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Pb. 3.35 The angle modulation scheme, which includes frequency modula-
tion (FM) and phase modulation (PM), has the modulated signal given by:

+
u t ( ) = A cos(2π f t k m t( ))
PM c c p
 t 
dτ
τ
u t ( ) = A cos2π f t + 2π k ∫ m( ) 
FM c  c f −∞ 
Assuming the same message as in Pb. 3.33:
a. Write the expression for the modulated signal in both schemes.
b. Plot the modulated signal in both schemes. Let k = k = 100.
f
p
Pb. 3.36 If f(x) = f(–x) for all x, then the graph of f(x) is symmetric with
respect to the y-axis, and the function f(x) is called an even function. If f(x) =
–f(–x) for all x, the graph of f(x) is anti-symmetric with respect to the origin,
and we call such a function an odd function.
a. Show that any function can be written as the sum of an odd func-
tion plus an even function. List as many even and odd functions
as you can.
b. State what conditions must be true for a polynomial to be even, or
to be odd.
c. Show that the product of two even functions is even; the product
of two odd functions is even; and the product of an odd and even
function is odd.
d. Replace in c above the word product by either quotient or power
and deduce the parity of the resulting function.
e. Deduce from the above results that the sign/parity of a function
follows algebraic rules.
f. Find the even and odd parts of the following functions:
4
7
(i) f(x) = x + 3x + 6x + 2
2
(ii) f(x) = (sin(x) + 3) sinh (x) exp(–x )
2
Pb. 3.37 Decompose the signal shown in Figure 3.7 into its even and odd
parts:
Pb. 3.38 Plot the function y defined through:

 x + 4 x + 4 for − ≤2 x < −1
2

2
yx() = 0 .16 x − 0 .48 x for − <1 x < 1 .5

 0 elsewhere
and find its even and odd parts.

© 2001 by CRC Press LLC