Discrete-time analytic signal using Hilbert transform
x = hilbert(xr)
x = hilbert(xr,n)
x = hilbert(xr) returns a complex helical sequence, sometimes called the analytic signal, from a real data sequence. The analytic signal x = xr + i*xi has a real part, xr, which is the original data, and an imaginary part, xi, which contains the Hilbert transform. The imaginary part is a version of the original real sequence with a 90° phase shift. Sines are therefore transformed to cosines and vice versa. The Hilbert transformed series has the same amplitude and frequency content as the original real data and includes phase information that depends on the phase of the original data.
If xr is a matrix, x = hilbert(xr) operates columnwise on the matrix, finding the Hilbert transform of each column.
The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the amplitude and frequency. The instantaneous amplitude is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle. For a pure sinusoid, the instantaneous amplitude and frequency are constant. The instantaneous phase, however, is a sawtooth, reflecting the way in which the local phase angle varies linearly over a single cycle. For mixtures of sinusoids, the attributes are short term, or local, averages spanning no more than two or three points.
Reference  describes the Kolmogorov method for minimum phase reconstruction, which involves taking the Hilbert transform of the logarithm of the spectral density of a time series. The toolbox function rceps performs this reconstruction.
For a discrete-time analytic signal, x, the last half of fft(x) is zero, and the first (DC) and center (Nyquist) elements of fft(x) are purely real.
Define a sequence and compute its analyutic signal using hilbert.
xr = [1 2 3 4]; x = hilbert(xr)
x = 1.0000 + 1.0000i 2.0000 - 1.0000i 3.0000 - 1.0000i 4.0000 + 1.0000i
The imaginary part of x is the Hilbert transform of xr, and the real part is xr itself.
imx = imag(x) rex = real(x)
imx = 1 -1 -1 1 rex = 1 2 3 4
The last half of the DFT of x is zero. (In this example, the last half of the transform is just the last element.) The DC and Nyquist elements of fft(x) are purely real.
dft = fft(x)
dft = 10.0000 + 0.0000i -4.0000 + 4.0000i -2.0000 + 0.0000i 0.0000 + 0.0000i
The analytic signal for a sequence x has a one-sided Fourier transform. That is, the transform vanishes for negative frequencies. To approximate the analytic signal, hilbert calculates the FFT of the input sequence, replaces those FFT coefficients that correspond to negative frequencies with zeros, and calculates the inverse FFT of the result.
In detail, hilbert uses a four-step algorithm:
1 for i = 1, (n/2)+1
2 for i = 2, 3, ... , (n/2)
0 for i = (n/2)+2, ... , n
If the input data xr is a matrix, hilbert operates in a similar manner, extending each step above to handle the matrix case.
 Claerbout, Jon F. Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting. Oxford, UK: Blackwell, 1985, pp. 59–62.
 Marple, S. L. "Computing the Discrete-Time Analytic Signal via FFT." IEEE Transactions on Signal Processing. Vol. 47, 1999, pp. 2600–2603.
 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.