DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

0 1 2 3
1.0
0.8
0.6
0.4
0.2
0
Time, multiple of ts -0.2 0 2π 4π 6π 8π 10π
Frequency, ω, rad·s
12π
-1
1.0
0.8
0.6
0.4
0.2
0
-1 4 5 6
Figure 16.7 Frequency response of a boxcar filter. The top graph shows a boxcar weighting function,
symmetrical around t = 0. Its frequency spectrum is a sinc function, shown underneath. The
solid line shows that at certain frequencies, the filter causes phase inversion. Filter response is
usually plotted as magnitude; phase inversion in the stopband is reflected as the absolute (magnitude)
values shown in dashed lines.
⎧1,
ω = 0
⎪
sinc ω = ⎨sin
ω
⎪ , ω ≠ 0
⎩ ω
Eq 16.1 sinc function is
pronounced sink. Formally, its
argument is in radians per
second (rad·s -1 ); here I use the
conventional symbol ω for that
quantity. The term (sin x)/x
(pronounced sine ecks over ecks)
is often used synonymously
with sinc, without mention of
the units of the argument. If
applied to frequency in hertz,
the function could be written
(sin 2πf)/2πf.
sinc is unrelated to sync
(synchronization).
Magnitude frequency response of a boxcar
The top graph of Figure 16.7 above shows the
weighting function of Point sampling on page 144, as
a function of time (in sample intervals). The Fourier
transform of the boxcar function – that is, the magnitude
frequency response of a boxcar weighting
function – takes the shape of (sin x)/x. The response is
graphed at the bottom of Figure 16.7, with the
frequency axis in units of ω =2πfS. Equation 16.1 in the
margin defines the function. This function is so important
that it has been given the special symbol sinc,
introduced by Phillip M. Woodward in 1953 as a
contraction of sinus cardinalis.
A presampling filter should have fairly uniform response
below half the sample rate, to provide good sharpness,
and needs to severely attenuate frequencies at and
above half the sample rate, to achieve low aliasing. The
bottom graph of Figure 16.7 shows that this requirement
is not met by a boxcar weighting function. The
graph of sinc predicts frequencies where aliasing can be
CHAPTER 16 FILTERING AND SAMPLING 147