of Microprocessors

DIGITAL FILTERING
519
AMPLITUDE, (dB)
-1.0
1-8 1-6 1-4
1-7 1-5
1+6 Its
ItS 1+7
1
SAMPLE NUMBER
Fig. 14-28. Impulse response of ideal low-pass interpolating filter
Figure 14-29 shows in detail how the impulse responses to a string of
samples combine to produce a very smooth curve that exactly passes through
the sample points that are being low-pass filtered. In this respect, the filter
output satisfies the requirements for an interpolation function of the sample
points.
At this time, we can write the equation of the smooth curve connecting
the data points. If we ignore the filter's delay and call the point at the center
of the cluster point 0 (X = 0, y = Yo) and scale the X axis in units of the
sampling period, the impulse response, 10, due to point Xo,Yo is:
10=Yosin(7TX)l7rX. Continuing, the response due to the point Xl, Yl is:
It= Y lsin(7T(X -l»/7T(X- 1), where the (X-1) accounts for shifting the
impulse response right one unit so that it is centered about Xl. In fact the
response due to a general point Xi, Yt is: 1i=Yisin(7T(X-i))/7T(X-i). The
INDIVIDUAL
IMPULSE
RESPONSE
"'..
/' INTERPOLATED
/ CURVE
Fig. 14-29. Action of low-pass interpolating filter