QUADRATURE SIGNALS IN THE FREQUENCY DOMAIN
In quadrature processing, by convention, the
real part of the spectrum is called the in-phase component and the imaginary
part of the spectrum is called the quadrature component. The signals whose
complex spectra are in Figure
8-12(a), (b), and (c) are real, and in the time
domain they can be represented by amplitude values having nonzero
real parts and zero-valued imaginary parts. We're not forced to use
complex notation to represent them in the time domainthe
signals are real only.
Figure 8-12. Quadrature representation
of signals: (a) real sinusoid cos(2pfot + ø); (b) real bandpass signal
containing six sinusoids over bandwidth B; (c) real bandpass signal containing
an infinite number of sinusoids over bandwidth B Hz; (d) complex bandpass signal of
bandwidth B Hz.
Real signals always have positive and negative
frequency spectral components. For any real signal, the positive
and negative frequency components of its in-phase (real) spectrum
always have even symmetry around the zero-frequency point. That is,
the in-phase part's positive and negative frequency components are
mirror images of each other. Conversely, the positive and negative
frequency components of its quadrature (imaginary) spectrum are
always negatives of each other. This means that the phase angle of
any given positive quadrature frequency component is the negative
of the phase angle of the corresponding quadrature negative
frequency component as shown by the thin solid arrows in Figure 8-12(a). This conjugate symmetry is the invariant
nature of real signals, and is obvious when their spectra are
represented using complex notation.
A complex-valued time signal, whose spectrum can
be that in Figure 8-12(d),
is not restricted to the above spectral conjugate symmetry
conditions. We'll call that special complex signal an analytic signal, signifying that it has
no negative-frequency spectral components.
Let's remind ourselves again: those bold arrows
in Figure 8-12(a) and (b) are not rotating phasors.
They're frequency-domain impulses indicating a single complex
exponential ej2pft. The directions in which
the impulses are pointing show the relative phases of the spectral
There's an important principle to keep in mind
before we continue. Multiplying a time signal by the complex
exponential , what we call quadrature
mixing (also called complex
mixing) shifts a signal's spectrum upward in frequency by
fo Hz, as shown in Figure 8-13(a) and (b). Likewise, multiplying a
time signal by (also called complex
down-conversion or mixing to baseband) shifts a signal's
spectrum down to a center frequency of zero Hz as shown in Figure 8-13(c). The process
of quadrature mixing is used in many DSP applications as well as
most modern-day digital communications systems.
Figure 8-13. Quadrature mixing of a
bandpass signal: (a) spectrum of a complex signal x(t);
(b) spectrum of ; (c) spectrum of .
Our entire quadrature signals discussion, thus
far, has been based on continuous signals, but the principles
described here apply equally well to discrete-time signals. Let's
look at the effect of complex down-conversion of a discrete