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function [Spec,f] = spectrum(varargin)
%SPECTRUM Power spectrum estimate of one or two data sequences.
% P=SPECTRUM(X,NFFT,NOVERLAP,WIND) estimates the Power Spectral Density of
% signal vector X using Welch's averaged periodogram method. The signal X
% is divided into overlapping sections, each of which is detrended and
% windowed by the WINDOW parameter, then zero padded to length NFFT. The
% magnitude squared of the length NFFT DFTs of the sections are averaged
% to form Pxx. P is a two column matrix P = [Pxx Pxxc]; the second column
% Pxxc is the 95% confidence interval. The number of rows of P is NFFT/2+1
% for NFFT even, (NFFT+1)/2 for NFFT odd, or NFFT if the signal X is comp-
% lex. If you specify a scalar for WINDOW, a Hanning window of that length
% is used.
%
% [P,F] = SPECTRUM(X,NFFT,NOVERLAP,WINDOW,Fs) given a sampling frequency
% Fs returns a vector of frequencies the same length as Pxx at which the
% PSD is estimated. PLOT(F,P(:,1)) plots the power spectrum estimate
% versus true frequency.
%
% [P, F] = SPECTRUM(X,NFFT,NOVERLAP,WINDOW,Fs,Pr) where Pr is a scalar
% between 0 and 1, overrides the default 95% confidence interval and
% returns the Pr*100% confidence interval for Pxx instead.
%
% SPECTRUM(X) with no output arguments plots the PSD in the current
% figure window, with confidence intervals.
%
% The default values for the parameters are NFFT = 256 (or LENGTH(X),
% whichever is smaller), NOVERLAP = 0, WINDOW = HANNING(NFFT), Fs = 2,
% and Pr = .95. You can obtain a default parameter by leaving it out
% or inserting an empty matrix [], e.g. SPECTRUM(X,[],128).
%
% P = SPECTRUM(X,Y) performs spectral analysis of the two sequences
% X and Y using the Welch method. SPECTRUM returns the 8 column array
% P = [Pxx Pyy Pxy Txy Cxy Pxxc Pyyc Pxyc]
% where
% Pxx = X-vector power spectral density
% Pyy = Y-vector power spectral density
% Pxy = Cross spectral density
% Txy = Complex transfer function from X to Y = Pxy./Pxx
% Cxy = Coherence function between X and Y = (abs(Pxy).^2)./(Pxx.*Pyy)
% Pxxc,Pyyc,Pxyc = Confidence range.
% All input and output options are otherwise exactly the same as for the
% single input case.
%
% SPECTRUM(X,Y) with no output arguments will plot Pxx, Pyy, abs(Txy),
% angle(Txy) and Cxy in sequence, pausing between plots.
%
% SPECTRUM(X,...,DFLAG), where DFLAG can be 'linear', 'mean' or 'none',
% specifies a detrending mode for the prewindowed sections of X (and Y).
% DFLAG can take the place of any parameter in the parameter list
% (besides X) as long as it is last, e.g. SPECTRUM(X,'none');
%
% See also PSD, CSD, TFE, COHERE, SPECGRAM, SPECPLOT, DETREND, PMTM,
% PMUSIC.
% ETFE, SPA, and ARX in the Identification Toolbox.
% Author(s): J.N. Little, 7-9-86
% C. Denham, 4-25-88, revised
% L. Shure, 12-20-88, revised
% J.N. Little, 8-31-89, revised
% L. Shure, 8-11-92, revised
% T. Krauss, 4-15-93, revised
% Copyright (c) 1988-98 by The MathWorks, Inc.
% $Revision: 1.1 $ $Date: 1998/06/03 14:43:54 $
% The units on the power spectra Pxx and Pyy are such that, using
% Parseval's theorem:
%
% SUM(Pxx)/LENGTH(Pxx) = SUM(X.^2)/LENGTH(X) = COV(X)
%
% The RMS value of the signal is the square root of this.
