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4.3 Fir Filter Structures 123

Figure 4.5 Direct form FIR structure (Transversal filter)

The direct form FIR filter structure of order M = 59 is described by a single
difference equation:

The required numbers of multiplications and additions are N and N-l, respec-
tively. This structure is suitable for implementation on processors that are efficient
in computing sum-of-products. Most standard signal processors provide special fea-
tures to support sum-of-product computations, i.e., a multiplier-accumulator and
hardware implementation of loops and circular memory addressing. The signal lev-
els in this structure are inherently scaled except for the output which, for short FIR
filters, is normally scaled using the "safe scaling criterion" (see Chapter 5).

4.3.2 Transposed Direct Form
The transposition theorem [3, 18] discussed in Chapter 3 can be used to generate
new structures that have the same transfer function as the original filter struc-
ture. The number of arithmetic operations as well as the numerical properties are
generally different from the original structure.
The transposed direct form structure, shown in Figure 4.6, is derived from the
direct form structure. The amount of required hardware can be significantly
reduced for this type of structure where many multiplications are performed with
the same input value. This technique is discussed in Chapter 11. The filter struc-
ture is a graphic illustration of the following set of difference equations: