Digital Signal Processing Reference
In-Depth Information
which means that
H
e
j
ω
)
e
j
ω
)
(
is an approximation of
H
(
obtained by us-
ing only 2
N
+
1 Fourier coefficients. Since
H
(
e
j
ω
)
has a jump discontinuity
c
,
H
e
j
ω
)
in
c
.The
Gibbs phenomenon states that, when approximating a discontinuous func-
tion with a finite number of Fourier coefficients, the maximum error in an
interval around the jump discontinuity is actually independent of the num-
ber of terms in the approximation and is always equal to roughly 9% of the
jump. In other words, we have no control over the maximum error in the
ω
(
incurs the well-known Gibbs phenomenon around
ω
l
g
r
,
y
i
d
.
,
©
,
L
s
magnitude response. This is apparent in Figure 7.5 where
H
e
j
ω
)
is plot-
ted for increasing values of
N
; the maximum error does not decrease with
increasing
N
and, therefore, there are no means to meet a set of specifica-
tions which require less than 9% error in either stopband or passband.
(
error
≈
0.09
1
0
0
π/
2
π
Figure 7.5
Gibbs phenomenon in lowpass approximation; magnitude of the ap-
proximated lowpass filter for
N
=
4(lightgray),
N
=
10 (dark gray) and
N
=
50
(black).
The Rectangular Window.
Another way to look at the resulting approx-
imation is to express
h
[
n
]
as
h
[
n
]=
h
[
n
]
w
[
n
]
(7.2)
with
1
rect
n
N
N
0 rwie
−
N
≤
n
≤
w
[
n
]=
=
(7.3)
w
[
n
]
is called a rectangular
window
of length
(
2
N
+
1
)
taps, which in this
case is centered at
n
=
0.
