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    %% Upper and Lower 95%-Confidence Bounds
% The following example illustrates the use of confidence bounds with the
% periodogram. While not a necessary condition for statistical
% significance, frequencies in the periodogram where the lower confidence
% bound exceeds the upper confidence bound for surrounding PSD estimates
% clearly indicate significant oscillations in the time series.

% Copyright 2015 The MathWorks, Inc.


%%
% Create a signal consisting of the superposition of 100 Hz and 150 Hz sine
% waves in additive white _N_(0,1) noise. The amplitude of the two sine
% waves is 1. The sampling frequency is 1 kHz.

fs = 1000;
t = 0:0.001:1-0.001;
x = cos(2*pi*100*t)+sin(2*pi*150*t)+randn(size(t));

%%
% Obtain the periodogram with 95%-confidence bounds. Plot the periodogram
% along with the confidence interval and zoom in on the frequency region of
% interest near 100 and 150 Hz.

[pxx,f,pxxc] = periodogram(x,rectwin(length(x)),length(x),fs,...
    'ConfidenceLevel', 0.95);

plot(f,10*log10(pxx))
hold on
plot(f,10*log10(pxxc),'r-.')

xlim([85 175])
xlabel('Hz')
ylabel('dB')
title('Periodogram with 95%-Confidence Bounds')

%%
% The lower confidence bound in the immediate vicinity of 100 and 150 Hz is
% significantly above the upper confidence bound outside the vicinity of
% 100 and 150 Hz.