8.6. BANDPASS
QUADRATURE SIGNALS IN THE FREQUENCY DOMAIN
In quadrature processing, by convention, the
real part of the spectrum is called the inphase component and the imaginary
part of the spectrum is called the quadrature component. The signals whose
complex spectra are in Figure
812(a), (b), and (c) are real, and in the time
domain they can be represented by amplitude values having nonzero
real parts and zerovalued imaginary parts. We're not forced to use
complex notation to represent them in the time domain—the
signals are real only.
Figure 812. Quadrature representation
of signals: (a) real sinusoid cos(2pf_{o}t + 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 inphase (real) spectrum
always have even symmetry around the zerofrequency point. That is,
the inphase 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 812(a). This conjugate symmetry is the invariant
nature of real signals, and is obvious when their spectra are
represented using complex notation.
A complexvalued time signal, whose spectrum can
be that in Figure 812(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 negativefrequency spectral components.
Let's remind ourselves again: those bold arrows
in Figure 812(a) and (b) are not rotating phasors.
They're frequencydomain impulses indicating a single complex
exponential e^{j}^{2}^{p}^{ft}. The directions in which
the impulses are pointing show the relative phases of the spectral
components.
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
f_{o} Hz, as shown in Figure 813(a) and (b). Likewise, multiplying a
time signal by (also called complex
downconversion or mixing to baseband) shifts a signal's
spectrum down to a center frequency of zero Hz as shown in Figure 813(c). The process
of quadrature mixing is used in many DSP applications as well as
most modernday digital communications systems.
Figure 813. 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 discretetime signals. Let's
look at the effect of complex downconversion of a discrete
signal's spectrum.
