The origins and uses of complex signals - Statistical Signal Processing of Complex-Valued Data - page 17

1

2 e − j w 0 t

e j w 0 t

g ( t ) LPF

∗

Re ( g ∗ p )( t )

p ( t ) LPF

e − j w 0 t

Figure 1.7 Passband filtering at baseband.

Hence, passband filtering can be performed at baseband. In the implementation shown
in Fig. 1.7 , the passband signal p ( t ) is complex-demodulated and lowpass-filtered to
produce the complex baseband signal x ( t ) (as discussed in Section 1.4.3 ), then filtered
using the complex baseband impulse response g b ( t ), and finally modulated back to
passband. This is what is done in most practical applications.
This concludes our discussion of complex signals for general modulation of the
amplitude and phase of a sinusoidal carrier. The essential point is that, once again, the
representation of two real modulating signals as a complex signal leads to insights and
economies of reasoning that would not otherwise emerge. Moreover, complex signal
theory allows us to construct bandwidth-efficient versions of amplitude modulation,
using the Hilbert transform and the complex analytic signal.

1.5 Complex signals for the efficient use of the FFT

There are four Fourier transforms: the continuous-time Fourier transform, the
continuous-time Fourier series, the discrete-time Fourier transform (DTFT), and the
discrete-time Fourier series, usually called the discrete Fourier transform (DFT). In
practice, the DFT is always computed using the fast Fourier transform (FFT). All of
these transforms are applied to a complex signal and they return a complex transform,
or spectrum. When they are applied to a real signal, then the returned complex spectrum
has Hermitian symmetry in frequency, meaning the negative-frequency half of the spec-
trum has been (inefficiently) computed when it could have been determined by simply
complex conjugating an efficiently computed positive-frequency half of the spectrum.
One might be inclined to say that the analytic signal solves this problem by placing a
real signal in the real part and the real Hilbert transform of this signal in the imaginary
part to form a complex analytic signal, whose spectrum no longer has Hermitian sym-
metry (being zero for negative frequencies). However, again, this special non-Hermitian
spectrum is known to be zero for negative frequencies and therefore the negative-
frequency part of the spectrum has been computed inefficiently for its zero values. So,
the only way to exploit the Fourier transform efficiently is to use it to simultaneously