Signals and Systems - Electrical Engineering

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

Remarks

n In general, given a direct form I realization one can easily obtain the difference equation and consequently

the transfer function of the filter from it.

n Theminimal realizationof qth-order discrete filter must use q delays. The direct form I is only capable

of providing these minimal realizations forall-pole filters(i.e., when the numerator in Equation (11.62)

is a constant), otherwise we need to use the direct form II to to obtain minimal realizations. If the transfer

function has only poles, then it is possible to obtain a minimal realization with direct form I. Indeed, if

H(z)=

Y(z)

X(z)

=

b 0

1 +

∑N− 1

k= 1 akz

−k

(11.64)

where Y(z)and X(z)are the Z-transforms of the output y[n]and the input x[n], the input–output

relationship is given by the difference equation

y[n]=−

N∑− 1

k= 1

aky[n−k]+b 0 x[n] (11.65)

which only requires N− 1 delays for the output, and none for the input. This is a minimal realization of

H(z)as only N− 1 delays are needed.

Direct Form II

If the polynomials

B(z)=

M∑− 1

k= 0

bkz−kandA(z)=

N∑− 1

k= 0

akz−k M≤N

represent the numerator and denominator of the transfer functionH(z)of the filter we wish to realize,

we have

H(z)=

Y(z)

X(z)

=

B(z)

A(z)

whereX(z)andY(z)correspond to the Z-transforms of the input and of the output of the filter. We

then have that

Y(z)=H(z)X(z)=B(z)

[

X(z)

A(z)

]

(11.66)

Defining an outputw[n] withW(z)=X(z)/A(z), corresponding to the second term in the last

equation, we obtain an all-pole filter with transfer function

W(z)

X(z)

=

1

A(z)

(11.67)

The outputy[n] is then obtained as the inverse of

Y(z)=B(z)W(z) (11.68)