Digital Signal Processing Reference
In-Depth Information
design
filter order
L
D
, the latter being used in the actual fabrication of the filter.
This will be explained later in Section 2.4, “Design Rules.”
2.1 IMPULSE RESPONSE FUNCTION
H
k
The continuous impulse response function
H
LP
(
k
) for the ideal low-pass filter in
the time (or spatial) domain is the inverse Fourier transform of the unit step
function
H
(
F
).
H
LP
(
k
) is well documented (see e.g. [2,3,9]) and is of the form
sin(
πF
k
)
c
H
(
k
)
=
<
k
<
(2.1)
LP
πk
This is discretized to
[
]
(
)
sin
πF
k
+
1
N
c
K
2
k
=
1
2
,
N
+
1
(2.2)
(
)
H
(
F
)
=
H
=
π
N
k
+
1
LP
c
k
2
KF
k
=
N
+
1
c
2
where
N
is a relatively large even number,
F
c
= f
c
/f
N
is the normalized frequency,
where
f
c
is the design low-pass cut-off frequency,
f
N
is the Nyquist frequency (=
f
s
/2),
f
s
being the sampling frequency, and
K
= 512 is a gain factor. Additionally,
note that
H
k
(or
H
LP
(
k
)) is the basis for development of the high-pass, band-pass,
and band-reject filters and will be used in the realization of such filters. Figure
2.1(a) shows part of the impulse response function of (2.2) for
N
= 1024, and
F
c
=
0.4, whereas Figure 2.1(b) shows its FFT using a radix two algorithm and floating
point arithmetic. The vertical scale in Figure 2.1(b) has been normalized to show
value 1 in the pass band. There is a 10% overshoot (from ringing) clearly visible
close to the transition point in Figure 2.1(a). Note that Figure 2.1(c) is the same as
Figure 2.1(b) but with its magnitude converted to a logarithmic scale (i.e., 20 log
|
H
ω
|).
2.2 WINDOW FUNCTION
W
k
Several window (or weighting) functions are available in signal processing
[4]
with the Kaiser, Blackman, Harris-Nutall, and Gaussian windows showing very
good attenuation properties in the rejection band [4,5,10]. Additionally, the
Blackman and Harris-Nutall windows are special cases of the short cosine series
window. However, in practice, there is usually a trade-off between main lobe
width and side lobe area. This usually translates into a trade-off between filter
length and edge transition sharpness using the window technique. In this regard,
the Kaiser window is optimal in terms of the trade-off between main lobe width
and side lobe area. The Gaussian window has the special property that it remains
