For many lowpass filter designs the peak passband excursion
where the design parameter
As before, SBR is the minimum stopband attenuation compared to the nominal passband power transmission level, measured in decibels.
Example 1: Continuing from Example I "Statement of the Optimal Linear FIR Filter Design Problem"
Suppose as before that the lowpass
filter of interest is to have a peaktopeak passband ripple (PBR) of 0.5 dB and
a minimum stopband attenuation of 60 dB. Since
Note that the required filter order
Table 1 provides the values of the design parameter
Stopband Attenuation (in dB) 

Maximum Passband Ripple (in dB)  Minimum Passband Ripple (in dB) 
45  1.87  1.74  1.0 
50  2.05  1.74  0.55 
55  2.23  1.74  0.31 
60  2.42  1.74  0.174 
65  2.60  1.74  0.098 
70  2.78  1.74  0.055 
Derivation of the Formula
This section describes the theoretical underpinnings of Equation 1 and Equation 2. A clear understanding of this section is not required to use the ParksMcClellan software routines or to enjoy the remainder of this technical note.
As discussed in Section 2, the ParksMcClellan synthesis algorithm uses the
Remez exchange algorithm to optimally select the values of the
Suppose we desire to design a highorder, FIR, linear phase filter for
which the passband is as narrow as possible. Looking again at
Figure 1 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem"
with this in mind reveals that all of
the ripple behavior for such a filter will occur in the stopband. Such
a filter, or a very close approximation to it, can be synthesized using
another FIR filter design method, that of multiplying a
sampled
We desire that the oscillatory portion of the polynomial correspond to the
stopband region of the filter response and the
If
When the argument of the hyperbolic cosine is large, the function can be approximated as
With suitable manipulation we find that
Substituting this expression for the inverse hyperbolic cosine
yields a simple formula for
Rewriting this equation shows that
where
Rewriting equation 4 from the module titled "Statement of the Optimal Linear Phase FIR Filter Design Problem",
Substituting this into Equation 9 yields
which can be recognized as Equation 2.
Caveats
The derivation just presented assumes that the filter of interest is a
lowpass design, the filter order is high (
This figure shows the smallest value of
While Figure 2 shows that