Digital filters - Continuous and Discrete Time Signals and Systems - page 658

The M ATLAB code to determine the poles and zeros of H (z) is given by

>>B=[01 − 2 1]; % The numerator of H(z)
>>A=[1 − 0.1 − 0.07 − 0.065]; % The denominator of H(z)
>> [Z, P, K] = tf2zp(B, A);

% Calculate poles and
% zeros

The locations of the poles and zeros are given by

Z=[011]
P = [0.5000 − 0.2000+0.3000j − 0.2000 − 0.3000j] andK=1.

The transfer function H ( z ) can therefore be expressed as follows:

− 1 )
(1 − 0 . 5 z − 1 )(1 − ( − 0 . 2 + j0 . 3) z − 1 )(1 − ( − 0 . 2 − j0 . 3) z − 1 )

− 1 )(1 − 1 z − 1 )(1 − 1 z

(1 − 0 z

H ( z ) = 1

(1 − z

− 1 ) 2

− 1 ) .
Combining the complex roots in the denominator, the cascaded configuration
is given by

=

− 1 )(1 − ( − 0 . 2 + j0 . 3) z − 1 )(1 − ( − 0 . 2 − j0 . 3) z

(1 − 0 . 5 z

− 1
1 + 0 . 4 z − 1 + 0 . 13 z − 2 .
The cascaded configuration is then implemented using the series form as shown
in Fig. 14.23.

− 1

1 − z 1 − z

H ( z ) =

1 − 0 . 5 z − 1

14.10 Summary

Chapter 14 defined digital filters as systems used to transform the frequency
characteristics of the DT sequences, applied at the input of the filter, in a prede-
fined manner. Based on the magnitude spectrum H ( Ω ) , Section 14.1 classifies
filters in four different categories. A lowpass filter removes the higher-frequency
components above a cut-off frequency Ω c from an input sequence, while retain-
ing the lower-frequency components Ω ≤ Ω c . A highpass filter is the converse
of the lowpass filter and removes the lower-frequency components below a cut-
off frequency Ω c from an input sequence, while retaining the higher-frequency
components Ω

Ω c . A bandpass filter retains a selected range of frequency
components between the lower cut-off frequency Ω c1 and the upper cut-off
frequency Ω c2 of the filter. A bandstop filter is the converse of the bandpass
filter, which rejects the frequency components between the lower cut-off fre-
quency Ω c1 and the upper cut-off frequency Ω c2 of the filter. All other frequency
components are retained at the output of the bandstop filter.
Section 14.2 introduces a second classification of digital filters based on the
length of the impulse response h [ k ] of the digital filter. Finite impulse response

≥