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%% Hilbert Transform % The Hilbert transform facilitates the formation of the analytic signal. The % analytic signal is useful in the area of communications, particularly in bandpass % signal processing. The toolbox function |hilbert| computes the Hilbert transform % for a real input sequence |x| and returns a complex result of the same length, % |y = hilbert(x)|, where the real part of |y| is the original real data and the % imaginary part is the actual Hilbert transform. |y| is sometimes called the % _analytic signal_, in reference to the continuous-time analytic signal. A key % property of the discrete-time analytic signal is that its Z-transform is 0 on % the lower half of the unit circle. Many applications of the analytic signal % are related to this property; for example, the analytic signal is useful in % avoiding aliasing effects for bandpass sampling operations. The magnitude of % the analytic signal is the complex envelope of the original signal. % % The Hilbert transform is related to the actual data by a 90-degree phase % shift; sines become cosines and vice versa. To plot a portion of data and its % Hilbert transform, use % Copyright 2015 The MathWorks, Inc. t = 0:1/1024:1; x = sin(2*pi*60*t); y = hilbert(x); plot(t(1:50),real(y(1:50))) hold on plot(t(1:50),imag(y(1:50))) hold off axis([0 0.05 -1.1 2]) legend('Real Part','Imaginary Part') %% % The analytic signal is useful in calculating instantaneous attributes % of a time series, the attributes of the series at any point in time. The procedure % requires that the signal be monocomponent.