DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Figure 16.20 Comb filter
response resembles the
teeth of a comb. This filter
has unity response at zero
frequency: It passes DC.
A filter having weights
[ 1 ⁄ 2 , 0, 0, …, 0, - 1 ⁄ 2 ]
blocks DC.
For details of the relationship
between the Dirac delta, the
Kronecker delta, and sampling in
DSP, see page 122 of Rorabaugh’s
book, cited on page 170.
1.0
0.5
π
Frequency, ω, rad·s-1 0
0 2π
Impulse response
I have explained filtering as weighted integration along
the time axis. I coined the term temporal weighting
function to denote the weights. I consider my explanation
of filtering in terms of its operation in the temporal
domain to be more intuitive to a digital technologist
than a more conventional explanation that starts in the
frequency domain. But my term temporal weighting
function is nonstandard, and I must now introduce the
usual but nonintuitive term impulse response.
An analog impulse signal has infinitesimal duration, infinite
amplitude, and an integral of unity. (An analog
impulse is conceptually equivalent to the Dirac or
Kronecker deltas of mathematics.) A digital impulse
signal is a solitary sample having unity amplitude amid
a stream of zeros; The impulse response of a digital filter
is its response to an input that is identically zero except
for a solitary unity-valued sample.
Finite impulse response (FIR) filters
In each of the filters that I have described so far, only
a few coefficients are nonzero. When a digital impulse
is presented to such a filter, the result is simply the
weighting coefficients scanned out in turn. The
response to an impulse is limited in duration; the examples
that I have described have finite impulse response.
They are FIR filters. In these filters, the impulse
response is identical to the set of coefficients. The
digital filters that I described on page 150 implement
temporal weighting directly. The impulse responses of
these filters, scaled to unity, are [ 1 ⁄2, 1 ⁄2], [ 1 ⁄2, -1 ⁄2],
[ 1 ⁄2, 0, 1 ⁄2], and [ 1 ⁄2, 0, -1 ⁄2], respectively.
CHAPTER 16 FILTERING AND SAMPLING 157