Finite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 257

Magnitude response of
type I FIR filter

Magnitude response of
type II FIR filter

4 2.5

2

3

1.5

2

1

1

0.5

0 0

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Normalized frequency Normalized frequency

Magnitude response of
type IV FIR filter

Magnitude response of
type III FIR filter

3

3.5
3
2.5
2
1.5
1
0.5
0

2.5

2

1.5

1

0.5

0

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Normalized frequency Normalized frequency

Figure 5.2 Magnitude responses of the four types of linear phase FIR filters.

explained later. For example, type I filters have a nonzero magnitude at ω = 0and

also a nonzero value at the normalized frequency ω/π 1 (which corresponds to

the Nyquist frequency), whereas type II filters have nonzero magnitude at ω

=

= 0
but a zero value at the Nyquist frequency. So it is obvious that these filters are
not suitable for designing bandpass and highpass filters, whereas both of them
are suitable for lowpass filters. The type III filters have zero magnitude at ω = 0

and also at ω/π = 1, so they are suitable for designing bandpass filters but not

lowpass and bandstop filters. Type IV filters have zero magnitude at ω = 0and

a nonzero magnitude at ω/π = 1. They are not suitable for designing lowpass

and bandstop filters but are candidates for bandpass and highpass filters.
In Figure 5.3a, the phase response of a type I filter is plotted showing the
linear relationship. When the transfer function has a zero on the unit circle in
the z plane, its phase response displays a jump discontinuity of π radians at the
corresponding frequency, and the plot uses a jump discontinuity of 2 π whenever
the phase response exceeds ± π so that the total phase response remains within

the principal range of π . If there are no jump discontinuities of π radians,

that is, if there are no zeros on the unit circle, the phase response becomes a

±