Signals and Systems - Electrical Engineering

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

z−^1

− + +

1

−0.4 0.2

x[n] v[n] y[n]

z−^1

++− ++

3

0.5 3

w[n]

FIGURE 11.29

Cascade realization ofH(z)=( 3 +3.6z−^1 +0.6z−^2 )/( 1 +0.1z−^1 −0.2z−^2 ).

It is also possible to expressH(z)as

H(z)=

[

1 +0.2z−^1

1 +0.5z−^1

]

︸ ︷︷ ︸

Hˆ 1 (z)

[

3 ( 1 +z−^1 )

1 −0.4z−^1

]

︸ ︷︷ ︸

Hˆ 2 (z)

which would give a different but equivalent realization ofH(z).

Since loading is not applicable, the product of the transfer functions always gives the overall trans-

fer function. As LTI systems these realizations can be cascaded in different orders with the same

result. n

nExample 11.17

Obtain a cascade realization of

H(z)=

1 +1.2z−^1 +0.2z−^2

1 −0.4z−^1 +z−^2 −0.4z−^3

Solution

The zeros ofH(z)arez=−1 andz=−0.2, while its poles arez=±jandz=0.4. We can thus

rewriteH(z)as the following two equivalent expressions:

H(z)=

[

1 +z−^1

1 +z−^2

] [

1 +0.2z−^1

1 −0.4z−^1

]

=

[

1 +0.2z−^1

1 +z−^2

] [

1 +z−^1

1 −0.4z−^1

]

where the complex-conjugate poles give the denominator of the first filter. Realizing each of these

components and cascading in any order would give different but equivalent representation ofH(z).

Figure 11.30 shows the realization of the top form ofH(z). n