Digital filters - Continuous and Discrete Time Signals and Systems - page 640

…

x [ k ] z −1 z −1 z −1 z −1

h [1] h [2] h [3] h [ N − 2] h [ N − 1] h [0]

+ + + + +

y [ k ]

There are several flow graph representations of the FIR filter. In the following,
we discuss some of them.

14.6.1 Direct form

The flow graph for direct form is achieved by implementing Eq. (14.15) directly.
In direct form, the constant multipliers are the same as the coefficients of the
difference equation, Eq. (14.15). The direct form of the flow graph for a causal
FIR filter is shown in Fig. 14.13. Since the cost of implementation of a filter is
directly proportional to the number of fundamental elements used, we include
a count of these elements for each flow graph. The number of the fundamental
elements used in Fig. 14.13 is shown in the second row of Table 14.5.
The flow graph for the direct form resembles a tapped delay line used fre-
quently in communication systems for channel equalization. The filter shown
in Fig. 14.13 is therefore referred to as a tapped delay line filter or sometimes
as a transversal filter.

14.6.2 Cascaded form

The flow graph for the cascaded form is achieved by expressing Eq. (14.14) in
terms of a product of quadratic terms:

⌈ N + 2 ⌉

(1 + b 1 n z − 1 + b 2 n z − 2 ) .

H ( z ) = h [0] (14.16)

n = 1

Factorizing H ( z ) in terms of quadratic terms ensures coefficients b 1 n and b 2 n
to be real-valued provided that the impulse response h [ k ] is also real-valued.
Had linear factors been considered in Eq. (14.16) there would be no guarantee
for the coefficients of the linear factors to be real-valued, even with real-valued
h [ k ]. The upper limit ⌈ ( N − 1) / 2 ⌉ in the summation in Eq. (14.16) represents
a ceiling operation, which equals ( N − 1) / 2if N is odd. If N is even, the upper
limit equals N / 2 with b 2 n = 0 for the last product term.
The flow graph of the cascaded form is achieved by considering ⌈ ( N − 1) / 2 ⌉
substructures and cascading the substructures together in a series configuration.
The resulting flow graph is shown in Fig. 14.14. The number of fundamental
elements used in Fig. 14.14 is shown in the third row of Table 14.5.