Basic Principles of Digital Modulation - Digital Video and Audio Broadcasting Technology - page 228

13.6 Use of the Hilbert transform in IQ modulation
In this section we will discuss the Hilbert Transform, which plays a major
role in some digital modulation methods such as OFDM or 8VSB (c.f.
ATSC, the U.S. version of digital terrestrial TV).
Let us start with sine and cosine signals. At the time t=0, the sine signal
has the value 0, the cosine signal the value 1. The sine signal is shifted 90°
relative to the cosine signal, i.e. it leads the cosine signal by 90°. We will
see later that the sine signal is the Hilbert Transform of the cosine signal.
Based on the sine and cosine functions, we can arrive at some important
definitions: the cosine function is an even function, i.e. it is symmetrical
about t=0, so that cos(x) = cos(-x) applies.
H()
j
j
Fig. 13.22. Fourier Transform of the Hilbert Transform
The sine function, on the other hand, is an odd function, i.e. it is half-
turn symmetrical about t=0, so that sin(x) = -sin(-x) applies. The spectrum
of the cosine, i.e. its Fourier Transform, is purely real and symmetrical
about f=0. The imaginary component is zero (s. Fig. 13.20.).
The spectrum of the sine, i.e. its Fourier transform, is purely imaginary
and half-turn symmetrical (Fig. 13.20.). The real component is zero. The
above facts are important for understanding the Hilbert Transform. For all
real time-domain signals, the spectrum of all real components versus f
(Re(f)) is symmetrical about f=0, and the spectrum of all imaginary com-
ponents versus f (Im(f)) is half-turn symmetrical about f=0 (s. Fig. 13.21.).
Any real time-domain signal can be represented as a Fourier series - the
superposition of the cosinusoidal and sinusoidal harmonics of the signal.
The cosine functions are even and the sine functions odd. Therefore, the