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Final Answers
© 2000-2015   Gérard P. Michon, Ph.D.

Linear Filters
Network Synthesis and Design

A filter is a device which allows signals at certain
frequencies while impeding the passage of others
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Related articles on this site:

 Electronic Design Net

Related Links (Outside this Site)

LC Filter Design  (Calculator)  by the late  Tony Fisher (1956-2000)
Filter Design in 30 seconds  (327 kB, pdf)  at  Texas Instrument.
Analog Filter Design Demystified  (Maxim AN 1795).
Filter Design Guide  by  Kevan Hashemi  (Brandeis University).
Circuit Sage: Filter Design and Analysis.
The Engineers' Club   |   Analog Filters  at  Corollary Theorems, Ltd.
Filter & Crossover Types for Loudspeakers  by  Gene DellaSala  (Audioholics).
Quality Factor, Peak Frequency and Bandwidth  by   Ruye Wang.
Butterworth filters  by   Richard Baraniuk  (Connexions, 2005-07-18)
A Bessel Filter Crossover  by   Ray Miller  (Rane Corporation).
Microwave Filters   |   The Legendre Filter  at  NuHertz Technologies, LLC.
Finite Impulse Response (FIR) Filters:  FAQ  by  Grant R. Griffin  (dspGuru).
Low-Pass Filters Using Ultraspherical Polynomials  by  D.E. & J.R. Johnson.
Network Synthesis using Genetic Algorithms  by  James B. Grimbleby  (1995).
Wikipedia:   Butterworth filter  |  Chebyshev filters  |  Cauer filters  |  Bessel filter  |  Minimum Phase  |  Transfer function  |  Bode plot  |  Nyquist plot  |  Q-factor

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Analog Filter Design
Linear Filters and Active Feedback

(2007-06-08)   Complex Pulsatance  and  Complex Amplitudes
Confusingly, complex pulsatance is often called complex frequency.

Technically, a (real) pulsatance is a rate of phase change per unit of time.  It's expressed in angular units (radians or degrees) per unit of time (second).  Pulsatance is also commonly called  angular frequency.  On the other hand,  frequency  is what pulsatance becomes when phases are expressed in  cycles  (one cycle is a phase change of 360° or  2p radians).

The modern convention is to express a pulsatance  (preferably denoted by the symbol w) in radians per second (rad/s) and the corresponding frequency (preferably denoted by the symbol n)  in hertz (Hz, or "cycle per second").

w   =   2p n

In electrical engineering, the letter "i" is often used to denote a current intensity.  It's thus unavailable as a name for the unit vector along the imaginary axis of the complex plane.  So, the letter "j" is used instead for that purpose.  The square of that imaginary number is -1.  It's to the number 1 (unity) what a step sideways  to the left is to a step forward.  Refrain from calling this  "the"  square root of -1.

j 2   =   -1

The value at time  t  of a pure sinewave signal of frequency  n  and/or of  pulsatance  w = 2pn  can be conveniently represented as the real part of the following expression, where  s  is equal to the  imaginary pulsatance  jw.

|A|  exp ( jq + s t )   =   |A|  cos ( q + wt )  +  j (...)

In this,  A  is a positive real number and  q  is called the  phase  of the signal.  The number  A = |A| exp(jq)  is called the  complex amplitude  of the signal and the value of the signal at time  t  is therefore simply the real part of:

A  exp ( s t )

Therefore, the  complex amplitude  of the signal's  derivative  is  A s.  Likewise, the complex amplitude of the second derivative is  A s 2, etc.

A slight generalization can be made by considering that the above remains true even if  s  is a complex number with a nonzero real part  s.

s   =   s + jw   =   -p + jw

s  is called the  damping constant.  A negative value of  s  (a positive p) does translate into a signal which is a  damped sinewave, like  e -pt cos(wt).

This observation may be construed as the basis for Oliver Heaviside's  operational calculus, which characterizes circuits by their reaction to nonoscillatory decaying signals  (p>0, w=0).  This approach rests on the co-called Laplace transform (and its inverse).  From a mathematical standpoint, such an analysis (which may be quite convenient) is just as sound as the more "physical" one based on sinewave signals, involving the Fourier transform (and its own inverse).  Either approach yields results applicable to  any  signal whatsoever.

