Digital Signal Processing Reference
In-Depth Information
dictated by the application. For example, if high-speed real-time filtering is
required and
L
is relatively small, say less than 150, then the time domain method
may be attractive, whereas in offline applications, as can occur in some medical
image reconstruction processes, the frequency domain approach may be a better
option. The method is also determined by whether a dedicated FFT hardware
processor is available.
There is another more fundamental issue between these two approaches: in
the time domain approach a significant amount of information is often lost at the
beginning and end of the process unless some measure is taken to mitigate the
filtering effects. This happens because implementation in the time domain is
equivalent to carrying out a weighted average over a selected set of data points,
equal to the length of the filter, then shifting the whole operation one datum step
forward. The filter length spans a period in time, and the calculated average must
correspond statistically to the center of the set of points. The first filtered result,
therefore, occurs midway along the length of the filter, and thus there is no
information on the data points that precede the first filtered datum. This point will
be illustrated further below.
In the frequency domain approach, an FFT is performed on both the data set
and the filter taps. The two sets are multiplied utilizing the scalar product then
inverse Fourier transformed (IFFT) to give the desired result. This technique
produces a signal in the time domain with the same length as that of the FFT
thereby retaining the number of data points, or even interpolating to produce a
longer data set. The disadvantage of this technique is that the data is partitioned
into time blocks and may be of consequence in some applications. Furthermore,
the number of operations and the use of additional memory may also be important.
However, this also opens up other more interesting ways of interpreting the
filtering process such as the use of overlapping time blocks for fast filtering (e.g.,
[6])
. This is discussed in Section 2.5.3, “Frequency Domain Method.”
2.5.1 Direct Method - Postprocessing
In the postprocessing method, signal processing is usually carried out on a block
of archived data of size
N
. The filtering process involves multiplying the filter
coefficients with the corresponding archived signal elements stored in the data
array. The
m
th-filtered output
y
is given by
/
2
+
m
=
∑
+
=
L
1
1
(2.10)
y
h
x
m
=
1
2
,
N
(
L
+
1
L
k
k
+
m
1
K
+
m
k
1
2
where,
k
h
k
is the input data to
the filter,
N
is the number of samples, and
K
is the gain factor, equal to 512 for
filters in this topic. In this implementation the first filtered datum is midway along
the filter at position
L
/2+1. Moreover, (2.10) reduces to the well-known moving-
are the
L
+1 filter taps as determined previously,
x
