Page 83 - Fundamentals of Communications Systems

Signals and Systems Review 2.35

It is useful to recall that multiplication in the time domain results in convo-
lution in the frequency domain, i.e.,

z(t) = x(t) × x(t) Z(f ) = X(f ) ∗ X(f ) (2.105)

(a) x(t) = Acos(2π f m t), which results in
2
2
y(t) = aAcos(2π f m t) + bA cos (2π f m t)
1 1

2
= aAcos(2π f m t) + bA + cos(4π f m t)
2 2
aA aA bA 2
= exp[ j 2π f m t] + exp[− j 2π f m t] +
2 2 2
bA 2 bA 2
+ exp[ j 4π f m t] + exp[− j 4π f m t] (2.106)
4 4
The frequency domain representation for the output signal is given as
aA aA bA 2 bA 2
Y (f ) = δ( f − f m ) + δ( f + f m ) + δ(f ) + δ( f − 2 f m )
2 2 2 4
bA 2
+ δ( f + 2 f m ) (2.107)
4
The first two terms in Eq. (2.107) are due to the linear term in Eq. (2.104)
while the last three terms are due to the square law term. It is interesting
to note that these last three terms can be viewed as being obtained by
convolving the two “delta” function frequency domain representation of a
cosine wave, i.e., 2 cos(2π f m t) = exp[ j 2π f m t] + exp[− j 2π f m t], with itself.
(b) The input signal is

A || f |− f c |≤ f m
X(f ) = (2.108)
0 elsewhere
Taking the inverse Fourier transform gives

sin(π f m t)
x(t) = 4Af m cos(2π f c t) (2.109)
π f m t
The output time signal of this quadratic nonlinearity is
2

sin(π f m t) sin(π f m t)
y(t) = 4aAf m cos(2π f c t) + b 4Af m cos(2π f c t)
π f m t π f m t
sin(π f m t) sin(π f m t) 2
= 4aAf m cos(2π f c t) + b 4Af m
π f m t π f m t

1 1
× + cos(4π f c t) (2.110)
2 2