Introduction to Acoustics

514 Part D Hearing and Signal Processing
Part D 14.6
third moment,
skewness = µ3/µ 3/2
2 . (14.85)
The fourth central moment leads to an impression
about how much strength there is in the wings of a probability
density compared to the standard deviation. The
14.6 Hilbert Transform and the Envelope
The Hilbert transform of a signal x(t) isH[x(t)] or
function xI(t), where
xI(t) = H[x(t)]= 1
π
�∞
−∞
dt ′ x(t′ )
. (14.87)
t − t ′
Some facts about the Hilbert transform are stated
here without proof. Proofs and further applications may
be found in appendices to [14.1].
First, the Hilbert transform is its own inverse, except
for a minus sign,
x(t) = H[xI(t)]=− 1
π
�∞
−∞
dt ′ xI(t ′ )
. (14.88)
t − t ′
Second, a signal and its Hilbert transform are orthogonal
in the sense that
�
dt x(t) xI(t) = 0 . (14.89)
Third, the Hilbert transform of sin(ωt + ϕ) is
− cos(ωt + ϕ), and the Hilbert transform of cos(ωt + ϕ)
is sin(ωt + ϕ).
Further the Hilbert transform is linear. Consequently,
for any function for which a Fourier transform exists,
� �
�
H An cos(ωnt) + Bn sin(ωnt)
or
n
= �
An sin(ωnt) − Bn cos(ωnt) (14.90)
n
� �
�
H Cn sin(ωnt + ϕn)
n
=− �
Cn cos(ωnt + ϕn)
n
= �
Cn sin(ωnt + ϕn − π/2) . (14.91)
n
normalized fourth moment is the kurtosis,
kurtosis = µ4/µ 2 2 . (14.86)
For instance, the kurtosis of a normal density, which has
significant wings, is 3. But the kurtosis of a rectangular
density, which is sharply cut off, is only 9/5.
Analytic signal
1
0.8
0.6
0.4
0.2
0
Real x(t)
–0.2
–0.4
–0.6
–0.8
–1
Imaginary xI(t) –100 –50 0 50
100
Time t (ms)
Fig. 14.6 A Gaussian pulse x(t) and its Hilbert transform
xI(t) are the real and imaginary parts of the analytic signal
corresponding to the Gaussian pulse
Comparing the two sine functions above makes it clear
why a Hilbert transform is sometimes called a 90-degree
rotation of the signal.
Figure 14.6 shows a Gaussian pulse, x(t), and its
Hilbert transform, xI(t). The Gaussian pulse was made
by adding up 100 cosine harmonics with amplitudes
given by a Gaussian spectrum per (14.35). The Hilbert
transform was computed by using the same amplitude
spectrum and replacing all the cosine functions by sine
functions.
Figure 14.6 illustrates the difficulty often encountered
in computing the Hilbert transform using the time
integrals that define the transform and its inverse. If we
had to calculate x(t) by transforming xI(t)using(14.88)
we would be troubled by the fact that xI(t) goes to zero
so slowly. An accurate calculation of x(t) would require
a longer time span than that shown in the figure.
14.6.1 The Analytic Signal
The analytic signal ˜x(t)forx(t) is given by the complex
sum of the original signal and an imaginary part equal