Signals and Systems - Electrical Engineering

694 C H A P T E R 11: Introduction to the Design of Discrete Filters

If we replace the first equation into the second we obtain an expression containingw[n]

andw[n−2] andx[n] so that we cannot expressy[n] directly in terms of the input. Instead, consider

the Z-transforms of the above equation,

( 1 +0.1z−^1 )W(z)=X(z)

Y(z)=( 1 +1.5z−^1 )W(z) (11.70)

Thus, we obtain from the top equation in Equation (11.70):

W(z)=

X(z)

1 +0.1z−^1

which when replaced in the bottom equation in Equation (11.70) gives

Y(z)=

X(z)( 1 +1.5z−^1 )

1 +0.1z−^1

giving the transfer functionH(z). n

Remarks

n Direct form II is more advantageous than direct form I because of the consequence of using fewer delays.

We will use direct form II to realize first- and second-order modules in the cascade and parallel realizations.

n The cascade and the parallel realizations will connect first- and second-order systems to realize a given

transfer function. General direct form II realizations for a first and second-order filter with respective

transfer functions

H 1 (z)=

b 0 +b 1 z−^1

1 +a 1 z−^1

(11.71)

H 2 (z)=

b 0 +b 1 z−^1 +b 2 z−^2

1 +a 1 z−^1 +a 2 z−^2

(11.72)

are given in Figure 11.28. The coefficients of the above transfer functions are real. The realization of H 1 (z)

is obtained by getting rid of the lower part of the realization (i.e., getting rid of the constant multipliers

FIGURE 11.28

Direct form II realization of first- and

second-order filters (for the first-order filters

leta 2 =b 2 = 0 thus eliminating the constant

multipliers and the lower delay).

x[n] y[n]

w[n]

z−^1

−

+

+

z−^1

− b 0

b 1

a 2 b 2

a 1