This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm, called the quick Fourier transform (QFT), that reduces the number of oating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime leng...

The nonuniform discrete Fourier transform (NDFT), used in many processing schemes, can be computed using a fast algorithm known as the non uniform fast Fourier transform (NFFT). The NFFT is not a new algorithm, but it is an approximation scheme that can be use to calculate an approximate spectrum. In one dimension, computational complexity of the NFFT is O(NlogN) which is a dramatic improvement...

The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. It is equivalent to a DFT of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sampl...

The symmetry and periodicity properties of the discrete Fourier transform (DFT) allow a variety of useful and interesting decompositions. In particular, by clever grouping and reordering of the complex exponential multiplications it is possible to achieve substantial computational savings while still obtaining the exact DFT solution (no approximation required). Many “fast” algorithms have been ...

The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. It is equivalent to a DFT of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sampl...

Discrete Fourier transform (DFT) is one of the most important tools used in almost all fields of science and engineering. DFT can be implemented with efficient algorithms generally classified as fast Fourier transforms (FFT). The most widely used approaches are so-called the algorithms for, such as radix-2, radix 4 and split radix FFT (SRFFT). Considerable researches have carried out and result...

Chirped-pulse millimeter-wave (CPmmW) spectroscopy is the first broadband (multi-GHz in each shot) Fourier-transform technique for high-resolution survey spectroscopy in the millimeter-wave region. The design is based on chirped-pulse Fourier-transform microwave (CP-FTMW) spectroscopy [G. G. Brown, B. C. Dian, K. O. Douglass, S. M. Geyer, S. T. Shipman, and B. H. Pate, Rev. Sci. Instrum. 79, 05...

Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can find more thorough discussions in Oppenheim's book. Please let me know if you find any typos. In this lecture, we discuss the Discrete Fourier Transform (DFT), which is a fourier representation for finite length signals. The main practical importance of thi...

Various transforms are employed for signal processing to obtain useful information, which is not explicitly available when the signal is in the time domain. Most of the real time signals such as speech, biomedical signals, etc., are non-stationary signals. The Fourier transform (FT), used for most of the signal processing applications, determines the frequency components present in the signal b...

Certain solutions to Harper’s equation are discrete analogues of (and approximations to) the Hermite–Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscill...