Page 138 - Fundamentals of Communications Systems

4.14 Chapter Four

4.5 Power of Carrier Modulated Signals
The output power of a carrier modulated signal is often important in evaluating
the trade-offs between analog communication options. For generality of the
exposition in this text, power will be computed as a time average power. The
time average power is defined as

1 T m /2
2
= lim x (t)dt (4.22)
P x c c
T m →∞ T m −T m /2
Using the Rayleigh energy theorem and Eq. (2.56) the time average power of a
bandpass modulated signal is
∞

= lim (f , T m ) df (4.23)
P x c S X c
T m →∞
−∞
Using the power spectrum analog to Eq. (4.15) gives
1 T m /2 2
= lim (4.24)
P x c |x z (t)| dt = P x z
T m →∞ T m −T m /2
Hence we have shown that the power contained in a carrier modulated signal
is exactly the power in the complex envelope. This result is the main reason
√
why the notation used in this text uses the 2 term in the carrier terms.

4.6 Linear Systems and Bandpass Signals
This section discusses methods for calculating the output of a linear, time-
invariant (LTI) filter with a bandpass input signal using complex envelopes.
Linear system outputs are characterized by the convolution integral given as
∞

y c (t) = x c (τ)h(t − τ)dτ (4.25)
−∞
where h(t) is the impulse response of the filter. Since the input signal is band-
pass, the effects of an arbitrary filter, h(t), in Eq. (4.25) can be modeled with an
equivalent bandpass filter, h c (t), with no loss in generality. The bandpass filter,
H c (f ), only needs to equal the true filter, H(f ) over the frequency support of the
bandpass signal and the two filters need not be equal otherwise. Because of this
characterization the bandpass filter is often simpler to model (and to simulate).

EXAMPLE 4.6
For example consider the lowpass filter given in Figure 4.11(a). Since the bandpass
signal only has a nonzero spectrum in a bandwidth of B T around the carrier frequency,
f c , the bandpass filter shown in Figure 4.11(b) would be input–output equivalent to the
filter in Figure 4.11(a).

This bandpass LTI system also has a canonical representation given as

h c (t) = 2h I (t) cos(2π f c t) − 2h Q (t) sin(2π f c t) (4.26)