen, “A memory based approach to design and

implement systolic arrays for DFTand DCT,” ICASSP-92, Vol. 5, pp. 621-624, San

Francisco, CA, March 1992.

    E. Bessalash, “VLSI architecture for fast orthogonal transforms on-line computation,”

ICSPAT, pp. 1607-1618, Santa Clara, CA, Sept./Oct. 1993.

    A. Ambardar, “Analog and digital processing,” (3.5” DOS diskette, IBM PC, PS/2

etc., compatible, requires MATLAB 3.5 or 4.0), Boston, MA: PWS Publishing Co.,

1995.

    H.V. Sorensen, C.S. Burrus and M.T. Heideman, “Fast Fourier transform database,”

(disk and hard copy), Boston, MA: PWS Publishing Co., 1995.

    Tektrum Development Corp., P.O. Box 1837, Alief, TX 77411, E-mail: 74171.3310@Compuserve.com (Ph: 713-265-8882, Fax: 713-265-8887)
    V for Windows, Digital Optics Ltd., Box 35-715, Browns Bay, Optics Ltd.,

Auckland 10, New Zealand, Ph: (65+9) 478-5779, (65+9) 479-4750, E-mail: 100237.423@Compuserve.com

    H.R. Wu and F.J. Paoloni, “Implementation of 2-D vector radix FFT algorithm using

the frequency domain processor A 41102, Proc. IASTED Intl. Symp. on signal Processing and digital filtering, June 1990.

    H.R. Wu and F.J. Paoloni, “Structured vector radix FFT algorithms and hardware

implementation,” J. of Electrical and Electronic Engineering, Australia, vol. 10, pp.

241-253, Sept. 1990.

    N.C. Geckinli and D. Yavuz, “Discrete Fourier transformation and its applications to

power spectra estimation,” Ansterdam, Netherlands: Elsevier Scientific Publishing

Co. 1983.

    V. Cizek, “Discrete Fourier transforms and their applications,” Bristol and Boston:

Adam Higler, 1986.

    M.T.Heideman, “Multiplicative complexity, Convolution and the DFT,” Heidelberg,

Germany: Springer – Verlag, 1988.

    W.W. Smith and J.M. Smith, “Handbook of real-time fast Fourier transforms:

Algorithms to product testing,” Piscataway, NJ, IEEE 1995 (Assembly language programming, Implement FFT algorithms on DSP chips, etc.)

    L. Mintzer, “The FPAG as FFT processor,” ICSPAT, Boston, MA, Oct. 1995
    R.M. Piedra, “Efficient FFT implementation on reduced memory DSPs,” ICSPAT,

Boston, MA, Oct. 1995.

    DADiSP Worksheet (soft package) (DADiSP 4.0) DSP Development Corpn., One

Kendall Square, Cambridge, MA 02139, Fax: 617-577-8211, Phone: 617-577-1133,

student edition on the web, Web site http://www.dadisp.com.

    C.V. Loan, “Computational frameworks for the fast Fourier transform,” (uses

MATLAB notation) SIAM, 1992.

    R.M. Gray and J.W. Goodman, “Fourier transforms,” Boston, MA: Kluwer

Academic, 1995.

    S.K. Lu, S.Y. Kuo and C.W. Wu, “On fault-tolerant FFT butterfly network design,”

ISCAS 96, Vol. 2, pp. 69-72, Atlanta, GA, May 1996.

    S. Rahardja and B.J. Falkowski, “Family of fast transforms for mixed arithmetic

logic,” ISCAS 96, Vol. 4, pp. 396-399, Atlanta, GA, May 1996.

    Mathematica 3.0, Wolfram Research, Inc. http://www.wolfram.com,

E-mail: info@wolfram.com, Ph:217-398-0700, Book: S. Wolfram, “The mathematica

Book,” III Edition, Cambridge Univ. Press, New York, NY: web site: http:// www.cup.org

    J.S. Walker, “Fast Fourier transform,” II Edition, Boca Raton, FL: CRC Press, 1996

(Software on disk).

