Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 230

which can be expressed in terms of the following partial fraction expansion:

1
1 + j ω

1
2 + j ω .

Y ( ω ) = −

Taking the inverse CTFT yields

− t − 2 t ) u ( t )

y ( t ) = (e − e

which is identical to the result obtained in Example 3.6 by direct convolution.

5.5.9 Parseval's energy theorem

Parseval's theorem relates the energy of a signal in the time domain to the
energy of its CTFT in the frequency domain. It shows that the CTFT is a
lossless transform as there is no loss of energy if a signal is transformed by the
CTFT.

For an energy signal x ( t ) , the following relationship holds true:

∞ ∞

= 1
2 π

x ( t ) 2 d t X ( ω ) 2 d ω. E x = (5.57)

−∞ −∞

Proof
To prove the Parseval's theorem, consider

∞ ∞

X ( ω ) 2 d ω =

∗

X ( ω ) X ( ω )d ω.

−∞ −∞

Substituting for the CTFT X ( ω ) using the definition in Eq. (5.10) yields

∗

∞ ∞ ∞ ∞

X ( ω ) 2 d ω =

− j ωα d α − j ωβ d β

x ( α )e x ( β )e d ω,

−∞ −∞ −∞ −∞

where we have used the dummy variables α and β to differentiate between the
two CTFT integrals. Taking the conjugate of the third integral and rearranging
the order of integration, we obtain
∞ ∞ ∞ ∞

X ( ω ) 2 d ω =

∗

e j ω ( β−α ) d ω x ( α ) x ( β ) d β d α.

−∞ −∞ −∞ −∞

Based on Eq. (5.15), we know that
∞

e j ω ( β−α ) d ω = 2 πδ ( β − α ) ,

−∞