(2007-06-10)   Resistance, Reactance, Complex Impedance
The complex number characterizing a linear dipole at a fixed frequency.

dipole is defined as a current-conserving two-terminal device  (the total electric charge inside the device does not change).  Each terminal may also be referred to as an  electrode  or a  pin.  One of them is (somewhat arbitrarily) called "input", the other is the "output" terminal.  Whatever current enters the input goes out the output; this quantity is the current (i)  through  the dipole.  The difference between the tension (voltage) of the input electrode and the output tension is called the voltage (u)  across  the dipole.

A dipole for which  u  is proportional to  i,  is called a  linear  dipole.  The coefficient of proportionality between  u  and  i  is the  impedance  (Z).

u   =   Z i

In this,  Z  is a  complex number  which may depend on the operating complex pulsatance  (s)  defined above.  For example, Z may be equal to s multiplied by a (real) constant L  when the voltage is proportional to the derivative of the current  (such is the case for a perfect inductor of inductance L, as discussed below).

Operating at a given  imaginary  pulsatance  s = jw,  the dipole's  resistance  is defined as the real part of its impedance  Z  (the aforementioned perfect inductor has zero resistance).  The imaginary part of an impedance is called  reactance.

A nonzero reactance at (imaginary) pulsatance  s  indicates that the current and the voltage are out of phase at the corresponding operating frequency.


A linear dipole whose impedance is a real number  R  which does not depend on the frequency of the signal is called a  pure resistor  of resistance R.

Practical resistors are never  quite  ideal, because  any  conducting element has a nonzero inductance which may become noticeable at very high frequencies.  Also, there may be a tiny dependence of  R  on the amplitude of the signal  (Ohm's law is a very good practical approximation, but it's not a strict law of nature).

For completeness, we may also mention that resistance may vary greatly with temperature, so that a high current (which heats up the resistor) may give the apparence of a change in resistance with the amplitude of "large signals".


Ideally, a  capacitor  (or electrical condenser) is a two-terminal device which stores opposite charges  (q and -q)  on two opposing  armatures,  connected to each terminal.  That charge (q) is proportional to the voltage (U) across the terminals and the coefficient of proportionality is the condenser's capacity (C).

q   =   C u

We discuss elsewhere the physical basis for that relation and how the capacity (C) can be computed from geometric parameters and/or from the characteristics of the dielectric material separating the conducting plates (armatures).

Strictly speaking, the charge on each armature is proportional to its absolute voltage  (with respect to an "infinitely distant" ground)  so there may be a bias in the actual charges stored on each armature.  However, the only important practical quantities are variations in the charges (i.e., currents) and/or differences in voltage, so the above fiction is an adequate description.

 Come back later, we're
 still working on this one...


(2007-06-07)   Quality factor  (Q)
The ratio of maximal energy stored to power dissipated.

The quality factor  Q  of a system reacting to a periodic excitation is the ratio of its maximum energy to the average energy it dissipates  (per radian of phase change).

Q  =  wL / R

 Come back later, we're
 still working on this one...

(2007-06-09)   Nullators and Norators
Strange dipoles embodied by active electronic components.

I first heard about the following approach to elementary analog electronic design in the late 1970's  at  Ecole Polytechnique  (X).  It was a novelty at the time.

In an electronic circuit, a dipole is defined as a two-terminal component; whatever current enters one terminal goes out the other.

Normally, such a dipole is characterized by how the current through it varies with time as a function of the voltage across it (or vice-versa).  The  characteristic  of an ordinary dipole thus imposes one constraint between current and voltage...

However, two types of extraordinary dipoles may be considered which greatly simplify the design of some active systems which could not otherwise be modelized by dipoles alone...  One such beast is called a  nullator  (symbol -o-)  and imposes two constraints:  Zero current, zero voltage.  On the other hand, a so-called  norator  dipole  (symbol -¥-)  imposes no constraints at all:  Any current, any voltage.  Neither of those can be realized by itself but they can appear in  complementary pairs  which make the total number of constraints just right (i.e., one constraint per dipole connecting two nodes).

For example, a short-circuit (zero voltage, any current) can be considered to consist of a nullator and a norator in parallel NPN transistor in 
 nullator-norator terms  An open circuit (zero current, any voltage) consists of a nullator and a norator in series.  Less trivially, a properly polarized high-gain transistor is approximately equivalent to a norator from collector (C) to emitter (E) and a nullator from base (B) to emitter (E).

nearly perfect  embodiment of a useful nullator-norator combination is the popular type of subsystem known as an  operational amplifier.  The gain of an operational amplifier is normally so large that some feedback must somehow occur which forces the two high-impedance inputs of the amplifier to be at nearly the same voltage  (or else the output "saturates" at either the lowest or the highest value allowed).  The amplifier's inputs may thus be construed as the two extremities of a nearly perfect  nullator.  Conversely, the amplifier's output can be viewed as one extremity of a  norator  connected to the system's ground.

