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306 Chapter 7 FIR FILTER DESIGN
Using (7.2), we obtain
δ 2
δ 2
A s =50=−20 log 10 = −20 log
1+δ 10 ⇒ δ2 =0.0032 □
1 1+0.0144
□ EXAMPLE 7.2 Given the passband tolerance δ 1 =0.01 and the stopband tolerance δ 2 =0.001,
determine the passband ripple R p and the stopband attenuation A s.
Solution
From (7.1) the passband ripple is
R p = −20 log 10
1 − δ 1
1+δ 1
=0.1737 dB
and from (7.2) the stopband attenuation is
A s = −20 log 10
δ 2
1+δ 1
=60dB □
Problem P7.1 develops MATLAB functions to convert one set of specifications
into another.
These specifications were given for a lowpass filter. Similar specifications
can also be given for other types of frequency-selective filters, such
as highpass or bandpass. However, the most important design parameters
are frequency-band tolerances (or ripples) and band-edge frequencies.
Whether the given band is a passband or a stopband is a relatively minor
issue. Therefore in describing design techniques, we will concentrate
on a lowpass filter. In the next chapter we will discuss how to transform
alowpass filter into other types of frequency-selective filters. Hence it
makes more sense to develop techniques for a lowpass filter so that we
can compare these techniques. However, we will also provide examples
of other types of filters. In light of this discussion our design goal is the
following.
Problem statement Design a lowpass filter (i.e., obtain its system
function H(z) orits difference equation) that has a passband [0,ω p ] with
tolerance δ 1 (or R p in dB) and a stopband [ω s ,π] with tolerance δ 2 (or
A s in dB).
In this chapter we turn our attention to the design and approximation
of FIR digital filters. These filters have several design and implementational
advantages:
• The phase response can be exactly linear.
• They are relatively easy to design since there are no stability problems.
• They are efficient to implement.
• The DFT can be used in their implementation.
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