function [w,e] = ezfft(varargin)
%EZFFT Easy to use Power Spectrum
% 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 pulsation 'omega'. W is *NOT* the frequency:
% the frequency is W/(2*pi). If T is considered as a space coordinate,
% W is a wave number (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 is displayed instead of the pulsation omega
% (this applies for the display only: the output argument
% remains the pulsation 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 pulsation, Delta W, given by 2*PI/Tmax
% W(end), the highest measurable pulsation, 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 pulsation increment 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.
%
% Example 1: 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 2: how to change the color of the plot
% h = ezfft(t,u,'disp','handle');
% set(h,'Color','red');
%
% Example 3: how to use the output of ezfft
% [w,e] = ezfft(t,u,'hann');
% loglog(w,e,'b*');
%
% See also FFT
% F. Moisy, moisy_at_fast.u-psud.fr
% Revision: 1.01, Date: 2008/11/20
% This function is part of the EzyFit Toolbox
% History:
% 2008/11/19: v1.00, first version.
% 2008/11/20: v1.01, includes property 'space'. Help text improved
error(nargchk(1,inf,nargin));
if nargin==1
u=varargin{1};
dt=1;
else
dt=varargin{1};
u=varargin{2};
end
if length(dt)>1
if length(u)~=length(dt)
error('Vectors T and U should be of equal length.');
end
dt = abs(dt(2)-dt(1));
end
% works only with vectors of even length
if mod(length(u),2)~=0;
u=u(1:(end-1));
end
u=real(u);
% works with line vector (otherwise transpose it)
if size(u,2)==1
u=u';
end
% signal appodization with a Hann window
if any(strcmpi(varargin,'hann'))
u=u.*hann(length(u))';
end
n = length(u)/2; % length of the output fft (half the signal length)
w = linspace(0,pi,n)/dt; % output pulsation
e = abs(fft(u)).^2; % power spectrum
e = 2*e(1:n) / (2*n)^2 / w(2); % normalisation
e(1)=e(1)/2; % mode w=0 was counted twice is the previous line
% display
if nargout==0 || any(strncmpi(varargin,'disp',1))
if any(strncmpi(varargin,'freq',1)) && any(strncmpi(varargin,'space',1))
error('Properties ''freq'' and ''space'' cannot be specified simultaneously');
elseif any(strncmpi(varargin,'freq',1)) && ~any(strncmpi(varargin,'space',1))
hh=loglog(w/(2*pi),2*pi*e);
xlabel('f = \omega / 2\pi');
ylabel('E(f)');
elseif ~any(strncmpi(varargin,'freq',1)) && any(strncmpi(varargin,'space',1))
hh=loglog(w,e);
xlabel('k');
ylabel('E(k)');
else
hh=loglog(w,e);
xlabel('\omega');
ylabel('E(\omega)');
end
end
% output
if nargout>0 && any(strncmpi(varargin,'handle',4)) && any(strncmpi(varargin,'disp',1))
w=hh;
clear e
return
end
if nargout==0
clear w e
end
function y=hann(n)
%HANN Window Hann function
% Y = HANN(N) returns a N-point Hann function (usually N is a power of
% 2), ie a discretization of Y(X) = (1 - COS (2*PI*X))/2.
%
% The HANN function is used to window (apodize) finite samples of signal
% in order to avoid high frequency oscillations in Fourier transform.
%
% Example: plot(hann(128));
%
% F. Moisy
% Revision: 1.03, Date: 2006/03/03
%
% See also FFT.
% History:
% 2004/09/17: v1.00, first version.
% 2005/23/02: v1.01, cosmetics
% 2005/09/03: v1.02, id.
% 2006/03/03: v1.03, bug fixed 0:n-1
error(nargchk(1,1,nargin));
y=0.5*(1-cos(2*pi*(0:n-1)/(n-1)))';