Digital Signal Processing Reference
In-Depth Information
because the
FT
of the impulse res
ponse
h(t)
represents
the transfer function
H(f)
.
ht
()
⎯⎯→
FT
H f
( )
For reasons of symmetry (see Cha
pter 5, Illustration 91
), the following must also be valid
IFT
Hf
()
⎯⎯→
ht
()
or overall
⎯⎯→
FT
IFT
h(t)
Hf
(
)
←⎯ ⎯
In order to observe the effects of the convolution in the frequency domain, the convolution
integral undergoes an
FT
∞∞
ª
º
∞
ª
∞
º
−
jt
ω
−
jt
ω
³³
³ ³
Yf
(
)
=
xht
( ) (
τ
−
τ
)
de t
τ
=
x
( )
τ
ht
(
−
τ
)
e t
d
τ
«
»
«
»
¬
¼
¬
¼
−∞
−∞
−∞
−∞
The
FT
of the signal
h(t
-
)
in the square brackets
.
This
FT
coincides with the
FT
of h(t),
excluding the phase shift. This phase angle equates to the rotation angle
e
-j
ωτ
.
This results
in
τ
∞
∞
³
−
j
ωτ
³
−
j
ωτ
Yf
()
=
xHf e d Hf
() ()
τ
τ
=
()
xe d
(
τ
)
τ
=
Hf Xf
()
⋅
(
)
−∞
−∞
The spectrum of the output signal thus can be calculated via multiplication of the trans-
mission function
H(f)
by the input signal spectrum
X(f)
.
Yf
()
=
Hf Xf
()
⋅
()
If the
IFT
is applied to the product
Y(f)
, the result again is the convolution. The convolu-
tion in the time domain equ
ates to a multiplication in the freq
uency domain.
FT
ht
()
∗⎯⎯→⋅
xt
()
H f
( )
X f
( )
For reasons of symmetry, t
he following must also be valid:
FT
ht
()
⋅
xt
()
⎯⎯→∗
H f
( )
X f
( )
Multiplication in the time domain equates to a convolution in the frequency domain which
can be best observed in the amplitude modulation AM in Chapter 8 (see for instance
Illustration 158 et seq.).
These are the two important convolution theorems. They are so
important in practical use, because you can calculate the convo-
lution quickly and conveniently by using the Fast FOURIER-
transformation (
FFT
) and the Inverse FFT (
IFFT
). This method
was used almost exclusively in Chapter 10 (“digital filters”)..
