Electronic Hardware
The first thing we do after we capture/collect an analog or a digital signal is to filter it. Filtering can be analog or digital: between the two, digital filters are way more efficient. However, digital filters do not work very well if we do not use an analog filter module first. In other words, analog filters are mandatory no matter what!

Analog filters are built with electronic hardware components, while digital filters are best implemented in firmware. However, there are also hardware digital filters built with PLG (Programmable Logic Gates) when the speed factor is imperative.

The most basic roots of Digital Signal Processing (DSP) technology is using filters--of course, that is digital filters. DSP, however, is a late child, therefore it works as an improved version of the good old analog filtering techniques. If you intend to start working with DSP, you need to study analog filters very, very well first.

RED LEAF RThe simplified structure employed to present analog filters on this page is:

1. Types of Filters
2. Low-Pass Filters
3. High-Pass Filters
4. Band-Pass Filters
5. Band-Reject Filters


The basic notions highlighted on this page are related to a few electronic design topics presented in the first part, Hardware Design, of




There are 2 main types of filters:

1. passive filters
, made of passive components: resistors, capacitors, and inductors;
2. active filters, employing Operational Amplifiers.

Passive filters are built with:

A. Resistor-Capacitor: these are RC filters, and they are the most used ones since they are cheaper to build;
B. Inductor-Capacitor: they are noted as LC filters, and they have (somewhat) better performances. However, the problems are the fact that inductors are expensive, they are difficult to tune up to exact/precise values, and they require shielding of their electromagnetic field.

Both passive and active filters may be serialized (cascaded): in this way we obtain 1, 2, 3 ... n order filters. Of course, the higher the order, the better is the analog filtering employed. Commonly, we build a 3 up to 9 order analog filter, and then we use a digital firmware filter (a DSP filter) of 500, 1000, or even of a 2000 order. Again, although digital filters--they are in fact firmware and software routines--are way more efficient, they do not work very well if you do not have a first, basic, analog hardware filter.

Now, depending on their functionality, both passive and active filters can be:

1. Low-Pass
2. High-Pass
3. Band-Pass
4. Band-Reject


Frequency response Filter Type
Low-Pass filter  Fig1: Low-Pass Filter

  Avp = attenuation in passband
   Avs = attenuation in stopband
   HPp = Half-Power point (0.707 of voltage)
   fc = cutoff frequency

High-Pass filter  Fig 2: High-Pass Filter

  Avp = attenuation in passband
   Avs = attenuation in stopband
   HPp = Half-Power point (0.707 of voltage)
   fc = cutoff frequency

Band-Pass attenuation  Fig 3: Band-Pass Filter

  Avp = attenuation in passband
   Avs = attenuation in stopband
   HPp = Half-Power point (0.707 of voltage)
   fcl = cutoff frequency low
   fch = cutoff frequency high

Band-Reject attenuation  Fig 4: Band-Reject Filter

Avp = attenuation in passband
   Avs = attenuation in stopband
   fr = frequency (band) reject

Active Filters
are a bit more complex since they are built using one of the following Schematics:

1. Butterworth
2. Chebichev
3. Inverse Chebichev
4. Eliptic Integral (or Zolatarev, or complete Chebichev)
5. Legendre
6. Bessel

Filter Response Specifications Comments
Butterworth filter Fig 5: Butterworth
The best amplitude flat response in passband.
Most popular, general-purpose filters.
Chebychev filter Fig 6: Chebyshev
Built for equal amplitude ripples in passband.
The transition slope attenuation is steeper than the Butterworth one.
Inverse Chebichev filter Fig 7: Inverse Chebyshev
Built for equal amplitude ripples in stopband.
No passband ripples.
Elliptic Integral filter Fig 8: Elliptic Integral (or Zolatarev, or complete Chebychev)
Equal amplitude ripples in both the passband and the stopband.
Some of the best analog filters.
Legendre filter Fig 9: Legendre
Similar to Butterworth with no ripple in passband, and steeper transition.
Good filters, though not very flat in passband.
Bessel filter Fig 10: Bessel
Almost linear in passband but very poor transient slope.
Excellent for pulse generator circuits since they minimizes ringing and overshooting.
Particularly good when combined with firmware digital filters.

In order to help designing active filters employing Operational Amplifiers, Microchip offers their FilterLab application software for free download. That is a nice useful program, easy to learn and master.


filters will stop all frequencies greater than the cutoff frequency.


Graphic Representation Description
Low-Pass response Fig 11: Low-Pass Attenuation curve

