Digital filters - Continuous and Discrete Time Signals and Systems - page 625

The pass-band gain is assumed to be unity. For bandpass and bandstop filters, there
are two cut-off frequencies, and Ω n 2 > Ω n 1

Filter Normalized cut-

Type off frequency Ideal filter impulse response

Lowpass h ilp [ k ] = Ω n sinc[ k Ω n ]

Ω n

Highpass h ilp [ k ] = δ [ k ] − Ω n sinc[ k Ω n ]

Ω n

Bandpass Ω n1 , Ω n2 h ibp [ k ] = Ω n2 sinc[ k Ω n2 ] − Ω n1 sinc[ k Ω n1 ]

Bandstop Ω n1 , Ω n2 h ibs [ k ] = δ [ k ] − Ω n2 sinc[ k Ω n2 ] + Ω n1 sinc[ k Ω n1 ]

The transfer function H ibs ( Ω ) of an ideal bandstop filter is related to the
transfer function H ibp ( Ω ) of an ideal bandpass filter by

H ibs ( Ω ) = 1 − H ibp ( Ω ) , (14.5b)

provided that the the cut-off frequencies Ω c1 and Ω c2 of both filters are the same.
Calculating the inverse DTFT of Eq. (14.5b), the impulse response h ibs [ k ]of
the ideal bandstop filter is obtained:

h ibs [ k ] = δ [ k ] − h ibp [ k ]

= Ω c2 , Ω c1

Ω c

= δ [ k ] = h ilp1 [ k ] − h ilp2 [ k ] (14.6)

Ω c = Ω c2 Ω c = Ω c1

k Ω c2
π

k Ω c1
π

= δ [ k ] − Ω c2
π

− Ω c1
π

sinc sinc .

Equation (14.6) shows that a bandstop filter can be formed by a parallel con-
figuration of two lowpass filters having cut-off frequencies Ω c2 and Ω c1 .
The impulse responses of the four types of frequency-selective ideal filters
discussed above are summarized in Table 14.1 in terms of the normalized cut-
off frequencies. It is observed that the impulse responses primarily include one
or two sinc functions and that all four types of ideal filters are non-causal.

14.2 FIR and IIR fil ters

A second classification of digital filters is made on the length of their impulse
response h [ k ]. The length (or width) of a digital filter is the number N of samples
k beyond which the impulse response h [ k ] is zero in both directions along the
k -axis. A filter of length N is also referred to as an N -tap filter.
A finite impulse response (FIR) filter is defined as a filter whose length N
is finite. On the other hand, if the length N of the filter is infinite, the filter is
called an infinite impulse response (IIR) filter. Below, we provide examples of
FIR and IIR filters with length N specified in the parentheses.