 Operational amplifier in 
 nullator-norator terms

In practice, of course, the circuit will only be stable with the proper choice of amplifier inputs for the extremities of the  nullator  ("inverting" vs. "non-inverting" input).  Nevertheless, the nullator-norator approach allows a quick preliminary design before final stability issues are addressed.

(2007-06-07)   Corner Frequency & first-order rolloff  (20 dB / decade)
First-order low-pass RC filter and its half-power bandwidth.
 First-order passive RC 
 low-pass filter   At left is the standard first-order passive RC low-pass attenuator  (usually,  G = 0).  At zero output current, the input voltage u is to R+Z what the output v is to Z.  In other words:
u / v  =  1 + R/Z  =  1 + R (G+jwC) 
 First-order low-pass filter 
 normalized Bode plot

The ratio  v/u = H(s) expressed as a function of the complex pulsatance (s)  is called the  transfer function.  In this case, it's equal to  1 / (1+RG+RC s).  Introducing the DC attenuation  A = 1 / (1+RG)  and the circuit's characteristic pulsatance  w0 = ARC, we obtain:

H   =   A / (1 + j x)

The normalized variable is   x   =   w / w0   =   2pn ( 1/RC + G/C ).

The normalized gain (in dB) of the first-order low-pass filter is obtained by plotting  20 log(|H|/A)  as a function of x, using a logarithmic scale for x, as shown above.  This diagram is called a  Bode plot  and is commonly used to chart the frequency response of any filter.

The above shape is the main reason why bandwidth is usually defined as the range of frequencies for which the signal's amplitude is attenuated by no more than a factor of  Ö2  (-3 dB)  from a reference gain  (corresponding to low-frequency signals and/or DC in the case of a low-pass filter).  As the power is the square of the amplitude, such an attenuation means that the power is divided by 2, so the above is best called  "half-power bandwidth".

This definition does gives  directly  the "corner frequency" of any low-pass Butterworth filter, including the above first-order lowpass, which is the simplest Butterworth filter...  The relation isn't so simple in other cases.

(2007-06-07)   Second-order low-pass filters
Second-order rolloff is  40 dB  per decade  (roughly  12 dB  per octave).
 Second-order passive RLC 
 low-pass filter   The second-order passive RLC low-pass filter at left is like its first order counterpart, except that the resistor R becomes the impedance  R+jwL.  Therefore, u/v is  1+(R+jwL)(G+jwC)
 Second-order low-pass filter 
 normalized Bode plot

u / v   =   (1+RG) + jw (RC+LG) - w2 LC

We may cast this in a  normalized  form:

v / u   =   A / [ 1 + l j w/w0 - (w/w0 ) 2 ]

A    =    1 / (1+RG)
2pn0   =   w0   =    Ö Vinculum

A = 1 / (1+RG)   is the low-frequency  attenuation,  used as the 0 dB reference level in the above normalized  Bode amplitude plot  which charts the variations of the gain  |v/u|  in decibels, against the ratio of the pulsatance w to the  nominal pulsatance  (w)  on a logarithmic horizontal scale.

So normalized, the response of a second-order lowpass filter is characterized by the so-called  damping  l.  For the above actual circuit, it's useful to express  l  by introducing the  characteristic resistance   R0 = Ö(L/C).

l   =   w0  RC+LG = R/R0 + R0 G
Vinculum Vinculum
1+RG ( 1+RG )½

For the  common  case where  G = 0,  this means that  l  is simply  R/R0.

