DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Clarke, R.J., Transform Coding of
Images (Boston: Academic Press,
1985).
0.4
0.2
0
0
1
P( x)
=
2
3
1
2π
2
x
-
2
Figure 16.11 Gaussian function
is shown here in its onesided
form, with the scaling that
is usual in statistics, where the
function (augmented with mean
and variance terms) is known as
the normal function. Its integral
is the error function, erf(x). The
frequency response of cascaded
Gaussian filters is Gaussian.
Fourier transform pairs
Figure 16.9 opposite shows Fourier transform pairs for
several different functions. In the left column is a set of
waveforms; beside each waveform is its frequency spectrum.
Functions having short time durations transform
to functions with widely distributed frequency components.
Conversely, functions that are compact in their
frequency representation transform to temporal functions
with long duration. (See Figure 16.10 overleaf.)
A Gaussian function – the middle transform pair in
Figure 16.9, detailed in Figure 16.11 in the margin – is
the identify function for the Fourier transform: It has
the unique property of transforming to itself (within
a scale factor). The Gaussian function has moderate
spread both in the time domain and in the frequency
domain; it has infinite extent, but becomes negligibly
small more than a few units from the origin. The
Gaussian function lies at the balance point between the
distribution of power in the time domain and the distribution
of power in the frequency domain.
Analog filters
Analog filtering is necessary prior to digitization, to
bring a signal into the digital domain without aliases.
I have described filtering as integration using different
weighting functions; an antialiasing filter performs the
integration using analog circuitry.
An analog filter performs integration by storing
a magnetic field in an inductor (coil) using the electrical
property of inductance (L), and/or by storing an
electrical charge in a capacitor using the electrical property
of capacitance (C). In low-performance filters, resistance
(R) is used as well. An ordinary analog filter has
an impulse response that is infinite in temporal extent.
The design of analog filters is best left to specialists.
Digital filters
Once digitized, a signal can be filtered directly in the
digital domain. Design and implementation of such
filters – in hardware, firmware, or software – is the
domain of digital signal processing (DSP). Filters like the
150 DIGITAL VIDEO AND HDTV ALGORITHMS AND INTERFACES