Information Technology Reference
In-Depth Information
Simply truncating the impulse response causes an abrupt transition from input samples which matter and those
which do not. Truncating the filter superimposes a rectangular shape on the time-domain impulse response. In the
frequency domain the rectangular shape transforms to a sin
x
/
x
characteristic which is superimposed on the desired
frequency response as a ripple. One consequence of this is known as Gibb's phenomenon; a tendency for the
response to peak just before the cut-off frequency.
[
2
]
[
3
]
As a result, the length of the impulse which must be
considered will depend not only on the frequency response but also on the amount of ripple which can be tolerated.
If the relevant period of the impulse is measured in sample periods, the result will be the number of points or
multiplications needed in the filter.
Figure 3.10
compares the performance of filters with different numbers of points.
A high-quality digital audio FIR filter may need in excess of 100 points.
Figure 3.10:
The truncation of the impulse in an FIR filter caused by the use of a finite number of points (
N
) results
in ripple in the response. Shown here are three different numbers of points for the same impulse response. The
filter is an LPF which rolls off at 0.4 of the fundamental interval. (Courtesy
Philips Technical Review
)
Rather than simply truncate the impulse response in time, it is better to make a smooth transition from samples
which do not count to those that do. This can be done by multiplying the coefficients in the filter by a window
function which peaks in the centre of the impulse.
Figure 3.11
shows some different window functions and their
responses. The rectangular window is the case of truncation, and the response is shown at I. A linear reduction in
weight from the centre of the window to the edges characterizes the Bartlett window II, which trades ripple for an
increase in transition-region width. At III is shown the Hann window, which is essentially a raised cosine shape. Not
shown is the similar Hamming window, which offers a slightly different trade-off between ripple and the width of the
main lobe. The Blackman window introduces an extra cosine term into the Hamming window at half the period of
the main cosine period, reducing Gibb's phenomenon and ripple level, but increasing the width of the transition
region. The Kaiser window is a family of windows based on the Bessel function, allowing various tradeoffs between
ripple ratio and main lobe width. Two of these are shown in IV and V.
Figure 3.11:
The effect of window functions. At top, various window functions are shown in continuous form. Once
