Documentation

hilbert

Discrete-time analytic signal using Hilbert transform

Syntax

x = hilbert(xr)
x = hilbert(xr,n)

Description

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 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.

If xr is a matrix, x = hilbert(xr) operates columnwise on the matrix, finding the analytic signal corresponding to each column.

x = hilbert(xr,n) uses an n point FFT to compute the Hilbert transform. The input data xr is zero-padded or truncated to length 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 [1] 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.

Examples

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Analytic Signal of a Sequence

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

Analytic Signal and Hilbert Transform

The 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.

pwelch([x;y].',256,0,[],fs,'centered')
legend('Original','hilbert')

Use the 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 hilbert result.

pwelch([y;[y2 zeros(1,grd)]].',256,0,[],fs,'centered')
legend('hilbert','FIR Filter')

More About

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Algorithms

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. It calculates the FFT of the input sequence, storing the result in a vector x.

  2. It creates a vector h whose elements h(i) have the values:

    • 1 for i = 1, (n/2)+1

    • 2 for i = 2, 3, ... , (n/2)

    • 0 for i = (n/2)+2, ... , n

  3. It calculates the element-wise product of x and h.

  4. It calculates the inverse FFT of the sequence obtained in step 3 and returns the first n elements of the result.

This algorithm was first introduced in [2]. The technique assumes that the input signal, x, 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.

References

[1] Claerbout, Jon F. Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting. Oxford, UK: Blackwell, 1985, pp. 59–62.

[2] Marple, S. L. "Computing the Discrete-Time Analytic Signal via FFT." IEEE Transactions on Signal Processing. Vol. 47, 1999, pp. 2600–2603.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

See Also

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