In the normalized lowpass transfer function   1 / ( 1 + l s + s 2 )   different values of the  damping  l  make the corresponding second-order filter a member of one of the general families discussed elsewhere on this page:

l =   0 Perfect (ideal) resonator, no damping.  R = 0   and   G = 0.
l =   1 Natural Chebyshev filter, with 1.25 dB ripple.  (Blue line.)
l = Ö2 Butterworth filter (= 0 dB Chebyshev filter).
l = Ö3 Bessel filter(Gold line.)
l =   2Linkwitz-Riley filter:  Two cascaded identical first-order filters.
l >   2Two first-order filters with distinct corner frequencies
(whose geometric mean is 1 and whose sum is l).
To clarify some of the technical literature pertaining to Chebyshev filters, it's important to distinguish the "corner" frequency (compatible with the above "nominal" frequency) from what's best called the "cutoff" frequency...  The cutoff frequency of a lowpass "equiripple" Chebyshev filter is defined as the highest frequency for which the gain is equal to one of the bandpass minima  (all such minima are equal in a Chebyshev filter).  The cutoff frequency coincides with the corner ("nominal") frequency only in the case of a "natural" Chebyshev filter  (like the 1.25 dB second-order Chebyshev filter plotted above).  For high-ripple Chebyshev filters, the cutoff frequency is higher than the corner frequency.  For low-ripple Chebyshev filters, it's lower  (and the term "cutoff frequency" is not recommended in that case).

(2014-05-20)   Two cascaded  passive  first-order filters
The resulting second-order filter can  almost  achieve critical damping.
For future reference  (and for easy comparison with the  next section)  we present this filter with a buffered output.  If the opamp used as a voltage-follower has JFET inputs  (featuring impedances measured in teraohms)  this circuit can effectively be used to hold a voltage for a very long time when the input goes into a high-impedance state  (as the charges in the capacitors have nowhere to go, then).

 Second-order passive
 RC low-pass filter The impedance of a capacitor  C  at pulsatance  w  is equal to  1 / (jw).  Using that,  it's left as an exercise for the reader to verify the following relation  (HINT:  obtain w from v and u from w):

u / v   =   1  +  jw  [ R1 C1 + R2 C2 + R1 C2 -  w2 R1 C1 R2 C2

Critical damping is  almost  achieved when the first two bracketed terms are equal  (RC1  =  RC)  and the third bracketed term  (R1 C2 )  is very small compared to that common value  (i.e,  the impedance of the first stage ought to be much lower than the impedance of the second one).

This illustrates a fairly general principle:  To cascade several RC filters, we usually want the early stages to work in a low impedance regime  (fairly high current)  so that they can feed small currents to later stages without being significantly affected.

This is a key advantage of active filters; they can always have a low impedance output.  So does the above filter when endowed with its voltage-follower.  Without some active section like this, the output of the filter would be of limited use.

(2007-06-16)   The Sallen-Key lowpass filter
Active second-order filters and/or resonators without inductors.

When active components are used for signal processing, the DC gain of a lowpass filter should be kept close to  unity.  A larger gain would impose limitations on the input amplitudes  (in order to prevent saturation of the output signal)  whereas a much smaller gain would worsen the signal-to-noise ratio (SNR or S/N).

This second-order active lowpass filter of unity gain was among the designs introduced in 1955 by R.P. Sallen and E. L. Key (Lincoln Labs  of MIT).

"A Practical Method of Designing Active Filters" by R.P. Sallen and E.L. Key.
IRE Transactions on Circuit Theory  CT-2,  74 -85  (1955)

It can be used as a building block (along with a first-order stage) to realize all the lowpass filters described on this page,  without  using any inductor.

 Second-order active 
 RC low-pass filter   2pn0   =   w0   =   1 / RC
l   =   ( x + 1/x )  y
v   =   u / [ 1 + l j w/w0 - (w/w0 ) 2 ]

The value of  l  in a normalized second-order factor  1/(1+ls+s 2 )  may thus be obtained from any convenient combination of the parameters  x  and  y.

For example, with equal resistors (x=1) we have  l = 2y  and a second-order Butterworth filter  (l=Ö2)  is obtained for  y=1/Ö2  (i.e., C1 = 2 C).

In practice, capacitors may only be available in a few standard values.  Picking coarse values for the capacitors, we may use the following formula to compute precise matching values for the two resistors  R-  and  R+

Vinculum Vinculum
R±   =   R ( z ±  Ö  z 2 -1 )   where   R   =   1 / w0 Ö C0 C1
  and   z    =   (½ l)   Ö C1 / C0

We just have to choose capacitor values so that   z > 1.

The voltage response does not depend on which resistor goes where, but you may want to make the  input impedance  larger  (and/or reduce the power involved)  by placing the larger resistance  R+  on the input side.

