Page 76 - Fundamentals of Communications Systems

2.28 Chapter Two

(a) If x(t) = Acos(2π f m t), what is y(t) and Y (f )?
(b) If

A || f |− f c |≤ f m
X(f ) = (2.88)
0 elsewhere
what is y(t) and Y (f )?
(c) A quadratic nonlinearity is often used in a frequency doubler. What com-
ponent would you need to add in series with this quadratic memoryless
nonlinearity such that you could put a sine wave in and get a sine wave out
of twice the input frequency?

Problem 2.9. Consider the following signal
x(t) = cos(2π f 1 t) + asin(2π f 1 t)

= X A (a) cos(2π f 1 t + X p (a)) (2.89)

(a) Find X A(a).
(b) Find X p (a).
(c) What is the power of x(t), P x ?
(d) Is x(t) periodic? If so, what is the period and the Fourier series
representation of x(t)?

Problem 2.10. Consider a signal and a linear system as depicted in Figure 2.12
where
x(t) = A+ cos(2π f 1 t)
and

⎧
1
⎪
0 ≤ t ≤ T p
⎨

h(t) = T p (2.90)
0 elsewhere
⎪
⎩
Compute the output y(t).
Problem 2.11. For the signal
sin(2π147t)
x(t) = 23 (2.91)
2π147t

x t() ht() yt() Figure 2.12 The system for
Problem 2.10.