Signals and Systems - Electrical Engineering

346 CHAPTER 5: Frequency Analysis: The Fourier Transform

after applying the sifting property ofδ(). The above shows that this Fourier transform is different

from the one for the cosine in the phase only. n

5.8.2 Differentiation and Integration

Ifx(t),−∞<t<∞, has a Fourier tranformX(), then

dNx(t)

dtN

⇔ (j)NX() (5.27)

∫t

−∞

x(σ)dσ ⇔

X()

j

+πX( 0 )δ() (5.28)

where

X( 0 )=

∫∞

−∞

x(t)dt

From the inverse Fourier transform given by

x(t)=

1

2 π

∫∞

−∞

X()ejtd

we then have that

dx(t)

dt

=

1

2 π

∫∞

−∞

X()

d ejt

dt

d

=

1

2 π

∫∞

−∞

[X()j]ejtd

indicating that

dx(t)

dt

⇔jX()

and similarly for higher derivatives.

The proof of the integration property can be done in two parts:

1. The convolution ofu(t)andx(t)gives the integral—that is

∫t

−∞

x(τ)dτ=

∫∞

−∞

x(τ)u(t−τ)dτ=[x∗u](t)