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ezfft
Easy to use Power Spectrum

Description ```ezfft(T,U) plots the power spectrum of the signal U(T) , where T is a 'time' and U is a real signal (T can be considered as a space coordinate as well). If T is a scalar, then it is interpreted as the 'sampling time' of the signal U. If T is a vector, then it is interpreted as the 'time' itself. In this latter case, T must be equally spaced (as obtained by LINSPACE for instance), and it must have the same length as U. If T is not specified, then a 'sampling time' of unity (1 second for instance) is taken. Windowing (appodization) can be applied to reduce border effects (see below). [W,E] = ezfft(T,U) returns the power spectrum E(W), where E is the energy density and W the angular frequency (or pulsation) 'omega' (e.g. in rad/s). Note that the frequency (e.g. in Hz) is W/(2*pi). If T is considered as a space coordinate (say, X), W is a wavenumber (usually noted K = 2*PI/LAMBDA, where LAMBDA is a wavelength). ezfft(..., 'Property1', 'Property2', ...) specifies the properties: 'hann' applies a Hann appodization window to the data (reduces aliasing). 'disp' displays the spectrum (by default if no output argument) 'freq' the frequency f=2*pi*w is displayed instead of the angular frequency omega (this applies for the display only: the output argument remains the angular frequency omega, not the frequency f). 'space' the time series is considered as a space series. This simply renames the label 'omega' by 'k' (wave number) in the plot, but has no influence on the computation itself. 'handle' returns a handle H instead of [W,E] - it works only if the properties 'disp' is also specified. The handle H is useful to change the line properties (color, thickness) of the plot (see the example below). The length of the vectors W and E is N/2, where N is the length of U (this is because U is assumed to be a real signal.) If N is odd, the last point of U and T are ignored. If U is not real, only its real part is considered. W(1) is always 0. E(1) is the energy density of the average of U (when plotted in log coordinates, the first point is W(2), E(2)). W(2) is the increment of angular frequency (Delta W), given by 2*PI/Tmax. W(end), the highest measurable angular frequency, is PI/DT, where DT is the sampling time (Nyquist theorem). Parseval Theorem (Energy conservation): For every signal U, the 'energy' computed in the time domain and in the frequency domain are equal, MEAN(U.^2) == SUM(E)*W(2) where W(2) is the increment of angular frequency Delta W. Note that, depending on the situation considered, the physical 'energy' is usually defined as 0.5*MEAN(U.^2). Energy conservation only applies if no appodization of the signal (windowing) is used. Otherwise, some energy is lost in the appodization, so the spectral energy is lower than the actual one. The amount of energy lost depends on the signal, it may be of order 0.5. As for FFT, the execution time depends on the length of the signal. It is fastest for powers of two. Example simple display of a power spectrum t = linspace(0,400,2000); u = 0.2 + 0.7*sin(2*pi*t/47) + cos(2*pi*t/11); ezfft(t,u); Example how to change the color of the plot h = ezfft(t,u,'disp','handle'); set(h,'Color','red'); Example how to use the output of ezfft [w,e] = ezfft(t,u,'hann'); loglog(w,e,'b*'); See Also FFT Published output in the Help browser showdemo ezfft ```
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