# The Papoulis Filter (aka Optimum "L" Filter)

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## Introduction

Figure 1: Athanasios Papoulis. (Source)

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.

## Background

Most active filters are designed using one of four filter functions:

• Butterworth
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.
• Chebyshev
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.
• Bessel
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.

### History

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 2:Comparison of L-Filter, Butterworth, and Chebyshev Magnitude Characteristics.

Figure 3 shows how the L-filter characteristics change as the L-filter order increases.

Figure 3: Comparison of L-Filter Magnitude Characteristics for 3rd, 5th, and 7th Orders.

## Analysis

### Function Definition

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.

#### Legendre Polynomials

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.

 Eq. 1 ${{P}_{i}}\left( x \right)=1\quad i=0$ ${{P}_{i}}\left( x \right)=x\quad i=1$ ${{P}_{i}}\left( x \right)=\frac{\left( 2\cdot i-1 \right)\cdot x\cdot {{P}_{i-1}}\left( x \right)-\left( i-1 \right)\cdot {{P}_{i-2}}\left( x \right)}{i}\quad \text{otherwise}$

#### n Odd

Equation 2 generates L(ω²) for filters of odd order.

 Eq. 2 ${{L}_{n}}\left( \omega _{n}^{2} \right)={{\int\limits_{-1}^{2\cdot {{\omega }_{n}}-1}{\left( \sum\limits_{i=0}^{k}{{{a}_{i}}\cdot {{P}_{i}}\left( x \right)} \right)}}^{2}}dx$

where

• Pi are the Legendre polynomials of the first kind.
• ωn is the normalized frequency variable $\left( {{\omega }_{n}}\triangleq \frac{\omega }{{{\omega }_{3\text{dB}}}},\text{ where }{{\omega }_{3\text{dB}}}=2\cdot \pi \cdot {{f}_{3\text{dB}}} \right)$
• f3dB is the 3 dB bandwidth of the filter.
• $\omega \triangleq 2\cdot \pi \cdot f,\text{ where }f\text{ is frequency}\text{.}$
• ${{a}_{i}}\triangleq \frac{2\cdot i+1}{\sqrt{2}\cdot \left( k+1 \right)},\text{ where }k=\frac{n-1}{2}$

#### n Even

Equation 3 generates L(ω²) for filters of even order.

 Eq. 3 ${{L}_{n}}\left( \omega _{n}^{2} \right)={{\int\limits_{-1}^{2\cdot {{\omega }_{n}}-1}{\left( x+1 \right)\cdot \left( \sum\limits_{i=0}^{k}{{{a}_{i}}\cdot {{P}_{i}}\left( x \right)} \right)}}^{2}}dx$

where ${{a}_{i}}\triangleq \frac{2\cdot i+1}{\sqrt{\left( k+1 \right)\cdot (k+2)}}, i\text{ even, 0 otherwise }\left( k=\frac{n-2}{2} \right)$.

### 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).

Figure 4: Mathcad Routine for Determination of the L-Function.

### 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.

Figure 5: Mathcad Routine for Generation of the Filter Function.

### L(ω2) Function

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).

 n $L\left( {{\omega }^{2}} \right)$ 1 ${{\omega }^{2}}$ 2 ${{\omega }^{4}}$ 3 $3\cdot {{\omega }^{6}}-3\cdot {{\omega }^{4}}+{{\omega }^{2}}$ 4 $6\cdot {{\omega }^{8}}-8\cdot {{\omega }^{6}}+3\cdot {{\omega }^{4}}$ 5 $20\cdot {{\omega }^{10}}-40\cdot {{\omega }^{8}}+28\cdot {{\omega }^{6}}-8\cdot {{\omega }^{4}}+{{\omega }^{2}}$ 6 $50\cdot {{\omega }^{12}}-120\cdot {{\omega }^{10}}+105\cdot {{\omega }^{8}}-40\cdot {{\omega }^{6}}+6\cdot {{\omega }^{4}}$ 7 $175\cdot {{\omega }^{14}}-525\cdot {{\omega }^{12}}+615\cdot {{\omega }^{10}}-355\cdot {{\omega }^{8}}+105\cdot {{\omega }^{6}}-15\cdot {{\omega }^{4}}+{{\omega }^{2}}$ 8 $490\cdot {{\omega }^{16}}-1680\cdot {{\omega }^{14}}+2310\cdot {{\omega }^{12}}-1624\cdot {{\omega }^{10}}+615\cdot {{\omega }^{8}}-120\cdot {{\omega }^{6}}+10\cdot {{\omega }^{4}}$ 9 $1764\cdot {{\omega }^{18}}-7056\cdot {{\omega }^{16}}+11704\cdot {{\omega }^{14}}-10416\cdot {{\omega }^{12}}+5376\cdot {{\omega }^{10}}-1624\cdot {{\omega }^{8}}+276\cdot {{\omega }^{6}}-24\cdot {{\omega }^{2}}+\cdot {{\omega }^{2}}$ 10 $5292\cdot {{\omega }^{20}}-23520\cdot {{\omega }^{18}}+44100\cdot {{\omega }^{16}}-45360\cdot {{\omega }^{14}}+27860\cdot {{\omega }^{12}}-10416\cdot {{\omega }^{10}}+2310\cdot {{\omega }^{8}}-280\cdot {{\omega }^{6}}+15\cdot {{\omega }^{4}}$