    S.A. Tretter, “Communication system design using DSP algorithms with laboratory

experiments for the TMS 320C30,” Plenum Publishing Corp., 1996 (Software on disk) (800-221-9369).

    J.C.D de Melo, “Partial FFT Evaluation,” ICSPAT 96, pp. 137-141, Boston, MA, Oct. 1996.

83. J.K. McWilliams and M.E. Fleming, “Small, Flexible, Low Power, DFT Filter Bank

for Channeled receivers,” ICSPAT 96, pp. 609-613, Boston, MA, Oct. 1996.

    C. Nagabhushan et al, “Design of Two High Performance Radix-4 FFT Processors

Using a Modular Architecture Family,” ICSPAT 96, Boston, MA, Oct. 1996.

    J. McCaskill, “TM-66 siwFFT Block Transform DSP Chip,”ICSPAT 95, pp. 689-693, Boston,MA, Oct. 1996.
    M. Langhammer and C. Crome, “Tool for Automated FFT Processor Design,”

ICSPAT 96, Boston, MA, Oct. 1996.

    T.J. Ding and J.V. McCANNY, “Synthesizable FFTs Cores,” 30th Asilomar Corf.

on Signals, Systems and Computers, Pacific Grove, CA, Nov. 1996.

    H. Kwan et al, “Three-dimensional FFTs on a digital-signal parallel processor with

no interprocessor communication,” 30th Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, CA, Nov. 1996.

    M. Cavadini, “A high performance memory and bus architecture for implementing

2D FFT on a SPMD machine,” ISCAS’97, Vol. 3. pp. 2032-2036, Hong Kong, June 1997.

    S.F. Hsiao and C.Y. Yen, “Power, speed and area comparison of several DFT

architectures,” ISCAS’97, Vol. 4, pp. 2577-2581, Hong Kong, June 1997.

    TMS 320C 6201 General-purpose programmable fixed-point-DSP chip (5ns cycle

time). Can compute 1024-point complex FFT in 70 sec. TI Inc., P.O. Box 172228,

Denver, CO 80217. TMS 320 (www.ti.com/dsps)

    M. Grajcar and B. Sick, “The FFT butterfly operation in 4 processor cycles on a 24 bit fixed-point DSP with a pipelined multiplier,” ICASSP 97, Vol. 1, pp. 611-614,

Munich, Germany, April 1997.

    S. Hsiao and C. Yen, “New unified VLSI architectures for computing DFT and

other transforms,” ICASSP 97, vol. 1, pp. 615-618, Munich, Germany, April 1997.

    D. Griesinger, “Beyond MLS-Occupied Hall measurement with FFT techniques,”

101th Convention of AES, Preprint 4403, Los Angeles, CA, Nov. 1996.

    R. Makowitz and M. Mayr, “Optimal pipelined FFT processing based on embedded static RAM,” ICSPAT 97, San Diego, CA, Sept. 1997.
    C.J. Ju, “FFT-Based parallel systems for array processing with low latency: Sub-40ns 4K butterfly FFT,” ICSPAT 97, San Diego, CA, Sept. 1997.
    T. Williams, “Case Study: Variable size, variable bit-width FFT engine offering DSP-Like performance with FPGA versatility,” ICSPAT 97, San Diego, CA, Sept. 1997.
    D. Ridge et al, “PLD based FFTs,” ICSPAT 97, San Diego, CA, Sept. 1997.
    M. Wintermantel, “Reducing the complexity of discrete convolutions and DFT by a linear transformation,” ECCTD 97, Budapest, Hungary, Sept. 1997.
    R. Stasinski, “Optimization of vector-radix-3 FFTs,” ECCTD 97, Budapest,

Hungary, Sept. 1997.

    “Fundamentals of FFT-based signal analysis and meaurement,” application note.

