Basic Principles of Digital Modulation - Digital Video and Audio Broadcasting Technology - page 229

characteristics previously stated for a single cosine function or a single
sine function also generally apply to a sum of cosine functions or a sum of
sine functions. Let us now discuss the Hilbert Transform itself. Fig. 13.22.
shows the transfer function of a Hilbert transformer. A Hilbert transformer
is a signal processing block with special characteristics. Its main purpose is
to phase shift a sine signal by 90°. This means that a cosine is converted to
a sine and a sine to a minus cosine. The amplitude remains invariant under
the Hilbert Transform. These characteristics apply to any type of sinusoi-
dal signal, i.e. of any frequency, amplitude, or phase. Hence, they also ap-
ply to all the harmonics of any type of time-domain signal. This is due to
the transfer function of the Hilbert transformer which is shown in
Fig. 13.22. - essentially it only makes use of the symmetry characteristics
of even and odd time-domain signals referred to above.
Examining the transfer function of the Hilbert transformer, we find:
• All negative frequencies are multiplied by j, all positive fre-
quencies by -j. j is the positive, imaginary square root of -1
• The rule j • j = -1 applies
• Real spectral components, therefore, become imaginary and
imaginary components become real
• Multiplication by j or -j may invert the negative or positive part
of the spectrum
Applying the Hilbert Transform to a cosine signal, the following is ob-
tained: A cosine has a purely real spectrum symmetrical about zero. If the
negative half of the spectrum is multiplied by j, a purely positive imagi-
nary spectrum is obtained for all negative frequencies. If the positive half
of the spectrum is multiplied by -j, a purely negative imaginary spectrum is
obtained for all frequencies above zero. The spectrum of a sine is obtained.
This applies analogously to the Hilbert Transform of a sine signal:
By multiplying the positive imaginary negative sine spectrum by j, the
latter becomes negative real (j • j = -1). By multiplying the negative
imaginary positive sine spectrum by -j, the latter becomes purely positive
real (-j • -j = -(√-1 • √-1) = 1). The spectrum of a minus cosine is obtained.
The cosine-to-sine and sine-to-minus-cosine mapping by the Hilbert
Transform also applies to all the harmonics of any type of time-domain
signal.
Summarizing, the Hilbert Transform shifts the phases of all harmonics
of any type of time-domain signal by 90°, i.e. it acts as a 90° phase shifter
for all harmonics.