### Filter Characteristic Function (s domain)

Table 2 shows the optimum L characteristic polynomials for n = 1 to 10.

 n $D\left( s \right)$ 1 $\text{s+1}\text{.0}$ 2 ${{s}^{2}}+1.4142 \cdot s+1$ 3 ${{s}^{3}}+1.3107\cdot {{s}^{2}}+1.359039\cdot s+0.57736522$ 4 ${{s}^{4}}+1.5628\cdot {{s}^{3}}+1.8878811\cdot {{s}^{2}}+1.2414681\cdot s+0.40821577$ 5 ${{s}^{5}}+1.5515\cdot {{s}^{4}}+2.2034703\cdot {{s}^{3}}+1.692534\cdot {{s}^{2}}+0.89817083\cdot s+0.22356064$ 6 ${{s}^{6}}+1.7262\cdot {{s}^{5}}+2.6898696\cdot {{s}^{4}}+2.4335933\cdot {{s}^{3}}+1.6332102\cdot {{s}^{2}}+0.67969552\cdot s+0.14143644$ 7 ${{s}^{7}}+1.72772\cdot {{s}^{6}}+2.9926117\cdot {{s}^{5}}+2.9244072\cdot {{s}^{4}}+2.3320509\cdot {{s}^{3}}+1.2307095\cdot {{s}^{2}}+0.43791348\cdot s+0.075589885$ 8 ${{s}^{8}}+1.86148\cdot {{s}^{7}}+3.4468776\cdot {{s}^{6}}+3.7236172\cdot {{s}^{5}}+3.3481719\cdot {{s}^{4}}+2.1192882\cdot {{s}^{3}}+0.99411199\cdot {{s}^{2}}+0.29972527\cdot s+0.045184527$ 9 ${{s}^{9}}+1.8665\cdot {{s}^{8}}+3.7418201\cdot {{s}^{7}}+4.249464\cdot {{s}^{6}}+4.2481258\cdot {{s}^{5}}+3.0122773\cdot {{s}^{4}}+1.7075194\cdot {{s}^{3}}+0.68048233\cdot {{s}^{2}}+0.1815741$ 10 ${{s}^{10}}+1.9744\cdot {{s}^{9}}+4.1713498\cdot {{s}^{8}}+5.0886545\cdot {{s}^{7}}+5.5135609\cdot {{s}^{6}}+4.3545176\cdot {{s}^{5}}+2.8312648\cdot {{s}^{4}}+1.3753686\cdot {{s}^{3}}+0.49627083\cdot {{s}^{2}}+0.11679607\cdot s+0.013743239$

### 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.

 n Factored D(s) 1 $\text{s+1}\text{.0}$ 2 ${{s}^{2}}+1.4142 \cdot s+1$ 3 $\left( \text{s+0}\text{.6203} \right)\cdot \left( {{s}^{2}}+0.6904\cdot s+0.9308 \right)$ 4 $\left( {{s}^{2}}+1.0994\cdot s+0.4308 \right)\cdot \left( {{s}^{2}}+0.4634\cdot s+0.9477 \right)$ 5 $\left( \text{s+0}\text{.4681} \right)\cdot \left( {{s}^{2}}+0.7762\cdot s+0.4971 \right)\cdot \left( {{s}^{2}}+0.3072\cdot s+0.9608 \right)$ 6 $\left( {{s}^{2}}+0.6180\cdot s+0.5830 \right)\cdot \left( {{s}^{2}}+0.8778\cdot s+0.2502 \right)\cdot \left( {{s}^{2}}+0.2304\cdot s+0.9696 \right)$ 7 $\left( \text{s+0}\text{.3821} \right)\cdot \left( {{s}^{2}}+0.6984\cdot s+0.3060 \right)\cdot \left( {{s}^{2}}+0.4748\cdot s+0.6621 \right)\cdot \left( {{s}^{2}}+0.1724\cdot s+0.9765 \right)$ 8 $\left( {{s}^{2}}+0.6006\cdot s+0.3829 \right)\cdot \left( {{s}^{2}}+0.3886\cdot s+0.7180 \right)\cdot \left( {{s}^{2}}+0.13788\cdot s+0.9809 \right)\cdot \left( {{s}^{2}}+0.7344\cdot s+0.1676 \right)$ 9 $\left( \text{s+0}\text{.3257} \right)\left( {{s}^{2}}+0.3146\cdot s+0.7666 \right)\cdot \left( {{s}^{2}}+0.6188\cdot s+0.2090 \right)\cdot \left( {{s}^{2}}+0.4972\cdot s+0.4635 \right)\cdot \left( {{s}^{2}}+0.1102\cdot s+0.9845 \right)$ 10 $\left( {{s}^{2}}+0.5548\cdot s+0.2702 \right)\left( {{s}^{2}}+0.0918\cdot s+0.9870 \right)\cdot \left( {{s}^{2}}+0.4284\cdot s+0.5282 \right)\cdot \left( {{s}^{2}}+0.2650\cdot s+0.8013 \right)\cdot \left( {{s}^{2}}+0.6344\cdot s+0.1218 \right)$

