DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Averaging neighboring samples
is the simplest form of moving
average (MA) filter.
0 1 t 0 2π
0 1 t 0 2π
0 1 t 0 2π
Figure 16.10 Waveforms of three temporal extents are shown on the left; the corresponding
transforms are shown on the right. Spectral width is inversely proportional to temporal extent, not
only for the Gaussians shown here, but for all waveforms.
ones that I have been describing are implemented digitally
by computing weighted sums of samples.
Perhaps the simplest digital filter is one that just sums
adjacent samples; the weights in this case are [1, 1].
Figure 16.12 on the facing page shows the frequency
response of such a [1, 1] filter. This filter offers minimal
attenuation to very low frequencies; as signal frequency
approaches half the sampling rate, the response follows
a cosine curve to zero. This is a very simple, very cheap
lowpass filter (LPF).
I have drawn in gray the filter’s response from 0.5f S to
the sampling frequency. In a digital filter, frequencies in
this region are indistinguishable from frequencies
between 0.5f S and 0. The gain of this filter at zero
frequency (DC) is 2, the sum of its coefficients.
Normally, the coefficients of such a filter are normalized
to sum to unity, so that the overall DC gain of the
filter is one. In this case the normalized coefficients
would be [ 1 ⁄2, - 1 ⁄2]. However, it is inconvenient to call
this a [ 1 ⁄2, - 1 ⁄2]-filter; colloquially, this is a [1, 1]-filter.
152 DIGITAL VIDEO AND HDTV ALGORITHMS AND INTERFACES
ω
ω
ω