Digital Signal Processing Reference
In-Depth Information
Chapter 2
Filter Design and Implementation
The aim of this chapter is to show how the filters presented in this topic were
designed, thereby providing a basis for understanding the limitations of the
design. The filters described here were designed using the well-known
windowing
technique. In this technique, the desired frequency response
H
(
F
), is first created
in the frequency domain and then its inverse Fourier transform (IFT) is found.
This yields the impulse response function
H
(
k
), which is then discretized,
truncated, and
windowed
to form
h
k
in the spatial or time domain
k
. Note that in
the following discussion both
H
(
k
) and
H
(
F
) should be treated as continuous
functions, whereas
H
k
and
H
F
are their respective discretized forms. The
windowing and truncation of
H
k
yield the finite impulse response (FIR) digital
filter,
h
k
.
In principle, the domain of the impulse response function
H
(
F
) is infinite, but
to be able to use it in a digital context it is truncated to some reasonable length. If
an FFT is performed on the truncated form
H
k
,
ringing
is observed on the response
H
F
in the frequency domain. Ringing is characterized by a damped oscillatory
behavior at the transition edges of the filter; this is a consequence of the well-
known Gibbs phenomenon
[1].
Moreover, truncation of the impulse response
function distorts the ideal filter characteristics in the frequency domain, which
were set initially at the beginning of the design process. In other words, the
frequency response of the discretized filter
H
F
bears only some resemblance to its
template
H
(
F
). In fact, truncation induces an inadvertent convolution of
H
(
F
) with
domain, the said function is usually convolved with special window functions,
W
k
,
which taper the ends smoothly to zero in the time domain. The latter functions
tend to preserve the central portion, or main lobe, of the symmetric impulse
response while carefully managing the values on the sides of the said response
smoothly to zero.
For discussion purposes, it is useful to distinguish between the filter order and
the filter length. The filter order
L
, is the highest exponent of the
z
-transform of
the FIR filter, and its length is always one greater than its order, that is, the filter
length is (
L
+1). Occasionally, the filter order
L
is made slightly greater than the
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