In Fig 11 you can see that everything is fine and perfect until we reach the HPp (Half-Power point) corresponding to fc (the cutoff frequency).
That is when our filter starts working, because its purpose is to cut all frequencies greater than fc.
Low-Pass RC 1 Fig 12: Low-Pass, first order, simple RC circuit
This circuit is going to give us the above attenuation curve. A few formulas are needed when working with RC filters:
   A = Xo /√(R2 + Xc2)
The above formula becomes:
   A = 1 /√[1 + (2*PI*f*R*C)2]
Note that A = 0.707 in HPp. This allows us to calculate:
   fc = 1/(2*PI*R*C)
Low-Pass LC 1 Fig 13: Low-Pass, first order, simple LC circuit
Using inductors and capacitors we obtain the same output attenuation curve pictured in Fig 11. The formulas used to calculate the filter are a bit different.
First of all, because we deal with AC signals, we have a Characteristic Equivalent Resistance
 Re = √(L/C)
In this case the cutoff frequency is:
   fc = 1/[2*PI*√(L*C)]
Low-Pass LC T Fig 14: Low-Pass, first order, "T" LC circuit
The "T" LC filter is a common circuit, and we would like to point out that 2C needs 2 times the value of C in the previous case. The formulas used to calculate the circuit are the same as above.
Note that at high frequencies L behaves like a capacitor, while C behaves like a resistor, due the reactance formulas presented in the previous Design Notes.
Low-Pass LC PI Fig 15: Low-Pass, first order, "PI" LC circuit
The "PI" LC filter is another common filter circuit. In order to simplify things 2L has double the value in previous circuit.
Low-Pass Active 1 Fig 16: Low-Pass, first order, active filter
This is the simplest possible active Low-Pass filter. Note that the OA is used only to amplify the output of a simple RC filter.
Low-Pass RC 2 Fig 17: Low-Pass, second order, RC circuit
Better filtering results are obtained if we cascade 2 or more filters--commonly, up to 7..9
Low-Pass LC 2 Fig 18: Low-Pass, second order, LC circuit
Same as the above.
Low-Pass Active Fig 19: Low-Pass, second order, active filter circuit
This is a simple, second order Butterworth filter. Again, for best results it is recommend using some professional design software, as is FilterLab.


filters stop al frequencies smaller than the cutoff frequency.
Graphic Representation Description
High-Pass attenuation Fig 20: High-Pass filter attenuation curve
The graph on left tells us that the High-Pass filters work to stop all frequencies up to the cutoff fc. The cutoff frequency appears when the attenuation reaches the Half-Power point (0.707*Vrms).
First order RC filter FIG 21: High-Pass, first order, simple RC filter
Two formulas are used to calculate this High-Pass simple RC circuit:
  A = 1 /√[1 + 1/(2*PI*R*C)2]
  fc = 1/(2*PI*R*C)
Above the cutoff A is almost 1 and A [db] appx = 0 [db]. Below the cutoff A is (2*PI*R*C) and
  A [db] appx = 20log(2*PI*R*C)
First order LC filter Fig 22: High-Pass, first order, simple LC filter
First we determine the Characteristic Equivalent Resistance: Re = √(L/C)
then the cutoff frequency: fc = 1/[2*PI*√(L*C)]
Re must have the same impedance as the source one; this allows us to calculate:
  L = Re/2*PI*fc
  C = 1/2*PI*fc*Re
First order LC T filter Fig 23: High-Pass, first order, "T" LC filter
Again it is improper to name this "T" circuit "a first order one", because it is in fact a second order in disguise. In order to facilitate calculations, the inductance is selected as L/2 of the previous circuit.
First order LC PI filter Fig 24: High-Pass, first order, "PI" LC filter
Same considerations as the above. This time C is half the value it had previously.
First order Active filter Fig 25: High-Pass, first order active filter
In this case the OA doesn't do too much in terms of amplification; however, first order active filters are almost never used. Things start being a bit more interesting beginning with the second order active filters up.
Second order RC filter Fig 26: High-Pass, second order, simple RC filter
We can improve filtering efficiency by using higher order filters.
Second order LC filter Fig 27: High-Pass. second order, simple LC filter
Same considerations as above.
Second order Active filter Fig 28: High-Pass, second order, active filter
In this particular case we can calculate:

   fc = 1/2*PI*√(C2*C3*R1*R2)
   Av = C2/C1


Logically, a High-Pass filter in series with a Low-Pass one results in a Band-Pass filter. The following table presents a few common instances.


Graphic Representation Description
Ban-Pass attenuation Fig 29: Band-Pass Attenuation

Note that we have 2 cutoff frequencies in this case:
  fcl = low-cutoff frequency
  fch = high-cutoff frequency
Simple Band-Pass RC schematic Fig 30: Simple RC Band-Pass Filter
  fch = 1/(2*PI*R2*C2)
  fcl = 1/(2*PI*R1*C1)

Note that R2 >10*R1
Simple Band-Pass LC schematic Fig 31: Simple LC Band-Pass Filter

  fch = 1/[2*PI*√(L2*C2)]
  fcl = 1/[2*PI*√(L1*C1)]
Simple Band-Pass Active schematic Fig 32: Second order active Band-Pass Filter
Use FilterLab to build your active filters in order to save time. That program has a Wizard for building filters; even more, you can adjust the value of each component and analyze the generated Baude diagram for changes.


There are many good Schematics used to build Band-Reject filters; following are presented two of the most basic.


Graphic Representation Description
Band-Reject attenuation Fig 33: Band-Reject Attenuation
Note that we have a frequency reject value here, marked as fr.
RC Band-Reject filter Fig 34: Simple Band-Reject Filter using RC components
Considering this particular case, we have:
   C1 = C2
   C3 = 2*C1
   R1 = R2
   R3 = R1/2
   fr = 1/2*PI*R2*C2
Active Band-Reject filter Fig 35: Simple Band-Reject Filter using OA (also known as "Twin T" circuit)

The gain is 1, and the reject frequency is:
   fr = 1/2*PI*R2*C2
with the same considerations as presented above

Last word: active filters are way more powerful, except they are fairly difficult to tune up. In addition, active filters may introduce some unwanted noise.

Passive filters need to be cascaded for some significant results: you need a minimum of 5..7 order filter. However, you can be certain your passive filter will not introduce any noise--excepting the unshielded LC filters, naturally.

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Page last updated on: February 04, 2017
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