Intel 2920 Design Handbook page 59

SUMMARY OF FILTER CHARACTERISTICS
The stage shown in Figure 5-7 behaves in a manner
analogous to a continuous analog stage which realizes a
complex conjugate pair of poles and zeroes. For exam-
ple, if the structure initially has values Y 1 and Y2 equal
to zero and is excited by a single impulse (i.e., one sam-
ple of unit value followed by zero-valued samples), the
output may take the form of samples of an exponen-
tially decaying sinusoid. The impulse response of the
stage may be expressed as:
when
h(O) D+A
h(iT)
e- a • T A cos ({3iT)+B sin ({3iT) for i>O
BI
2e- aT cos {3T
B2
-e- 2aT
AO
D+A
Al
-(2D+A)e- aT cos{3T+Be-aTsin{3T
A2
De- 2aT
a real part of pole
{3
imaginary part of pole
T sample period
ith sample
Figure
5-8.
Digital Filter Module
(First Order Section)
The diagram of Figure 5-8 corresponds to a stage realiz-
ing a single real pole. Its impulse response takes the
form:
5-8
h(O) D+A
h(iT)
Ae- aiT
when
BI
e- aT
AO
D+A
Al
-De-aT
From the equations, it can be seen the impulse responses
consist of (optional) initial delta functions, followed by
a series of samples which are equivalent to having
sampled an exponential decay, or an exponentially
decaying sinusoid.
Therefore, if we have a continuous filter FI that has an
impulse response which consists of a sum of decaying
exponentials or exponentially decaying sinuoids, we can
realize a digital filter F2 that has an impulse response
whose values at each sample time are identical to those
we would expect from Fl. This impulse response may be
achieved by building a network of the structures shown
in Figure 5-7 and 5-8, and summing their outputs.
This procedure defines a type of transform from the
continuous domain to the sampled domain, that is, the
sampled domain structure implements an impulse
response equivalent to having sampled the impulse
response of the corresponding continuous filter. This
transform is known as the "impulse invariant"
transform, and is one of several which may be used to
relate the sampled world and the continuous world.
Because of the nature of the sampling process and the
corresponding frequency folding about the sample rate,
it is not possible for a digital filter to duplicate exactly
the characteristics of a continuous analog filter. As the
frequencies of interest approach and exceed half the
sample rate, the frequency characteristics of the digital
filter differ radically from those of its continuous
counterpart. These differences may be shown by solving
for the frequency response of the second order digital
filter section as shown below:
F(jw)
= Ao
+
AI
(cos wT-j sin
wT)
+
A2
(cos 2wT-j sin
2wT)
I-B
I
(cos
wT-j
sin
wT)-B
2
(cos
2wT-j
sin
2wT)
Note that a periodic function of frequency results,
unlike the continuous case.
Sampled systems can be described as functions of a
complex variable z, where z=e
sT ,
T is the sample period,
and s is the Laplace complex frequency. In Figure 5-7,
each of the blocks labeled
z-I
corresponds to a unit delay