Signals and Systems - Electrical Engineering

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

The optimal approach designs the filter directly, setting the rational approximation as a nonlinear

optimization. The added flexibility of this approach is diminished by the need to ensure stability of

the designed filter. Stability is guaranteed, on the other hand, in the transformation approach.

11.4.1 Transformation Design of IIR Discrete Filters

To take advantage of well-understood analog filter design, a common practice is to design dis-

crete filters by means of analog filters and mappings of thes-plane into thez-plane. Two mappings

used are:

n The sampling transformationz=esTs.

n The bilinear transformation,

s=K

1 −z−^1

1 +z−^1

Recall the transformationz=esTswas found when relating the Laplace transform of a sampled signal

with its Z-transform. Using this transformation, we convert the analog impulse responseha(t)of an

analog filter into the impulse responseh[n] of a discrete filter and obtain the corresponding transfer

function. The resulting design procedure is called theimpulse-invariant method. Advantages of this

method are:

n It preserves the stability of the analog filter.

n Given the linear relation between the analog and the discrete frequencies the specifications for

the discrete filter can be easily transformed into the specifications for the analog filter.

Its drawback is possible frequency aliasing. Sampling of the analog impulse response requires that

the analog filter be band limited, which might not be possible to satisfy in all cases. Due to this we

will concentrate on the approach based on the bilinear transformation.

Bilinear Transformation

The bilinear transformation results from the trapezoidal rule approximation of an integral. Suppose

thatx(t)is the input andy(t)is the output of an integrator with transfer function

H(s)=

Y(s)

X(s)

=

1

s

(11.16)

Sampling the input and the output of this filter using a sampling periodTs, we have that the integral

at timenTsis

y(nTs)=

∫nTs

(n− 1 )Ts

x(τ)dτ+y((n− 1 )Ts) (11.17)

wherey((n− 1 )Ts)is the integral at time(n− 1 )Ts. Consider then the approximation of the integral. If

Tsis very small, the integral between(n− 1 )TsandnTscan be approximated by the area of a trapezoid