Numerically,  when  z  is large, the above expression yields a mediocre way to compute  R-  with ordinary floating-point arithmetic  (because subtracting nearly equal quantities entails a great loss of precision).  Instead, we compute  R+  first  (full precision is retained when quantities of like signs are added)  then obtain  R-  from the following formula  (no precision is lost in multiplications or divisions).

R-   =   R 2 / R+

(2007-06-09)   Low-pass Butterworth filters
The lowpass filters with the flattest low-frequency responses.

Such filters are named after the British radio engineer  Stephen Butterworth (1885-1958)  who first described them in 1930.

"On the Theory of Filter Amplifiers" (1930)  by  Stephen Butterworth
Experimental Wireless and the Radio Engineer, vol. 7, pp. 536-541.

Little is known  [ 1 | 2 ]  about the life of Stephen Butterworth (MSc, OBE).  He served in the British  National Physical Laboratory  (NPL)  and joined the Admiralty scientific staff in 1921.  He retired from the  Admiralty Research Laboratory in 1945 and passed away in 1958.

The normalized transfer function of an order-n  lowpass  Butterworth filter  is of the form  1/Bn(s)  where Bn is a  Butterworth polynomial  of order  n.

  n   Normalized Butterworth Polynomial   Bn(s)
11 + s
21 + s Ö2 + s 2
3( 1 + s )  ( 1 + s + s 2 )
4( 1 + s Ö(2-Ö2) + s 2 )  ( 1 + s Ö(2+Ö2) + s 2 )
5( 1 + s )  ( 1 + s (Ö5-1)/2 + s 2 )  ( 1 + s (Ö5+1)/2 + s 2 )
6  ( 1 + s (Ö6-Ö2)/2 + s 2 )  ( 1 + s Ö2 + s 2 )  ( 1 + s (Ö6+Ö2)/2 + s 2 )  
  m     [ 1   +   2 s sin p(2k-1)/2n   +   s 2 ]
(1+s) m     [ 1   +   2 s sin p(2k-1)/2n   +   s 2 ]

For any n,  | Bn (x) |  is  Ö2,  so the attenuation of a  Butterworth filter  at its corner frequency is always  -3 dB   (well, -3.0103 dB, to be more precise).

(2007-06-20)   Linkwitz-Riley crossover filter
2 cascaded lowpass Butterworth filters and 2 cascaded highpass filters.

Cascading two identical lowpass Butterworth filters of order n gives a lowpass filter of order 2n with a 6 dB attenuation at the corner frequency.

This is particularly useful in combination with a similar highpass filter tuned to the same frequency...  Since both output amplitudes are halved at that crossover frequency, their sum remains at the 0 dB level.

Such a feature is desirable in the design of audio systems, where low frequencies are directed to one loudspeaker and high frequencies to another.  Modern professional active audio crossovers are often based on a fourth-order Linkwitz-Riley design (LR-4).  With digital signal processing (DSP) Linkwitz-Riley crossovers of order 8 are available (LR-8).

The basic idea was credited to  Russ Riley  in a paper published by  Siegfried Linkwitz  in 1976  (both Linkwitz and Riley were HP R&D engineers).

  • "Active Crossover Networks for Non-coincident Drivers"
    Siegfried H. Linkwitz, J. Audio Eng. Soc., vol. 24, pp. 2-8  (1976).
  • Linkwitz-Riley Crossovers: A Primer  by  Dennis Bohn  (Rane, 2005).

Linkwitz-Riley active crossovers were first made commercially available by Sundholm and Rane in 1983.  Nowadays, this may well be the most popular design for professional audio crossovers.

(2007-06-11)   Chebyshev and inverse Chebyshev filters

The basic properties of  Chebyshev polynomials can be put to good use in filter design, by explicitly allowing ripples of amplitude  e  in the frequency response.

 Come back later, we're
 still working on this one...

T0(x)  =   1
T1(x)  =   x Tn+2(x)   =   2x Tn+1(x) - Tn(x)
T2(x)  =   -1+2x2
T3(x)  =   -3x+4x3
T4(x)  =   1-8x2+8x4
T5(x)  =   5x-20x3+16x5
T6(x)  =   -1+18x2-48x4 +32x6
T7(x)  =   -7x+56x3-112x5 +64x7
T8(x)  =   1-32x2+160x4 -256x6+128x8

(2007-06-11)   Cauer filters (elliptic  filters)

The parametrization of  Cauer filters  is general enough to include Butterworth filters and  both  types of Chebyshev filters.