## Conclusion

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.

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### 13 Responses to The Papoulis Filter (aka Optimum "L" Filter)

1. Charles says:

Nice article!

Some of us would like to have seen the filter gain vs. frequency on a dB scale, but still a very helpful primer!

I would be great if I could get the MathCad source files for figures 3 and 4. Is there any chance you could provide them?

Thanks again.

2. eric.h.fletcher@gmail.com says:

I have been using the Legendre-Papoulis filters since ...1973. They are very well documented in
the book of Paul Bildstein, Filtres actifs, Editions Radio / 1972. Pages 44-47 & 204-209.
I recently designed an 8th order Legendre universally-programmable filter with multiplying
CMOS-DAC's. The adjustment range of the cut-off frequency extends over 12 Octaves :
40 Hz to 163.84 kHz. Dynamic adjustment point of the filter is performed via a so-called
"Retroactive-Calibration-&-Simulation loop". This circuit is also used to adjust gains in a
Data-Acquisition system, by generating logarithmic signals as well as trigonometric.

3. ... says:

"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."

Can you expand more on why you would use this filter instead of the Chebyshev?

"Chebyshev
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."

So you're only talking about the Chebyshev Type I, right? How does the Chebyshev Type II compare to the L filter?

• mathscinotes says:

I am talking about the Chebyshev Type 1 filter. I have had to design many filters. My decisions usually are driven by the following questions:

(1) Can I tolerate passband ripple?
If the answer is yes, then I almost always use an elliptic filter. It gives me the sharpest transition region for a given order filter. The fact that it does not go to zero in the stopband is not a big deal. I usually have a specification that states what my stopband attenuation has to be. I just have to meet that level of attenuation.
(2) If I cannot tolerate passband ripple, do I need a smooth passband?
If I need a smooth passband, then I use Butterworth.
(3) Do I need good pulse transmission characteristics?
If the answer is yes, then I use Bessel. This is very commonly done in optical systems.
(4) What if I need a sharper transition than the Butterworth filter gives, and I can tolerate a passband that is monotonically decreasing and with no ripple?
Then I use an L filter.

These are the filter design questions that I have had to address during my career. Other engineers would probably have different experience.

mathscinotes

• ... says:

Ok, but Chebyshev II fits criteria 4 also (monotonic passband with no ripple, sharp transition), so I'm just curious the benefits of one over the other. What need wasn't met that required the invention of the L filter?

• mathscinotes says:

The differences are not very significant, but they have made me reluctant to use the Chebyshev II, and I used the L filter in those cases. Let me give you an example. I work a fair number of problems where I need a relatively flat passband and I have a small number of sinusoids that are interfering at a much higher frequency. A typically example is an RF video signal with a gigabit Ethernet port running right near it. The presence of this gigabit Ethenet signal generates a major interference component at 155 MHz (1/8 its rate). This has been extremely annoying because 155 MHz is also a frequency used for emergency communications in the US (fire and police). I have taken the conservative approach that I am going to suppress that interference as much as possible because I do not want to take any chances. So I want to use a filter whose response goes to zero at high frequency rather than just below some fixed level. I even wrote a blog post about one of 155 MHz adventures. I want to make sure that I avoid any possible chance of interference.

4. ... says:

Polynomial 9 has ω^2 listed twice, should be ω^4

5. F4EOB says:

Hello,
thanks a lot, I was looking for the D(s) pôlynoms. Not easy to find until 10. I had them until only 4. Great !

Thank you.

6. Brian L says:

Thank you for the article. I have been researching what a Legendre filter is and advantages or not over the Butterworth and Bessel filters. The Butterworth was usdd by Marantz in developing hks 10b, 20, and 20b tuners and the 18 and 19 receivers. Not sure if it was used after he departed the company. I have also seem it referenced with other higher end well regarded tuners and receivers. The Besel is also mentioned in in relation to other makes. But only Sherwood mentions the Legendre and it is only apparently used in the rare SEL200 receiver and SEL300 tuner but, none other. These tuners are based on their later variant of the 1st generation ss tuner, the S3000 that jses the Micro Circuit. No version of the S3000 uses the filter including the version the SEL tuner and receiver are derived from.

