Digital filters - Continuous and Discrete Time Signals and Systems - page 622

697 Hz 1 2 ABC

3 DEF

h [ k ] and is covered in Section 14.2. Yet another classification of digital filters
is made on the basis of the linearity of the phase < H ( Ω ), which is presented
in Section 14.3. The impulse response of the ideal frequency-selective filters,
considered in Section 14.1, is infinite, which makes them physically unreal-
izable. Section 14.4 describes realizable implementations of the ideal filters,
which are causal. Sections 14.5-14.7 cover physical implementations of digital
filters using special-purpose hardware confined to delays, adders, and scalar
multipliers. During the actual implementation of digital filters in software or
hardware, the filter coefficients can only be represented with finite precision.
The impact of finite-precision arithmetic on the performance of digital filters
is covered in Section 14.8. Important M ATLAB library functions used in the
analysis of digital filters are presented in Section 14.9. Finally, Section 14.9
concludes the chapter with summary of the important concepts.

4 GHI 5 JKL 6 MNO

770 Hz

7 PQRS 8 TUVW 9 WXYZ

852 Hz

7 PQRS 8 TUVW 9 WXYZ

941 Hz

Fig. 14.1. Dual-tone
multifrequency (DTMF) signals
used in digital telephone
systems.

14.1 Filter classification

A digital filter is often classified on the basis of the magnitude and phase spectra
derived from its transfer function. In this section, we consider a classification
based on the shape of the magnitude spectrum of the filter. In the case of ideal
filters, the shape of the magnitude spectrum is rectangular with a sharp transition
between the range of frequencies passed and the range of frequencies blocked
by the filter. The range of frequencies passed by the filter is referred to as the
pass band of the filter, while the range of blocked frequencies is referred to as
the stop band.

14.1.1 Ideal lowpass filter

The transfer function H ilp ( Ω ) of an ideal lowpass filter, with a cut-off frequency
of Ω c ,isgivenby

1

Ω ≤ Ω c
0 Ω c < Ω ≤π,

H ilp ( Ω ) = (14.1a)

which has a pass band of Ω ≤ Ω c and a stop band of Ω c ≤ Ω ≤π . Since the
frequency Ω = π is the highest frequency present in the DTFT, the lowpass filter
removes the higher frequencies in the range of Ω c < Ω ≤π . The magnitude
response of the lowpass filter is shown in Fig. 14.2(a). It is observed that the
lowpass filter has a unity gain in the pass band and zero gain in the stop band.
Sometimes, a lowpass filter has a pass band gain different from unity. If the
gain is greater than one, the pass band signal is amplified, if the gain is less than
one, the pass band signal is attenuated.