For example, on a slow machine, a 4096-point FFT takes 2.1 s, but requires 7 s if the sequence is 4095 points long and 58 s if the sequence is 4097 points long. The algorithm is fastest if the data length is a power of 2, or if the length has many prime factors. Although the MATLAB routine will interpolate if need be, calculation time will go up and how much depends on data length. The FFT algorithm requires the data length to be a power of 2. There is one downside to the fft routine. If n is greater than the length of x, then the signal is padded with trailing zeros to equal n. Presented in extremely rare Spectrum Blue that was only available as a colour for a limited time with chrome accents and Dove Grey Leather, this is a gorgeous. The argument n is optional and is used to modify the length of data analyzed: if n is less than the length of x, then the analysis is performed over the first n points. The second term in Xf is the complex representation of the fundamental sinusoidal component the third term represents the second harmonic and so on. The main advantage of the FX correlator is that dividing the spectrum into a number of frequency bands (say n bands) reduces the number of the cross multiply/accumulates2 (XMACs) required by n. (Recall, it is common to use capital letters for the Fourier transform variable.) The first term of Xf is real and is the un-normalized DC component you need to divide by the signal length, N, to get the actual DC component. Where x is the input waveform and Xf is a complex vector providing the sinusoidal coefficients. Measurements are always gathered in the time domain, after which they can be. Such a signal could be a broadband noise measurement, a harmonic analog signal, or a wideband signal of any type. Xf = fft(x,n) % Calculate the Fourier Transform Power spectrum and power spectral density are agnostic to the type of signal that is used to generate an intensity distribution in the frequency domain.
MATLAB has a routine that uses the FFT algorithm to implement Equation 3.30 very quickly: John Semmlow, in Circuits, Signals and Systems for Bioengineers (Third Edition), 2018 3.4.2 MATLAB Implementation of the Discrete Fourier Transform
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