Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 321

within the pass and stop bands of the ideal filters. Section 7.3 designs three
realizable implementations of an ideal lowpass filter. These implementations
are referred to as the Butterworth, Chebyshev, and elliptic filters. Section 7.4
transforms the frequency characteristics of the highpass, bandpass, and band-
stop filters in terms of the characteristics of the lowpass filters. These transfor-
mations are exploited to design the highpass, bandpass, and bandstop filters.
Finally, the chapter is concluded with a summary of important concepts in
Section 7.5.

7.1 Filter classifica tion

An ideal frequency-selective filter is a system that passes a prespecified range
of frequency components without any attenuation but completely rejects the
remaining frequency components. As discussed earlier, the range of input fre-
quencies that is left unaffected by the filter is referred to as the pass band of the
filter, while the range of input frequencies that are blocked from the output is
referred to as the stop band of the filter. In terms of the magnitude spectrum, the
absolute value of the transfer function H ( ω ) of the frequency filter, therefore,
toggles between the values of A and zero as a function of frequency ω . The
gain H ( ω ) is A , typically set to one, within the pass band, while H ( ω ) is
zero within the stop band. Depending upon the range of frequencies within the
pass and stop bands, an ideal frequency-selective filter is categorized in four
different categories. These categories are defined in the following discussion.

7.1.1 Lowpass filters

The transfer function H lp ( ω ) of an ideal lowpass filter is defined as follows:

ω≤ω c

A

H lp ( ω ) = (7.1)

0 ω >ω c ,

where ω c is referred to as the cut-off frequency of the filter. The pass band of
the lowpass filter is given by ω≤ω c , while the stop band of the lowpass filter
is given by ω c < ω < ∞ . The frequency characteristics of an ideal lowpass
filter are plotted in Fig. 7.1(a), where we observe that the magnitude H lp ( ω )
toggles between the values of A within the pass band and zero within the stop
band. The phase < H lp ( ω ) of an ideal lowpass filter is zero for all frequencies.

7.1.2 Highpass filters

The transfer function H hp ( ω ) of an ideal highpass filter is defined as follows:

0 ω≤ω c

H hp ( ω ) = (7.2)

ω >ω c ,

A