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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 finite impulse response (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')
xlabel('Time (s)')

Figure contains an axes object. The axes object with title hilbert Function contains 2 objects of type line. These objects represent real, imaginary.

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')

Figure contains an axes object. The axes object with title Power Spectral Density contains 2 objects of type line. These objects represent 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.

fo = 60;

d = designfilt('hilbertfir','FilterOrder',fo, ...
       'TransitionWidth',400,'SampleRate',fs); 

freqz(d,1024,fs)

Figure Filter Visualization Tool - Magnitude Response (dB) and Phase Response contains an axes object and other objects of type uitoolbar, uimenu. The axes object with title Magnitude Response (dB) and Phase Response contains an object of type line.

Filter the sinusoidal sequence to approximate the imaginary part of the analytic signal.

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')
xlabel('Time (s)')

Figure contains an axes object. The axes object with title FIR Filter contains 2 objects of type line. These objects represent real, imaginary.

Estimate the power spectral density (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')

Figure contains an axes object. The axes object with title Power Spectral Density contains 2 objects of type line. These objects represent hilbert, FIR Filter.

See Also

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