Digital Signal Processing Reference
In-Depth Information
Suppose that we only possess a sampled version of
f
(
t
)
,thatis,weonly
know the numeric value of
f
at times multiples of a sampling interval
T
s
and that we want to obtain an approximation of the Fourier transform
above.
Assume we do not know about the DTFT; an intuitive (and standard) place
to start is to write out the Fourier integral as a Riemann sum:
(
t
)
l
g
r
,
y
i
d
.
,
©
,
L
s
∞
Ω)
≈ F
e
−jT
s
n
Ω
F
(
j
(
j
Ω)=
T
s
f
(
nT
s
)
(9.40)
n
=
−∞
indeed, this expression only uses the known sampled values of
f
.Inor-
der to understand whether (9.40) is a good approximation consider the pe-
riodization of
F
(
t
)
(
j
Ω)
:
F
j
T
s
n
∞
2
F
(
j
Ω)=
Ω+
(9.41)
n
=
−∞
in which
F
(
j
Ω)
is repeated with overlap with period 2
π/
T
s
. We will show
that:
F
F
(
j
Ω)=
(
j
Ω)
that is, the Riemann approximation is equivalent to a periodization of the
original Fourier transform; in mathematics this is known as a particular
form of the
Poisson sum formula
.
To see this, consider the periodic nature of
F
(
j
Ω)
and remember that any
periodic function
f
(
x
)
of period
L
admits a
Fourier series
expansion:
∞
A
n
e
j
2
L
nt
f
(
t
)=
(9.42)
n
=
−∞
where
L
/
2
1
L
e
−j
2
L
nt
dt
=
(
)
A
n
f
t
(9.43)
/
−
L
2
Here's the trick: we regard
F
as an anonymous periodic complex func-
tion and we compute its Fourier series expansion coefficients. If we replace
L
by 2
(
j
Ω)
π/
T
s
in (9.43) we can write
π/
T
s
T
s
2
F
e
−jnT
s
Ω
d
A
n
=
(
j
Ω)
Ω
π
−
π/
T
s
π/
T
s
F
j
2
T
s
k
e
−jnT
s
Ω
d
+
∞
T
s
2
=
Ω+
Ω
π
−
π/
T
s
k
=
−∞
