Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 253

(9) The Parseval's theorem states that the total energy in a function is the same
in the time and frequency domains. Therefore, the energy in a function can
be obtained either in the time domain by calculating the energy per unit time
and integrating it over all time, or in the frequency domain by calculating
the energy per unit frequency and integrating over all frequencies.

In Section 5.6 we derived the following condition for the existence of the CTFT
of the signal x ( t ):

∞

x ( t ) d t < ∞,

−∞

while in Sections 5.7 and 5.8 we discussed the relationship between the CTFS
and CTFT of periodic signals. In particular, the CTFT of a periodic signal x ( t )
is obtained by the relationship

n =−∞ D n δ ( ω − n ω 0 ) ,

∞

CTFT
←−−→ x ( t ) 2 π

where D n denotes the exponential CTFS coefficients and ω 0 is the fundamental
frequency. Conversely, the CTFS of a periodic signal is obtained by sampling
the CTFT of one period of the periodic signal at frequencies ω = n ω 0 . Section
5.9 showed that the three representations (linear, constant-coefficient differen-
tial equation; impulse response; and transfer function) for LTIC systems are
equivalent. Given one representation, it is straightforward to derive the remain-
ing two representations based on the CTFT and its properties. The transfer
function H ( ω ) plays an important role in the analysis of LTIC systems, and is
typically the preferred model for representing LTIC systems. In Section 5.10,
we concluded the chapter by showing the steps involved in computing the CTFT
of a CT signal using M ATLAB .

Problems

5.1 For each of the following CT functions, calculate the expression for the
CTFT directly by using Eq. (5.10). Compare the CTFT with the corre-
sponding entry in Table 5.2 to confirm the validity of your result.
(a) x 1

t
τ

= (1 − t /τ ) u ( t + τ ) − u ( t − τ ) ;

(b) x 2 ( t ) = t 4 e

− at u ( t ) , with a

∈ℜ + ;

− at cos( ω 0 t ) u ( t ) , with a ,ω 0

∈ℜ +

(c) x 3 ( t ) = e ;

− t 2 / 2 σ 2

(d) x 4 ( t ) = e , with σ ∈ℜ.

5.2 Calculate the CTFT of the functions shown in Figs. P5.2 (a)-(e).

5.3 Three functions x 1 ( t ), x 2 ( t ), and x 3 ( t ) have an identical magnitude
spectrum X ( ω ) but different phase spectra denoted, respectively, by