As a lay person I was not sure if it was just a marketing ploy or not and wanted to know. And, if it was an actual component why it was better or not compared to the much worshipped Butterworth held up as the best FM if filter. While I did not understand any of the math; I am only a retired CPA and lawyer for beaven's sake, you laid out exactly what I wanted to find out very plainly.

The maximum frequency response outputed from the if section is 15khz as mandated by the FCC and 15khz from the demultiplexer. Because of this mandate the 15khz can not ne exceeded requiring the manufacturer to either start a rolloff before that frequency to meeting the requirement or create a brick wall having little or no rolloff.

It appears the Cauer filter may be popular as High Q filters by various names are frequently touted in the literature. It seems the design may form the basis of most tube based and early ss if circuits. I find it interesting when new if filters were introduced with the new fm stereo specification, these tuned coil systems were attached as deficient or no longer valid for fm radio.

The Butterworth filter deficiency is a bit surprising as Marantz spared no expense in developing his tuner, the 10b and 20/20b. He lost money on every 10b shipped, reportedly as mush as $200 per unit and was the reason he was ousted by the investors. Only reason may be the filter was developed in 1958, the year the sterel specification was released and the design of the original 10 started only a couple of years later by Saul who eventually brought in Sid Smith and Richard Sequerra to develop it in final form. They jist may nlt have been familiar with it or Saul may have already locked down the if. Sherwood stayed with the if used in their tube tuner, just converted to ss. It may have been more a production and financial consideration than anything else. Sherwood was a 2nd tier company in reputation but 1st tier in both rf and audio engineering. The owners were all the top engineers from Radio Craftsman one of the top radio coanies in its day. After the defection, RC quickly went downhill and disappeared. Today vintage collectors pay big money for their units. The Sherwood group started as a engineering company and only later entered manufacturing. It was a major contributor to the multiplex proponents during the stereo specification competition and supplied both broadcasting and receiving equipment during the period. When the FCC ruled, Sherwood had an advantage as it already had multiplex ready tuners. It had only produced a single short lived am-fm stereo tuner. It was nothing more than a reconfigured version of the existing am-fm tuner. They did not bank on the am-fm standard as did Scott and to a lesser degree Fisher. Every model tuner they issued had the mpx output or was mpx convertible. In designing and sourcing components, the engineers were critical of a constantly weak component every company had reliabilty issues with, the IF coils. In the States virtually every tuner manufacturer used Miller coils. The coils continued to break down regardless of what Miller did. Sherwood and later McIntosh decided to set up their own coil production and designed coils unique to their needs. They also would their own transformers which was another source of problems for many manufacturers as they had to use an off the shelf transformer unless they paid an outsource company the capital needed for that vender to tool up and set up production runs. When SS came out, retooling for new coils may at the time been a capital expenditure that was too costly. The engineers had sunk huge sums into the multiplex competition and then ss using germanium transistors hit the market. The Sherwood engineers tried to work with these new transistors but soon discovered these transistors were too electrically fragile and decided to withhold germainium based products from the market. Instead they continued offering only tube product and developed circuits around the more robust silicon transistor. When prices of transistors became economical, they released the ss component line but fully retained the tube line. The if circuit was a converted tube assembly so the coils did not change. They continued tube unit manufacturing long after others ended. They needed a line to steal the thunder and put them back in the game. They pulled out the stops to make their existing well proven ss systems as good as they could be. The result was the relatively short lived SEL tuner and receiver. The series also included a turntable and amp based on a new circuit that employed ICs. The tuner and receiver employed the Legrande filters. I am not sure if it was used on the later Sherwood Draco tuner. The SEL series cost the company a good deal of r&d budget. Though the series were excellent products; easily as good as anything on the market until that time and thereafter. Sadly, sales were just not there in any quantity with the majority being the tuner and receiver. The receiver was old world as it hit the market just as the power war started and$700 30 watt rms receivers were dead. The tuners fetch good money when one hits the market but the receiver goes for no more than the average price of a Sherwood imported receiver.

7. Matt says:

It looks like the Optimum L "Characteristic Polynomials" for n=2 is the same than the butterworth https://en.wikipedia.org/wiki/Butterworth_filter.
Is a second order L-filter the same than a second order butterworth ?

8. Dan says:

In Table 2, for order 9, the last coefficient is actually the coefficient of s, and you are missing one last coefficient 0.0238121. Thus the last two terms should be 0.1815741*s+0.0238121