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250 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
Finally, we note that the SP toolbox also provides functions similar to
the ones discussed in this section—the complementary functions, tf2latc
and latc2tf, compute all-pole lattice, all-zero lattice, and lattice-ladder
structure coefficients, and vice versa. Similarly, the function latcfilt
(the same name as the book function) implements the all-zero lattice
structure. The SP toolbox does not provide a function to implement the
lattice-ladder structure.
6.5 OVERVIEW OF FINITE-PRECISION NUMERICAL EFFECTS
Until now we have considered digital filter designs and implementations
in which both the filter coefficients and the filter operations such as additions
and multiplications were expressed using infinite-precision numbers.
When discrete-time systems are implemented in hardware or in software,
all parameters and arithmetic operations are implemented using finiteprecision
numbers and hence their effect is unavoidable.
Consider a typical digital filter implemented as a direct-form II structure,
which is shown in Figure 6.24a. When finite-precision representation
is used in its implementation, there are three possible considerations that
affect the overall quality of its output. We have to
1. quantize filter coefficients, {a k ,b k },toobtain their finite word-length
representations, {â k , ˆb k },
2. quantize the input sequence, x(n) toobtain ˆx(n), and
3. consider all internal arithmetic that must be converted to their next
best representations.
Thus, the output, y(n), is also a quantized value ŷ(n). This gives us a new
filter realization, Ĥ(z), which is shown in Figure 6.24b. We hope that this
x(n)
H(z)
y(n)
x(n) ˆ
H(z) ˆ
y(n) ˆ
x(n)
b 0 y(n) x(n) ˆ
bˆ 0 y(n) ˆ
z −1
z −1
a 1 b 1
aˆ
bˆ
1
1
z −1
z −1
a 2 b 2
aˆ
2 bˆ
2
z −1
z −1
a 3 b 3
ˆ
ˆ
a 3
b 3
(a)
(b)
FIGURE 6.24 Direct-form II digital filter implementation: (a) Infinite precision,
(b) Finite precision
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