Page 127 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.3

A bandpass signal has a representation of
√ √
x c (t) = x I (t) 2 cos(2π f c t) − x Q (t) 2 sin(2π f c t) (4.1)
√
= x A(t) 2 cos(2π f c t + x P (t)) (4.2)

where f c is denoted the carrier frequency with f C − B T /2 ≤ f c ≤ f C + B T /2.
The signal x I (t) in Eq. (4.1) is normally referred to as the in-phase (I) compo-
nent of the signal and the signal x Q (t) is normally referred to as the quadrature
(Q) component of the bandpass signal. x I (t) and x Q (t) are real valued lowpass
signals with a one-sided non–negligible energy spectrum no larger than B T Hz.
Two items should be noted

■ The center frequency of the bandpass signal, f C , (see Figure 4.1) and the
carrier frequency, f c are not always the same. While f c can theoretically take
a continuum of values in Eq. (4.1), in most applications an obvious value of
1
f c will give the simplest representation .
√
■ The 2 term is included in the definition of the bandpass signal to ensure that
the bandpass signal and the baseband signal have the same power/energy.
This will become apparent in Section 4.4.

The carrier signal is normally thought of as the cosine term, hence the I
component is in-phase with the carrier. Likewise the sine term is 90 out-of-
◦
phase (in quadrature) with the cosine or carrier term, hence the Q component
is quadrature to the carrier. Equation (4.1) is known as the canonical form of a
bandpass signal. Equation (4.2) is the amplitude and phase form of the band-
pass signal, where x A (t) is the amplitude of the signal and x P (t) is the phase
of the signal. A bandpass signal has two degrees of freedom. A communication
engineer can use either the I/Q representation or the amplitude and phase rep-
resentation to denote a bandpass signal. The transformations between the two
representations are given by

−1
2
x A (t) = x I (t) + x Q (t) 2 x P (t) = tan [x Q (t), x I (t)] (4.3)
and
x I (t) = x A(t) cos(x P (t)) x Q (t) = x A(t) sin(x P (t)) (4.4)

Note that the inverse tangent function in Eq. (4.3) has a range of [−π, π] (i.e.,
both the sign of x I (t) and x Q (t) and the ratio of x I (t) and x Q (t) are needed
to evaluate the function). This inverse tangent function is different than the
single argument function that is on most calculators. The particulars of the
communication design analysis determine which form for the bandpass signal
is most applicable.

1 This idea will become more obvious in Chapter 6.