% If the input signal is in Volts as a function of time, then
% the units on Pxx are Volts^2*seconds = Volt^2/Hz.
%
% Here are the covariance, RMS, and spectral amplitude values of
% some common functions:
% Function Cov=SUM(Pxx)/LENGTH(Pxx) RMS Pxx
% a*sin(w*t) a^2/2 a/sqrt(2) a^2*LENGTH(Pxx)/4
%Normal: a*rand(t) a^2 a a^2
%Uniform: a*rand(t) a^2/12 a/sqrt(12) a^2/12
%
% For example, a pure sine wave with amplitude A has an RMS value
% of A/sqrt(2), so A = SQRT(2*SUM(Pxx)/LENGTH(Pxx)).
%
% See Page 556, A.V. Oppenheim and R.W. Schafer, Digital Signal
% Processing, Prentice-Hall, 1975.
error(nargchk(1,8,nargin))
[msg,x,y,nfft,noverlap,window,Fs,p,dflag]=specchk(varargin);
error(msg)
if isempty(p),
p = .95; % default confidence interval even if not asked for
end
n = length(x); % Number of data points
nwind = length(window);
if n < nwind % zero-pad x (and y) if length less than the window length
x(nwind)=0; n=nwind;
if ~isempty(y), y(nwind)=0; end
end
x = x(:); % Make sure x and y are column vectors
y = y(:);
k = fix((n-noverlap)/(nwind-noverlap)); % Number of windows
% (k = fix(n/nwind) for noverlap=0)
index = 1:nwind;
KMU = k*norm(window)^2; % Normalizing scale factor ==> asymptotically unbiased
% KMU = k*sum(window)^2;% alt. Nrmlzng scale factor ==> peaks are about right
if (isempty(y)) % Single sequence case.
Pxx = zeros(nfft,1); Pxx2 = zeros(nfft,1);
for i=1:k
if strcmp(dflag,'linear')
xw = window.*detrend(x(index));
elseif strcmp(dflag,'none')
xw = window.*(x(index));
else
xw = window.*detrend(x(index),0);
end
index = index + (nwind - noverlap);
Xx = abs(fft(xw,nfft)).^2;
Pxx = Pxx + Xx;
Pxx2 = Pxx2 + abs(Xx).^2;
end
% Select first half
if ~any(any(imag(x)~=0)), % if x and y are not complex
if rem(nfft,2), % nfft odd
select = [1:(nfft+1)/2];
else
select = [1:nfft/2+1]; % include DC AND Nyquist
end
else
select = 1:nfft;
end
Pxx = Pxx(select);
Pxx2 = Pxx2(select);
cPxx = zeros(size(Pxx));
if k > 1
c = (k.*Pxx2-abs(Pxx).^2)./(k-1);
c = max(c,zeros(size(Pxx)));
cPxx = sqrt(c);
end
ff = sqrt(2)*erfinv(p); % Equal-tails.
Pxx = Pxx/KMU;
Pxxc = ff.*cPxx/KMU;
P = [Pxx Pxxc];
else
Pxx = zeros(nfft,1); % Dual sequence case.
