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Basic Elements 213
characteristics due to coefficient quantization. In Chapter 10, we will consider
the round-off noise effects in the digital filter implementations.
We begin with a description of basic building blocks that are used
to describe filter structures. In the subsequent sections, we briefly describe
IIR, FIR, and lattice filter structures, respectively, and provide
MATLAB functions to implement these structures. This is followed by
a brief treatment of the representation of numbers and the resulting error
characteristics, which is then used to analyze coefficient quantization
effects.
6.1 BASIC ELEMENTS
Since our filters are LTI systems, we need the following three elements to
describe digital filter structures. These elements are shown in Figure 6.1.
1. Adder: This element has two inputs and one output and is shown in
Figure 6.1a. Note that the addition of three or more signals is implemented
by successive two-input adders.
2. Multiplier (gain): This is a single-input, single-output element and is
shown in Figure 6.1b. Note that the multiplication by 1 is understood
and hence not explicitly shown.
3. Delay element (shifter or memory): This element delays the signal
passing through it by one sample, as shown in Figure 6.1c. It is
implemented by using a shift register.
Using these basic elements, we can now describe various structures of
both IIR and FIR filters. MATLAB is a convenient tool in the development
of these structures that require operations on polynomials.
x 1 (n)
x 1 (n) + x 2 (n)
x 2 (n)
(a) Adder
x(n)
a
(b) Multiplier
ax(n)
z −1
x(n) x(n − 1)
(c) Delay element
FIGURE 6.1
Three basic elements
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