HILBERT Function

Standard Library function that constructs a Hilbert transformation matrix.

result = HILBERT(x [, d])

x The vector to be transformed. Can be of either floating-point or complex data type, and can contain any number of elements.

d (optional) A flag to indicate the direction of rotation:

Returned Value

result The value of the Hilbert transform of x. Result is of a complex data type, with the same dimensions as x.

None.

A Hilbert transform is a series of numbers in which all periodic components have been phase-shifted by 90 degrees. Angle shifting is accomplished by multiplying or dividing by the complex number i = (0.000, 1.000).

A Hilbert series has the interesting property that the correlation between it and its own Hilbert transform is mathematically zero.

The HILBERT function generates a Hilbert matrix by generating the Fast Fourier Transform of the data with the FFT function and shifting the first half of the transform products by +90 degrees and the second half by -90 degrees. The constant elements of the transform are not changed.

The shifted vector is then submitted to the FFT function for the transformation back to the "time" domain. Before it is returned, the output is divided by the number of elements in the vector to correct for the multiplication effect characteristic of the FFT algorithm.

a = FINDGEN(1000)
sine_wave = SIN(a/(MAX(a)/(2 * !pi)))

; Create a sine wave.

PLOT, sine_wave

; Plot the sine wave.

OPLOT, HILBERT(sine_wave, -1)

; Plot the sine wave phase-shifted to the right by 90 degrees.

rand = RANDOMN(seed, 1000) * 0.05

; Create an array of random numbers to mimic a noisy signal.

PLOT, rand

; Plot the random numbers.

sandwich = [sine_wave, rand, sine_wave]

; Sandwich the random data between two sine waves.

PLOT, sandwich, XStyle=1

; Plot the two sine waves with the random noise in the middle, thereby
; turning them into a single signal.

OPLOT, HILBERT(sandwich, -1)

; Plot the sandwiched wave forms. Note that the sine waves are
; phase-shifted to the right by 90 degrees, while the noise data has
; not shifted at all, but rather has been distorted vertically (its 
; amplitude) by the effect of the two adjacent phase-shifted sine
; waves. This is because the sine waves and the noise data were set
; up to be a single signal.

FFT, HANNING