Pyy = Pxx; Pxy = Pxx; Pxx2 = Pxx; Pyy2 = Pxx; Pxy2 = Pxx;
for i=1:k
if strcmp(dflag,'linear')
xw = window.*detrend(x(index));
yw = window.*detrend(y(index));
elseif strcmp(dflag,'none')
xw = window.*(x(index));
yw = window.*(y(index));
else
xw = window.*detrend(x(index),0);
yw = window.*detrend(y(index),0);
end
index = index + (nwind - noverlap);
Xx = fft(xw,nfft);
Yy = fft(yw,nfft);
Yy2 = abs(Yy).^2;
Xx2 = abs(Xx).^2;
Xy = Yy .* conj(Xx);
Pxx = Pxx + Xx2;
Pyy = Pyy + Yy2;
Pxy = Pxy + Xy;
Pxx2 = Pxx2 + abs(Xx2).^2;
Pyy2 = Pyy2 + abs(Yy2).^2;
Pxy2 = Pxy2 + Xy .* conj(Xy);
end
% Select first half
if ~any(any(imag([x y])~=0)), % if x and y are not complex
if rem(nfft,2), % nfft odd
select = [1:(nfft+1)/2];
else
select = [1:nfft/2+1]; % include DC AND Nyquist
end
else
select = 1:nfft;
end
Pxx = Pxx(select);
Pyy = Pyy(select);
Pxy = Pxy(select);
Pxx2 = Pxx2(select);
Pyy2 = Pyy2(select);
Pxy2 = Pxy2(select);
cPxx = zeros(size(Pxx));
cPyy = cPxx;
cPxy = cPxx;
if k > 1
c = max((k.*Pxx2-abs(Pxx).^2)./(k-1),zeros(size(Pxx)));
cPxx = sqrt(c);
c = max((k.*Pyy2-abs(Pyy).^2)./(k-1),zeros(size(Pxx)));
cPyy = sqrt(c);
c = max((k.*Pxy2-abs(Pxy).^2)./(k-1),zeros(size(Pxx)));
cPxy = sqrt(c);
end
Txy = Pxy./Pxx;
Cxy = (abs(Pxy).^2)./(Pxx.*Pyy);
ff = sqrt(2)*erfinv(p); % Equal-tails.
Pxx = Pxx/KMU;
Pyy = Pyy/KMU;
Pxy = Pxy/KMU;
Pxxc = ff.*cPxx/KMU;
Pxyc = ff.*cPxy/KMU;
Pyyc = ff.*cPyy/KMU;
P = [Pxx Pyy Pxy Txy Cxy Pxxc Pyyc Pxyc];
end
freq_vector = (select - 1)'*Fs/nfft;
if nargout == 0, % do plots
newplot;
c = [max(Pxx-Pxxc,0) Pxx+Pxxc];
c = c.*(c>0);
semilogy(freq_vector,Pxx,freq_vector,c(:,1),'--',...
freq_vector,c(:,2),'--');
title('Pxx - X Power Spectral Density')
xlabel('Frequency')
if (isempty(y)), % single sequence case
return
end
pause
newplot;
c = [max(Pyy-Pyyc,0) Pyy+Pyyc];
c = c.*(c>0);
semilogy(freq_vector,Pyy,freq_vector,c(:,1),'--',...
freq_vector,c(:,2),'--');
title('Pyy - Y Power Spectral Density')
xlabel('Frequency')
pause
newplot;
semilogy(freq_vector,abs(Txy));
title('Txy - Transfer function magnitude')
xlabel('Frequency')
pause
newplot;
plot(freq_vector,180/pi*angle(Txy)), ...
title('Txy - Transfer function phase')
xlabel('Frequency')
pause
newplot;
plot(freq_vector,Cxy);
title('Cxy - Coherence')
xlabel('Frequency')
elseif nargout ==1,
Spec = P;
elseif nargout ==2,
Spec = P;
f = freq_vector;
end
function [msg,x,y,nfft,noverlap,window,Fs,p,dflag] = specchk(P)
%SPECCHK Helper function for SPECTRUM
% SPECCHK(P) takes the cell array P and uses each cell as
% an input argument. Assumes P has between 1 and 7 elements.
% Author(s): T. Krauss, 4-6-93
msg = [];
if length(P{1})<=1
msg = 'Input data must be a vector, not a scalar.';
x = [];
y = [];
elseif (length(P)>1),
if (all(size(P{1})==size(P{2})) & (length(P{1})>1) ) | ...
length(P{2})>1, % 0ne signal or 2 present?