Those filters are named after the German scientist Wilhelm Cauer (1900-1945).  They're also called  elliptic filters, complete Chebyshev filters or  Zolotarev filters  to honor the work of Egor Zolotarev (1847-1878) whose results were applied to filter theory by Wilhelm Cauer in 1933.

 Come back later, we're
 still working on this one...

 Adrien-Marie Legendre 
 (1753-1833) (2007-06-12)   Legendre filters
"Optimum L filters".

The Optimum "L" filter, or Legendre filter, was introduced in 1958 by Athanasios Papoulis (1921-2002).  Among all filters with a  monotonic  frequency response, the Legendre filter has the maximal roll-off rate.  Its features are thus intermediate between the slow roll-off of a Butterworth filter (which is monotonic with unimodal derivatives) and the faster roll-off of a (non-monotonous) Chebyshev filter.

 Come back later, we're
 still working on this one...

(2007-06-13)   Gegenbauer's Ultraspherical Polynomials

The Gegenbauer polynomials are a generalization of the Legendre polynomials (which correspond to the special case  l = ½).  They are named after Leopold Gegenbauer (1849-1903).

For a given value of  l,  the Gegenbauer polynomials are recursively defined:

  • C0(x)   =   1
  • C1(x)   =   2l x
  • Cn(x)   =   1/n [ (2n+2l-2)  x Cn-1(x)  -  (n+2l-2) Cn-2(x) ]

The generating function of those Gegenbauer polynomials is:

( 1 - 2xt + t2 ) -l   =     ån   Cn(x)  t n

Pochhammer symbols  are used below for orders beyond  n = 5
  n   Ultraspherical Gegenbauer Polynomial   Cn(x)
1 2 l x
2 -l  +  2 l(l+1) x 2
3 -2 l(l+1) x  +  4/3 l(l+1)(l+2) x 3
4 1/2 l(l+1)  -  2 l(l+1)(l+2) x 2  +  2/3 l(l+1)(l+2)(l+3) x 4
5 l(l+1)(l+2) x - 4/3 l(l+1)(l+2)(l+3) x 3 + 4/15 l(l+1)(l+2)(l+3)(l+4) x 5
(l)3 x  -  4/3 (l)4 x 3  +  4/15 (l)5 x 5
6 -1/6 (l)3  +  (l)4 x 2  -  2/3 (l)5 x 4  +  4/45 (l)6 x 6
7 -1/3 (l)4 x  +  2/3 (l)5 x 3  -  4/15 (l)6 x 5  +  8/315 (l)7 x 7
8 1/24 (l)4  -  1/3 (l)5 x 2  +  1/3 (l)6 x 4  -  4/45 (l)7 x 6  +  2/315 (l)8 x 8

 Come back later, we're
 still working on this one...

(2007-06-10)   Bode phase plot.  Bayard-Bode relations.
The correlation between phase delay and attenuation slope

If   G = |G| exp(jj)   is the  complex gain  of a discrete low-pass filter, the following approximative relation holds, far from its  corner frequencies,  because it holds far from the corner frequency of every elementary such filter  (the transfer function of higher-order filters is the product of transfer functions of order 1 or 2).

j   »   p/2  d ( Log |G| ) / d ( Log w )

 Come back later, we're
 still working on this one...

The Bayard-Bode relations where developed in 1936 by Marcel Bayard (1895-1956, X1919-S). 

(2007-06-10)   Group delay and Bessel-Thomson filters
Optimizing phase linearity and group delay to preserve signal shape.

The class of orthogonal polynomials named after the German mathematician and astronomer (Friedrich) Wilhelm Bessel (1784-1846) was only introduced in 1948 by H.L. Krall and O. Fink.  The filters themselves were first presented by W.E. Thomson in 1949 and are best called  Bessel-Thomson filters  (BT for short).

"Delay Networks Having Maximally Flat Frequency Characteristics"
W.E. Thomson.  Proc. IEEE, part 3, vol. 96, pp. 487-490 (Nov. 1949).

The  group delay  of a filter whose gain is   G = |G| exp(jj)   is defined to be:

tg   =   - dj / dw

 Come back later, we're
 still working on this one...

