Quote of the Day
Two things prevent us from happiness; living in the past and observing others.
Analog engineers often have to design filters, which generally entails a lot of polynomial manipulation. Since I am currently designing some active filters, I thought it would be worthwhile documenting a filter function that I am using right now, but that is not widely known. This is the Papoulis or Optimum "L" filter. For this article, I will refer to this filter as the "L-filter." The "L" stands for Legendre, a mathematician whose like named polynomials are used in the derivation of the function.
This particular post made extensive use of Mathcad's symbolic processor. I left pretty impressed with what it could do.
Most active filters are designed using one of four filter functions:
This is a very friendly filter function. It has the flattest of all passbands and tends to have a well-behaved transient response because its poles are relatively low Q. Unfortunately, its roll-off rate is rather poor.
This filter is covered in every filter textbook, but I have actually never been able to use it. I have never been able to tolerate the ripple it creates in the passband. It does have very good roll-off characteristics.
I use this one a lot. The Bessel filter's linear phase characteristic gives it the best behaved transient response of all the filter functions, but its roll-off rate is very poor.
- Cauer (Elliptic)
I have used the Cauer filter quite a bit. It allows ripple in the stopband, which I usually can tolerate. It is a bit complicated to design and it has high-Q poles, which can create transient response problems.
These four filters have always been more popular than the L-filter. However, I certainly have had more use for the L-filter than for the Chebyshev. Since I was going through a design, I thought I would document it here.
The L-filter was developed by Athanasios Papoulis in a pair of articles published in 1958 and 1959:
- odd-ordered polynomial: "Optimum Filters with Monotonic Response," Proc. IRE, 46, No. 3, March 1958, pp. 606-609
- even-ordered polynomial:"On Monotonic Response Filters," Proc. IRE, 47, No. 2, Feb. 1959, 332-333 (correspondence section)
For a given filter order, the L-filter has the fastest roll-off rate of all filters with a monotonic magnitude response (i.e. the low-pass filter magnitude function always decreases with increasing frequency). I have used this filter a number of times in situations where I needed a relatively flat filter response and a sharp roll-off.
Comparison with Other Filter Functions
Figure 2 shows how the L-filter compares to the Butterworth and Chebyshev filters (all 3rd order in the figure).
Figure 3 shows how the L-filter characteristics change as the L-filter order increases.
The definition of the L-filter function is rather complicated. Because I am mathematically lazy, I will be using Mathcad's symbolic processor to perform all the manipulations.
For detailed information on the Legendre polynomials, the Wikipedia is about as good as it gets. I chose to generate the polynomials using the recurrence relation shown in Equation 1.
Equation 2 generates L(ω²) for filters of odd order.
- Pi are the Legendre polynomials of the first kind.
- ωn is the normalized frequency variable
- f3dB is the 3 dB bandwidth of the filter.
Equation 3 generates L(ω²) for filters of even order.
Mathcad Routine for Determination of the L(ω²) function
Figure 4 shows the Mathcad routine that I developed to generate the Ln(ω²)(Equation 3).
Mathcad Routine for Determination of Filter Function
Figure 5 shows the Mathcad routine that I used to generate the actual filter function and my graphs.
Table 1 shows the L(ω²) polynomials. Some folks have documented these polynomials instead of the characteristic polynomials. These polynomials are not directly useful for designs, but some web sites list them and they are useful for comparison purposes (i.e. verifying that my algorithm implementation is working correctly).
Filter Characteristic Function (s domain)
Table 2 shows the optimum L characteristic polynomials for n = 1 to 10.
Factored D(s) for Single and Double Pole Realizations
When I design active filters, I usually implement them as quadratic and simple pole sections. This means the factored form of the denominator polynomial is easier for me to design with. Table 3 shows the factored form of the polynomials from Table 2.
My hope is that this post will help folks who find the need for another filter function option. I have found this function useful in a number of cases and maybe you will too.