Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 193

CHAPTER

5

Continuous-time Fourier transform

In Chapter 4, we introduced the frequency representations for periodic sig-
nals based on the trigonometric and exponential continuous-time Fourier series
(CTFS). The exponential CTFS is useful in calculating the output response of
a linear time-invariant (LTI) system to a periodic input signal. In this chapter,
we extend the Fourier framework to continuous-time (CT) aperiodic signals.
The resulting frequency decompositions are referred to as the continuous-time
Fourier transform (CTFT) and are used to express both aperiodic and periodic
CT signals in terms of linear combinations of complex exponential functions.
We show that the convolution in the time domain is equivalent to multiplication
in the frequency domain. The CTFT, therefore, provides an alternative analysis
technique for LTIC systems in the frequency domain.
Chapter 5 is organized as follows. Section 5.1 considers the CTFT as a
limiting case of the CTFS and formally defines the CTFT and its inverse. In
Section 5.2, we provide several examples to illustrate the steps involved in the
calculation of the CTFT for a number of elementary signals. Section 5.3 presents
the look-up table and partial fraction methods for calculating the inverse CTFT.
Section 5.4 lists the symmetry properties of the CTFT for real-valued, even, and
odd signals, while Section 5.5 lists the CTFT properties arising due to linear
transformations in the time domain. The condition for the existence of the
CTFT is derived in Section 5.6, while the relationship between the CTFT and
the CTFS for periodic signals is discussed in Sections 5.7 and 5.8. Section 5.9
applies the convolution property of the CTFT to evaluate the output response of
an LTIC system to an arbitrary CT input signal. The gain and phase responses
of LTIC systems are also defined in this section. Section 5.10 demonstrates how
M ATLAB is used to compute the CTFT, and Section 5.11 concludes the chapter.

5.1 CTFT for aperio dic signals

Consider the aperiodic signal x ( t ) shown in Fig. 5.1(a). In order to extend
the Fourier framework of the CTFS to aperiodic signals, we consider several

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