The transfer function is  qn(0)/qn(s) where qn is the n-th reverse Bessel polynomial, as tabulated below:

q0(s)  =  1
q1(s)  =   1+ s qn   =   (2n-1) qn-1  +  s2 qn-2
q2(s)  =   3+ 3 s+ s2
q3(s)  =   15+ 15 s+ 6 s2+ s3
q4(s)  =   105+ 105 s+ 45 s2+ 10 s3 + s4
q5(s)  =   945+ 945 s+ 420 s2+ 105 s3 + 15 s4+ s5
q6(s)  =   10395+ 10395 s+ 4725 s2+ 1260 s3 + 210 s4+ 21 s5+ s6

 Come back later, we're
 still working on this one...

 Carl Friedrich Gauss 
 (1777-1855) (2007-06-11)   Gaussian Pulses and Gaussian Filters
Preserving digital pulses in the "time domain".

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(2007-06-13)   Linear Phase Equiripple Filters
Ripples allow better group delay flatness than with Bessel filters.

These filters are to Bessel filters with respect to  group delay  what Chebyshev filters are to Butterworth filters with respect to  amplitude gain.  In either case, better pass-band flatness of the frequency response for the desired property is achieved by allowing some ripples, foregoing the strict monotonicity featured in Butterworth filters (for amplitude gain) or Bessel filters (for group delay).

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(2007-06-14)   DSL filters   (ADSL over POTS)
Allowing POTS bellow 3400 Hz and blocking digital data above 25 kHz.

"Plain Old Telephone Service" (POTS) requires only the  voiceband  (300 Hz to 3400 Hz)  corresponding to the spoken human voice.  PCM digitalized voice corresponds to the 0-4 kHz range  (8 kHz sampling rate).

This is strictly for standard telephony (voice).  By contrast, "CD quality" digital audio involves a 44.1 kHz sampling rate, corresponding to an upper audio limit of  22.05 kHz.  The "audio range" is most often quoted as going from  20 Hz  to 20 kHz, although you've certainly not heard a 20 kHz tone since you were an infant  (and never will again, if you ever did)...  The highest vocal note in classical repertoire is G7 (3136 Hz).  The last key on an 88-key grand piano is at  4186 Hz.

The final "twisted pair" which goes to the telephone subscriber is able to carry a much broader signal, up to 1.1 MHz  or more.  ADSL service makes use of that entire 0-1104 kHz band by dividing it into 256 channels, each 4.3125 kHz wide.

Those channels are numbered from 0 to 255.  The lowest one is the  voiceband  reserved for POTS.  Next are 5 silent channels which provide a wide gap (from 4 kHz to 25 kHz) so a simple so-called "DSL filter" can safely block the digital frequencies (above 25.875 kHz) for POTS devices  (telephone and/or FAX).

The remaining 250 channels, from 25.875 kHz to 1104 kHz, are used specifically for digital service.  With ADSL, there's typically much more traffic downstream (downloading) than upstream (uploading).  Only a small portion of the bandwidth is allocated to upstream traffic  (normally, the 26 channels from 25.875 kHz to 138 kHz, but this can be increased to 276 kHz per "Annex M" of the ADSL2 standard).  This explains the "A" for "asymmetric" in the ADSL acronym;  Such an  Asymmetric Digital Subscriber Line  is nominally  8.92 times faster  one way (223+1 download channels) than the other (25+1 upload channels).  In practice, a 4 to 1 ratio seems more common nowadays

A third-order lowpass filter with a nominal corner frequency of 3243.375 Hz  will produce an attenuation at 25.875 kHz roughly equal to the cube of the frequency ratio (1/8).  This means an amplitude ratio of less than  0.002  (-54 dB).  A typical  fourth order  filter will provide  -72 dB.

The characteristic impedance of a telephone line is 600 W.

 Order-4 DSL Filter Order-4 DSL filter by Ben Kamen.

 Switched capacitor (2014-05-14)   Switched Capacitors
A capacitor C switched between A and B at frequency  n  is like a resistor   R  =  1 / n.C   between A and B.

It's not necessary to have an  SPDT  switch  (single pole, double throw)  as in the conceptual sketch show above.  Instead, we can drive two ordinary switches  (SPST)  by two non-overlapping signals which  never turn on both switches at the same time.

This allows a charge  C(u-v)  to be transferred at each switching cycle, which yields an average current equal to  n.C (u-v).  This is precisely the current that would flow if the switched capacitor assembly was replaced by a resistor of value  1 / n.C

More generally, a single capacitor can be connected via switches to any number of points.

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Take the Mystery Out of the Switched-Capacitor Filter  by  Richard Markell.
Wikipedia :   Switched capacitor

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