B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

100 ANALYSIS AND TRANSMISSION OF SIGNALS
Example 3 . 15 The input x(t) and the output y(t) of a certain nonlinear channel are related as
y(t) = x(t) + 0.000158x 2 (t)
Find the output signal y(t) and its spectrum Y (f) if the input signal is x(t) = 2000
sine (2000nt). Verify that the bandwidth of the output signal is twice that of the input signal.
This is the result of signal squaring. Can the signal x(t) be recovered (without distortion)
from the output y(t)?
Since
x(t) = 2000 sine (2000nt)
X(f) - TI (_j_)
2000
We have
y(t) = x(t) + 0.000158x 2 (t) = 2000 sine (2000nt) + 0.316 • 2000 sinc 2 (2000nt)
{=}
Yif ) = TI ( 2 t oo ) + 0·316 fl ( 4 t oo )
Observe that 0.316 • 2000sinc 2 (2000n t) is the unwanted ( distortion) term in the received
signal. Figure 3.30a shows the input (desired) signal spectrum X (f); Fig. 3.30b shows
the spectrum of the undesired (distortion) term; and Fig. 3.30c shows the received signal
spectrum Y(f). We make the following observations.
1. The bandwidth of the received signal y(t) is twice that of the input signal x(t) (because
of signal squaring).
2. The received signal contains the input signal x(t) plus an unwanted signal
632 sinc 2 (2000nt). The spectra of these two signals are shown in Fig. 3.30a and b.
Figure 3.30c shows Y(f), the spectrum of the received signal. Note that spectra of
the desired signal and the distortion signal overlap, and it is impossible to recover the
signal x(t) from the received signal y(t) without some distortion.
3. We can reduce the distortion by passing the received signal through a low-pass filter
of bandwidth 1000 Hz. The spectrum of the output of this filter is shown in Fig. 3.30d.
Observe that the output of this filter is the desired input signal x(t) with some residual
distortion.
4. We have an additional problem of interference with other signals if the input signal x(t)
is frequency-division-multiplexed along with several other signals on this channel. This
means that several signals occupying nonoverlapping frequency bands are transmitted
simultaneously on the same channel. Spreading the spectrum X (f) outside its original
band of 1000 Hz will interfere with the signal in the band of 1000 to 2000 Hz. Thus,
in addition to the distortion of x(t), we have an interference with the neighboring
band.