% two signals, x and y, present
x = P{1}; y = P{2};
% shift parameters one left
P(1) = [];
else
% only one signal, x, present
x = P{1}; y = [];
end
else % length(P) == 1
% only one signal, x, present
x = P{1}; y = [];
end
% now x and y are defined; let's get the rest
if length(P) == 1
nfft = min(length(x),256);
window = hanning(nfft);
noverlap = 0;
Fs = 2;
p = [];
dflag = 'linear';
elseif length(P) == 2
if isempty(P{2}), dflag = 'linear'; nfft = min(length(x),256);
elseif isstr(P{2}), dflag = P{2}; nfft = min(length(x),256);
else dflag = 'linear'; nfft = P{2}; end
window = hanning(nfft);
noverlap = 0;
Fs = 2;
p = [];
elseif length(P) == 3
if isempty(P{2}), nfft = min(length(x),256); else nfft=P{2}; end
if isempty(P{3}), dflag = 'linear'; noverlap = 0;
elseif isstr(P{3}), dflag = P{3}; noverlap = 0;
else dflag = 'linear'; noverlap = P{3}; end
window = hanning(nfft);
Fs = 2;
p = [];
elseif length(P) == 4
if isempty(P{2}), nfft = min(length(x),256); else nfft=P{2}; end
if isstr(P{4})
dflag = P{4};
window = hanning(nfft);
else
dflag = 'linear';
window = P{4}; window = window(:); % force window to be a column
if length(window) == 1, window = hanning(window); end
if isempty(window), window = hanning(nfft); end
end
if isempty(P{3}), noverlap = 0; else noverlap=P{3}; end
Fs = 2;
p = [];
elseif length(P) == 5
if isempty(P{2}), nfft = min(length(x),256); else nfft=P{2}; end
window = P{4}; window = window(:); % force window to be a column
if length(window) == 1, window = hanning(window); end
if isempty(window), window = hanning(nfft); end
if isempty(P{3}), noverlap = 0; else noverlap=P{3}; end
if isstr(P{5})
dflag = P{5};
Fs = 2;
else
dflag = 'linear';
if isempty(P{5}), Fs = 2; else Fs = P{5}; end
end
p = [];
elseif length(P) == 6
if isempty(P{2}), nfft = min(length(x),256); else nfft=P{2}; end
window = P{4}; window = window(:); % force window to be a column
if length(window) == 1, window = hanning(window); end
if isempty(window), window = hanning(nfft); end
if isempty(P{3}), noverlap = 0; else noverlap=P{3}; end
if isempty(P{5}), Fs = 2; else Fs = P{5}; end
if isstr(P{6})
dflag = P{6};
p = [];
else
dflag = 'linear';
if isempty(P{6}), p = .95; else p = P{6}; end
end
elseif length(P) == 7
if isempty(P{2}), nfft = min(length(x),256); else nfft=P{2}; end
window = P{4}; window = window(:); % force window to be a column
if length(window) == 1, window = hanning(window); end
if isempty(window), window = hanning(nfft); end
if isempty(P{3}), noverlap = 0; else noverlap=P{3}; end
if isempty(P{5}), Fs = 2; else Fs = P{5}; end
if isempty(P{6}), p = .95; else p = P{6}; end
if isstr(P{7})
dflag = P{7};
else
msg = 'DFLAG parameter must be a string.'; return
end
end
% NOW do error checking
if (nfft<length(window)),
msg = 'Requires window''s length to be no greater than the FFT length.';
end
if (noverlap >= length(window)),
msg = 'Requires NOVERLAP to be strictly less than the window length.';
end
if (nfft ~= abs(round(nfft)))|(noverlap ~= abs(round(noverlap))),
msg = 'Requires positive integer values for NFFT and NOVERLAP.';
end
if ~isempty(p),
if (prod(size(p))>1)|(p(1,1)>1)|(p(1,1)<0),
msg = 'Requires confidence parameter to be a scalar between 0 and 1.';
end
end
if min(size(x))~=1,
msg = 'Requires vector (either row or column) input.';
end
if (min(size(y))~=1)&(~isempty(y)),
msg = 'Requires vector (either row or column) input.';
end
if (length(x)~=length(y))&(~isempty(y)),
msg = 'Requires X and Y be the same length.';
end