Discrete-time analytic signal using Hilbert transform
a complex helical sequence, sometimes called the analytic
signal, from a real data sequence. The analytic signal
i*xi has a real part,
which is the original data, and an imaginary part,
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 conversely. The Hilbert transformed
series has the same amplitude and frequency content as the original
sequence and includes phase information that depends on the phase
of the original.
xr is a matrix,
x = hilbert(xr) operates columnwise on the matrix,
finding the analytic signal corresponding to each column.
n point FFT to compute the Hilbert transform.
The input data
xr is zero-padded or truncated to
n, as appropriate.
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 how 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. See Hilbert Transform and Instantaneous Frequency for examples.
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
For a discrete-time analytic signal,
the last half of
fft(x) is zero, and the first
(DC) and center (Nyquist) elements of
Define a sequence and compute its analytic signal using
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
imx = imag(x)
imx = 1 -1 -1 1
rex = real(x)
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
hilbert function finds the exact analytic signal for a finite block of data. You can also generate the analytic signal by using an FIR Hilbert transformer filter to compute an approximation to the imaginary part.
Generate a sequence composed of three sinusoids with frequencies 203, 721, and 1001 Hz. The sequence is sampled at 10 kHz for about 1 second. Use the
hilbert function to compute the analytic signal. Plot it between 0.01 seconds and 0.03 seconds.
fs = 1e4; t = 0:1/fs:1; x = 2.5+cos(2*pi*203*t)+sin(2*pi*721*t)+cos(2*pi*1001*t); y = hilbert(x); plot(t,real(y),t,imag(y)) xlim([0.01 0.03]) legend('real','imaginary') title('hilbert Function')
Compute Welch estimates of the power spectral densities of the original sequence and the analytic signal. Divide the sequences into Hamming windowed nonoverlapping sections of length 256. Verify that the analytic signal has no power at negative frequencies.
designfilt function to design a 60th-order Hilbert transformer FIR filter. Specify a transition width of 400 Hz. Visualize the frequency response of the filter. Filter the sinusoidal sequence to approximate the imaginary part of the analytic signal.
fo = 60; d = designfilt('hilbertfir','FilterOrder',fo, ... 'TransitionWidth',400,'SampleRate',fs); freqz(d,1024,fs)
hb = filter(d,x);
The group delay of the filter,
grd, is equal to one-half the filter order. Compensate for this delay. Remove the first
grd samples of the imaginary part and the last
grd samples of the real part and the time vector. Plot the result between 0.01 seconds and 0.03 seconds.
grd = fo/2; y2 = x(1:end-grd) + 1j*hb(grd+1:end); t2 = t(1:end-grd); plot(t2,real(y2),t2,imag(y2)) xlim([0.01 0.03]) legend('real','imaginary') title('FIR Filter')
Estimate the PSD of the approximate analytic signal and compare it to the
pwelch([y;[y2 zeros(1,grd)]].',256,0,,fs,'centered') legend('hilbert','FIR Filter')
The analytic signal for a sequence
a one-sided Fourier transform. That is, the transform
vanishes for negative frequencies. To approximate the analytic signal,
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.
hilbert uses a four-step algorithm:
It calculates the FFT of the input
sequence, storing the result in a vector
It creates a vector
h(i) have the values:
i = 1,
i = 2,
3, ... ,
(n/2)+2, ... ,
It calculates the element-wise product
It calculates the inverse FFT of the
sequence obtained in step 3 and returns the first
of the result.
This algorithm was first introduced in .
The technique assumes that the input signal,
is a finite block of data. This assumption allows the function to
remove the spectral redundancy in
x exactly. Methods
based on FIR filtering can only approximate the analytic signal, but
they have the advantage that they operate continuously on the data.
See Single-Sideband Amplitude Modulation for another example
of a Hilbert transform computed with an FIR filter.
 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.