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...

Ruthenium dioxide nanoparticles (RuO2 ) with an average particle size of 50 nm are readily synthesized from RuCl3.xH2O via the precipitation process in the presence of in-situ sodium octanoate as anionic surfactant. The phase composition, morphology, lattice parameters and size of nanoparticles in these products are characterized by Fourier transform infrared spectroscopy (FT-IR), X-ray diffrac...

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...

We have used Fourier transform infrared spectroscopy (FTIR) to characterise the chemical and structural composition of the tendons of the rotator cuff and to identify structural differences among anatomically distinct tears. Such information may help to identify biomarkers of tears and to provide insight into the rates of healing of different sizes of tear. The infrared spectra of 81 partial, s...

The formation of the hydrogel poly(N-isopropylacrylamide)-clay (LAPONITE®) by redox polymerization was investigated, and the main factors governing the gel build-up were determined. The significant effect of the redox initiating system ammonium peroxodisulfate (APS) and tetramethylethylenediamine (TEMED) on gel formation and structure was established, making it possible to control the structure...