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Fourier Transform
Published in David C. Swanson, ®, 2011
The FFT is an engineered DFT such that the number of multiplications and additions are minimized. This is done by cleverly arranging the multiplies in the summation so that nothing is repeated. The FFT frequency bins are orthogonal for input frequencies which are exactly aligned with the bin frequencies, and thus, produce no leakage into adjacent FFT bins. The sine and cosine components of the FFT and DFT can be precomputed into tables to minimize redundant computation. Computationally, the FFT is significantly more efficient than the more explicit DFT requiring only Nlog2N multiplies as compared with the DFT’s N2. For a 1024-point FFT this difference is 10,240–1,048,576 or a reduction of 102.4:1. However, in order to use the FFT’s marvelous efficiency, the input samples must be regularly spaced and the number of input samples must equal the total number of FFT bins. While an FFT typically has N bins where N is a power of 2, any length of input buffer can be processed by adding zeros (zero padding) to bring the input buffer up to the next power of 2, but the effective frequency resolution is determined by the inverse of the length in time of the nonzero-padded input samples.
Fundamentals of Speech Processing
Published in Shaila Dinkar Apte, Random Signal Processing, 2017
The time domain approaches have been studied for pitch period measurement in the previous section. This section deals with a frequency domain approach for pitch period measurement. The frequency domain pitch detection algorithms operate in the spectrum domain. The periodic signal has a harmonic structure. The algorithm tracks the distance between successive harmonics. The main drawback of frequency domain methods is that computational complexity increases. Let us study the use of fast Fourier transform (FFT) for pitch measurement. FFT is a fast algorithm for the computation of discrete Fourier transform (DFT). DFT coefficient X(k) of any sampled signal (sequence) x(n) is given by Equation 5.8.
Quasi-Phase Matching
Published in Peter E. Powers, Joseph W. Haus, Fundamentals of Nonlinear Optics, 2017
Peter E. Powers, Joseph W. Haus
A second issue is that an FFT assumes the data being analyzed are a single period of a periodic function. Strategies to come up with an FFT that is more representative of the analytic Fourier transform include padding the array with zeroes at the beginning and end of the function and choosing an appropriate point spacing. A final issue with FFT algorithms is that they generate an output with the frequencies ordered in a nonintuitive way specific to the mathematical package. Hence, it is important to study the details of a given mathematical package's FFT operation.
Control techniques for renewable energy integration with shunt active filter: a review
Published in International Journal of Ambient Energy, 2023
Rajendran Boopathi, Vairavasundaram Indragandhi
The Fourier analysis can be utilised to calculate the distorted waveform of amplitude and phase angle. After sensing the harmonic component of the current and voltage, the fundamental component of the waveform is extracted from this waveform. The FFT is a powerful technique for calculating the discrete Fourier Transform of discrete signals. The FFT minimises calculation time by utilising the total count of selected sampling points N, which is a power of two. The FFT significantly decreases the amount of time it takes to calculate (Vardar, Akpinar, and Sürgevil 2009). Such a technique is preferable only in certain digital signal processing application domains where the waveform is analyzed using microcontrollers with a relatively high clock frequency (Patel, Patel, and Patel 2017).
Hardware chip performance analysis of different FFT architecture
Published in International Journal of Electronics, 2021
Amit Kumar, Adesh Kumar, Aakanksha Devrari
The discrete Fourier transform (DFT) is a widely used tool in several applications of digital signal processing (DSP) systems. It has a vital role in several applications such as signal analysis, speech processing, image processing, audio processing, video processing, communication systems and many others. DFT is a Fourier representation of signal over finite length sequence. The DFT is achieved by decomposing the valued sequences into different frequency components. It converts time domain signal to a frequency domain signal for the same length while IDFT converts frequency domain signal to time domain signal. The FFT (Cooley & Tukey, 1965) is an algorithm used to compute ‘N’ point DFT of a sequence while inverse FFT is used to compute IDFT. The FFT computes fast by factorising the DFT matrix into a product of sparse factors, mostly zero. An FFT can easily reduce the complexity of DFT hardware. The brute-force calculation of ‘N’ length DFT requires O(N2) multiplications, whereas FFT can reduce the complexity from O(N2) to O(N log2N) for a DFT of length ‘N’. The general equation of DFT for input sequence x(n) over a length ‘N’ is given by
Control of wave propagation response using quasi crystals: A formulation based on spectral finite element
Published in Mechanics of Advanced Materials and Structures, 2019
Vinita Chellappan, S. Gopalakrishnan, V. Mani
where μ is the mean and σ2 is the variance. The pulse width is essentially controlled by the variance parameter σ and A is the amplitude. The forcing function, and the corresponding FFT amplitude is shown in Figure 1b. The frequency response methods require zero padding in order to avoid signal wrap around effects due to a finite time window [12], [13]. Hence, the force is applied after a definite time interval at . In wave propagation studies, frequency domain methods are extensively used. The advantage of using this is due to its ease in computation while going back and forth between the time and frequency domain. FFT (Fast Fourier transforms) assumes periodicity in the time domain and hence induces periodicity in frequency domain. Thus, FFT is always associated with time windows. Hence, if the measured signal response does not die out within the chosen time window, the signal is completely distorted. In finite structures, the signal distortion becomes severe because of reflections from the boundaries that do not die out within the chosen time window. This problem is called signal wrap around.