Image Processing Reference
In-Depth Information
Figure 2.a shows a second-order Infinite Impulse Response (IIR) filter realized in direct
form, whose transfer function is
1
1
Hz
()
.
(8)
1
2
1
2
1
az
az
1
z
0.75
z
1
2
It is initially assumed that the filter is implemented using infinite precision, which implies
that the quantization effects are negligible and that all signals are generated as linear
combinations of the input and the state variables. This assumption allows: (i) to perform a
separate analysis of the mean and the width of the intervals; and (ii) to generalize the results
obtained in the simulation of a normalized interval to larger or smaller ones.
Figure 2.b shows the oversizing that occurs in the IA simulation. The input is set to the
normalized interval [-1, 1], and the state variables are initially set to zero. Here, the
representations are based on oriented intervals to keep track of the position of the samples
in each interval, and to detect the overestimations. The initial values and the evolution of the
intervals are:
t
= [1, -1]
t
= [1, -1]
sv
= [1, -1]
a1
sum
1
x
= [-1, 1]
y
= [-1, 1]
(9)
sv
= [0.75, -0.75]
2
and in the next sampled time the values are:
sv
= [1, -1]
y
= [1, -1]
t
= [-1, 1]
1
a1
t
= [-1.75, 1.75]
(10)
sum
sv
= [0.75, -0.75]
2
instead of t sum = [-0.25, 0.25], which is the correct value. Figure 2.b also shows that this
oversizing occurs because signal t sum depends on the input signal through two different
paths.
Since AA includes a separate signed identifier per uncertainty source, it avoids such
overestimations and provides the smallest intervals. In this case, the initial values and the
evolution of the affine forms are:
t
= 2
t
= 2
sv
= 2
a1
sum
1
x
= 2
y
= 2
(11)
sv
= -1.5
2
and in the next sampled time
sv
= 2
y
= 2
t
= 2
1
a1
t
= 0.5
(12)
sum
sv
= -1.5
2
which corresponds to the most accurate interval [-0.25, 0.25].
This simple example confirms the selection of AA instead of IA, particularly in structures with
feedback loops. Although the cancellation effect is not necessarily present in all the structures,
it commonly appears in most DSP realizations. For this reason, it is highly recommended to
use this arithmetic when performing interval-based analysis of DSP systems.
When there are multiple simultaneous uncertainty sources, it is necessary to use an oriented
identifier for each source, in addition to the average value of the signals, which are the
elements offered by AA to perform the computations. Moreover, the objective of AA is to
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