Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 216

Since x 2 ( t ) is an odd function, its CTFT, based on Eq. (5.41), is given by

∞ 1 2

X 2 ( ω ) =− j2 x ( t ) sin( ω t )d t =− j2 (2 t ) sin( ω t )d t − j2 2 sin( ω t )d t ,

0 0 1

which simplifies to

1 2

− t cos( ω t )
ω

+ 1 sin( ω t )
ω 2

− cos( ω t )
ω

=− j4 − j4

X 2 ( ω )

0 1

or

cos( ω )
ω

− sin( ω )
ω 2

cos(2 ω )
ω

− cos( ω )
ω

X 2 ( ω ) = j4 + j4

4
ω 2

= j [ ω cos(2 ω ) − sin( ω )] . (5.43)

The above result validates the symmetry property for real-valued odd functions.
Property 5.2 states that the CTFT of a real-valued odd function is imaginary
and odd. This is indeed the case for X 2 ( ω ) in Eq. (5.43).

5.5 Properties of t he CTFT

In Section 5.4, we covered the symmetry properties of the CTFT. In this section,
we present the properties of the CTFT based on the transformations of the
signals. Given the CTFT of a CT function x ( t ), we are interested in calculating
the CTFT of a function produced by a linear operation on x ( t ) in the time
domain. The linear operations being considered include superposition, time
shifting, scaling, differentiation and integration. We also consider some basic
non-linear operations like multiplication of two CT signals, convolution in the
time and frequency domain, and Parseval's relationship. A list of the CTFT
properties is included in Table 5.4.

5.5.1 Linearity

Often we are interested in calculating the CTFT of a signal that is a linear
combination of several elementary functions whose CTFTs are known. In such
a scenario, we use the linearity property to show that the overall CTFT is
given by the same linear combination of the individual CTFTs used in the time
domain. The linearity property is defined below.

If x 1 ( t ) and x 2 ( t ) are two CT signals with the following CTFT pairs:

CTFT
←−−→ x 1 ( t ) X 1 ( ω )

and

CTFT
←−−→ x 2 ( t ) X 2 ( ω )