National Instruments, 800-433-3488, E-mail: info@natinst.com

    MultiDSP, 4865 Linaro Dr., Crypress, CA 90630, Ph: 714-527-8086, Fax: 714-527-

8287, E-mail: multidsp@aol.com. Filters, windows, etc., also DCT/IDCT, FFT/IFFT,

Average FFT.

    W.W. Smith and J.W. Smith, “Handbook of real-time fast Fourier transforms,”

Piscataway, NJ: IEEE Press, 1995.

    G. Angelopoulos and I. Pitas, “Fast parallel DSP algorithms on barrel shifter

computers,” IEEE Trans. SP, vol. 41, pp. 2126-2127, Aug. 1993.

    B.M. Baas, “A 9.5mw 330 sec 1024-point FFT processor,” IEEE CICC, Santa

Clara, CA, May 1998.

    S. He and M. Torkelson, “Design and Implementation of a 1024 point pipeline

FFT,” IEEE CICC, Santa Clara, CA, May 1998.

    STV 0300 VLSI chip can be programmed to perform FFT or IFFT (up to 8192 point

complex FFT) with input from 1KHz to 30KHz. (8192 point FFT in 410 sec),

SGS-Thomson Microeletronics News & Views, issue #7, Dec. 1997. (www.st.com)

    A.Y. Wu and T.S. Chan, “Cost-effective parallel lattice VLSI architecture for the

IFFT/FFT in DMT transceiver technology,” ICASSP 98, pp. 3517-3520, Seattle,

WA, May 1998.

    L.S Ronga, “Fast phase sequences spreading codes for CDMA using FFT,”

EUSIPCO-98, Island of Rhodes, Greece, Sept. 1998.

    A. Jbira, “Performance of discrete Fourier transform with small overlap in

transform-predictive-coding-based coders,” EUSIPCO-98, Island of Rhodes,

Greece, Sept. 1998.

    A. Petrovsky, “Automated parallel-pipeline structure of FFT hardware design for

real-time multidimensional signal processing,” EUSIPCO-98, Island of Rhodes,

Greece, Sept, 1998.

    C. Becchetti, “DFT based optimal blind channel identification,” EUSIPCO-98,

Island of Rhodes, Greece, Sept. 1998.

    A.M. Krot, “Fast algorithm for reduction a modulo polynomial and Vandermonde

transform using FFT,” EUSIPCO-98, Island of Rhodes, Greece, Sept. 1998.

    I.W. Selesnick and C.S. Burrus, “Automatic generation of prime length FFT

programs,” IEEE Trans. SP, vol. 44, pp. 14-24, Jan. 1996.

    G. Naveh et al, “Optimal FFT implementation on the Carmel DSP Core,”

ICSPAT 98, Toronto, Canada, Sept. 1998.

    G. Chiassarini et al, “Implementation in a single ASIC chip, of a Winograd FFT for

a flexible demultiplexer of frequency demultiplexed signals,” ICSPAT 98, Toronto,

Canada, Sept. 1998.

    L. Mintzer, “A 100 megasample/sec FPGA-based DFT processor,” ICSPAT 98,

Toronto, Canada, Sept. 1998.

    S. Nag and H.K. Verma, “An efficent parallel design of FFTs in FPGAs,” ICSPAT

98, Toronto, Canada, Sept. 1998.

    C. Dick, “Computing multidimensional DFTs using Xilinx FPGAs,” ICSPAT 98,

Toronto, Canada, Sept. 1998.

    B. Bramer, M. Ibrahim and S. Rumsby, “An FFT implementation using almanet,”

ICSPAT 98, Toronto, Canada, Sept. 1998.

    J.S. Matos and J.M. Silva, “Board-level DFT techniques using the P1149.4 mixed-

signal test bus,” IEEE ICECS 98, Lisbon, Portugal, Sept. 1998.

    J.M. Rius and R. De Povrata-Doria, “New FFT bit reversal algorithm,” IEEE Trans. SP, vol. 43, pp. 991-994, April 1995.
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