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Fourier Transform
Published in David C. Swanson, ®, 2011
Both the FFT and DFT give exactly the same result when the DFT has N equally spaced output frequency bins over the Nyquist band. In the early days of digital signal processing with vacuum tube flip-flops and small magnetic core memories, the FFT was an absolutely essential algorithm. It is (and likely always will be) the standard way to compute digital Fourier transforms of evenly spaced data primarily for economical reasons. What is to be emphasized here is that the resolution of the DFT or FFT is always limited by the length of the input buffer. The spectral leakage within the DFT or FFT is caused by a breakdown of the periodicity assumptions in the Fourier transform when a noninteger number of full wavelengths appear in the input buffer. One can use an FFT to evaluate more frequency bins than input samples by “zero padding” the input buffer to fill out the larger-sized FFT buffer. This is sometimes a useful alternative to synchronized sampling as a means of finding an accurate peak level even with the underlying sinc function envelope. One can also perform a “zoom-FFT” by band limiting the input signal of interest centered at some high center frequency fc rather than zero Hz, and essentially performing a complex demodulation down to 0 Hz as described at the end of Section 1.3 and Figure 1.7 before computing the FFT.
Introduction to Digital Signal Conditioning in Instrumentation
Published in Robert B. Northrop, Introduction to Instrumentation and Measurements, 2018
is computed in the time domain, where wk is the windowing function. The purpose of a windowing function is to reduce a phenomenon known as spectral leakage. If a pure, sinusoidal x(t) is sampled and DFTed, we generally observe symmetrical, nonzero side lobes in {Xk}N around the main peak at the sinusoid’s frequency. This phenomenon was called spectral leakage because spectral energy at the sinusoid’s frequency effectively leaks into the adjacent frequency terms. What windows do in general is to reduce the magnitude of the leakage side lobes at the expense of the sharpness of the main spectral peak at the input sinusoid’s frequency.
Application of Windows in Spectral Analysis
Published in K. M. M. Prabhu, Window Functions and Their Applications in Signal Processing, 2018
Using Hamming window: In this case, the main-lobe width is twice that of the rectangular window, but the side-lobe attenuation is more than −40 dB. Therefore, the spectral resolution obtained due to this window is poorer when compared to the rectangular window case, but it has less spectral leakage. Another additional feature of this window is that it provides a smoother spectrum when compared to a spiky spectrum exhibited by the rectangular window. Hence, it is mainly used in speech applications as a trade-off between resolution and spectral leakage. In the spectrogram, we can see that the frequency lines have concentrated energy and the spectral leakage is less.
Novel frequency-based approach for detection of steady-state visual evoked potentials for realization of practical brain computer interfaces
Published in Brain-Computer Interfaces, 2022
Mehrnoosh Neghabi, Hamid Reza Marateb, Amin Mahnam
The aim of frequency detection algorithms in SSVEP BCI applications is to detect SSVEP responses that are intermingled with background EEG activity and noise. Generally, for high-frequency flickering stimuli, SSVEP response is low in amplitude, which makes SSVEP detection challenging. The main ideas behind the proposed SAoRS algorithm are to improve frequency detection performance by improving the signal-to-noise ratio and decreasing spectral leakage. When a finite length of a signal is used to estimate its frequency spectrum, the result would be the convolution of the actual spectrum and the spectrum of the window function, which results in spectral leakage. It is known that if the length of processing window for a periodic signal is selected to be a multiple of its period, then the leakage would be minimized [47]. Here, EEG was first resampled to a multiple of the targeted frequency and then divided into segments for averaging. This makes sure that the SSVEP response peak in the frequency domain remains sharp. Synchronous averaging is a well-known technique to improve the signal-to-noise ratio. However, in our application, the phase of the stimulus in the EEG signal is not known; therefore, the averaging was performed in the frequency domain.
Inverse method identification of thermophysical properties based on solotone effect analysis for discontinuous Sturm–Liouville systems
Published in Inverse Problems in Science and Engineering, 2019
Figure 6(a) and (b) displays the evidence of spectral leakage. The reason for this phenomenon is well documented, as are methods to reduce its effect. Furthermore, the frequency resolution is not sufficient to capture small changes in the solotone period; that is, incorrectly equals for sufficiently small δ and ε. In general, when using the Discrete Fourier Transform, increasing the sampling rate will improve matters, but this is clearly not applicable in our situation. However, both deficiencies may be alleviated by using spectral peak interpolation. Here we use Jacobsen's method [20] which uses three values: the maximum peak and adjacent values either side.
Development of a vibration sensor-based tool for online detection of roping in small-diameter hydrocyclones
Published in Mineral Processing and Extractive Metallurgy Review, 2022
S. Mishra, M. H. Tyeb, A. K. Majumder
Based on the aforementioned algorithm, the time length in this work was decided by the sampling rate 1 kHz used for collecting the vibration data. A ‘Hanning’ window function was used as a windowing function to reduce the spectral leakage with 50% overlap and 512 samples per second. However, it is to be noted that in the context of the current work, the choice of windowing function, percentage overlap, and the number of samples per second will not impact significantly on the final data analyses provided they remain unaltered throughout all the operating conditions as the relative response of hydrocyclone vibration characteristics concerning changes in its operating conditions are only being monitored.