Share |

Analog Filters

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

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

The structure employed to present analog filters is:
1. Types of filters
2. Low-Pass filters
3. High-Pass filters
4. Band-Pass filters
5. Band-Reject filters

TYPES OF FILTERS

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 may be realized either with:
A. Resistors-Capacitors: these are RC filters, and they are the most used ones, since they are easier and cheaper to build.
B. Inductors-Capacitors: they are noted as LC filters, and they have better performances. The problems are: inductors are expensive, very difficult to "tailor" to exact values, and they require shielding of their electromagnetic field.

Both the passive and the 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 filtering. It is common to build 3 to 9 order analog filters, and then to use digital firmware filters (DSP filters) of 500, 1000 or even 2000 order.

Again, although digital filters--they are in fact firmware and software routines--are way more efficient, they do not work properly if you do not have a first, basic, analog hardware filter.

Depending on their functionality, both passive and active filters can be:
1. Low-Pass
2. High-Pass
3. Band-Pass
4. Band-Reject

FILTERS CLASSIFICATION BASED ON FUNCTIONALITY

Frequency response

Filter Type

Fig1: Low-Pass Filter

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

Fig 2: High-Pass Filter

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

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

Fig 4: Band-Reject Filter

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

Active Filters are a bit more complex and they are constructed as one of the following:
1. Butterworth
2. Chebichev
3. Inverse Chebichev
4. Eliptic Integral (or Zolatarev, or complete Chebichev)
5. Legendre
6. Bessel

ACTIVE FILTERS COMPARISON CHART

Filter Response

Specifications

Fig 5: Butterworth

The best amplitude flat response in passband.

LOW-PASS FILTERS

Low-Pass filters will stop all frequencies greater than cutoff frequency.

Fig 11: Low-Pass Attenuation curve

In Fig 11 you can see that everything is fine and perfect until we reach the HP_p (Half-Power point) corresponding to f_c (cutoff frequency).
That is when our filter starts working, because its purpose is to cut all frequencies greater than f_c.

Fig 12: Low-Pass, first order, simple RC circuit
This circuit is going to give us the above attenuation curve. Few formulas are needed when working with RC filters:
A = Xo /√(R² + Xc²)
The above formula becomes:
A = 1 /√[1 + (2*π*f*R*C)²]
Note that A = 0.707 in HP_p. This allows us to calculate:
f_c = 1/(2*π*R*C)

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*π*√(L*C)]

Fig 14: Low-Pass, first order, "T" LC circuit
The "T" LC filter is a common circuit, and I would like to point out that C 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 the C behaves like a resistor, due the reactance formulas presented in previous Design Notes.

Fig 15: Low-Pass, first order, "PI" LC circuit
The "PI" LC filter is another common filter circuit. In order to simplify things L has double the value in previous circuit.

Fig 16: Low-Pass, first order, Active filter
This is the simplest possible active Low-Pass filter. Note that OA is used only to amplify the output of the simple RC filter.

Fig 17: Low-Pass, second order, RC circuit
Better filtering results are obtained if we cascade 2 or more filters--normally up to 9.

Fig 18: Low-Pass, second order, LC circuit
Same as the above.

Fig 19: Low-Pass, second order, Active filter circuit
This is a simple, second order Butterworth filter. Again, for best design results I recommend the use of specialized software programs like FilterLab.

HIGH-PASS FILTERS

High-Pass filters stop al frequencies smaller than cutoff frequency.

Fig 20: High-Pass filter attenuation curve
The simple graph on left tells us the High-Pass filters work intensely to stop all frequencies up to cutoff f_c. The cutoff frequency is when the attenuation reaches Half-Power point (0.707*Vrms).

FIG 21: High-Pass, first order, simple RC filter
Two formulas are used to calculate High-Pass simple RC circuit:
A = 1 /√[1 + 1/(2*π*R*C)²]
f_c = 1/(2*π*R*C)
Above cutoff A is almost 1 and A [db] appx = 0 [db]
Below cutoff A is almost (2*π*R*C) and
A [db] appx = 20log(2*π*R*C)

Fig 22: High-Pass, first order, simple LC filter
First we determine the Characteristic Equivalent Resistance: Re = √(L/C)
and the cutoff frequency: f_c = 1/[2*π*√(L*C)]
Re must have the same (impedance) as the source (V_i) one, and this allows us to calculate:
L = Re/2*π*fc
C = 1/2*π*fc*Re

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 chosen as L/2 of the previous circuit.

Fig 24: High-Pass, first order, "PI" LC filter
Same considerations as the above. This time C is half the value it had previously.

Fig 25: High-Pass, first order Active filter
In this case OA doesn't do much; however, first order filters are almost never used. Things start being more interesting beginning with the second order filters.

Fig 26: High-Pass, second order, simple RC filter
We can improve performances by using higher order filters.

Fig 27: High-Pass. second order, simple LC filter
Same considerations as the above.

Fig 28: High-Pass, second order, Active filter
In this particular case we can calculate:

f_c = 1/2*π*√(C₂*C₃*R₁*R₂)
A_v = C₂/C₁

BAND-PASS FILTERS

Logically, by using a High-Pass filter in series with a Low-Pass one we realized a Band-Pass filter. The following table presents few simple particular cases:

Fig 29: Band-Pass Attenuation

Note that we have 2 cutoff frequencies in this case:
f_cl = low-cutoff
f_ch = high-cutoff

Fig 30: Simple RC Band-Pass Filter

f_ch = 1/(2*π*R₂*C₂)
f_cl = 1/(2*π*R₁*C₁)
with R2 >10*R1

Fig 31: Simple LC Band-Pass Filter

f_ch = 1/[2*π*√(L₂*C₂)]
f_cl = 1/[2*π*√(L₁*C₁)]

Fig 32: Second order Active Band-Pass Filter

BAND-REJECT FILTERS

There are many good schematics used to build Band-Reject filters, and presented are only two of them.

Fig 33: Band-Reject Attenuation
Note that we have a frequency reject value here, marked fr

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*π*R₂*C₂

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**π*R₂*C₂
with the same considerations as presented above

Autonopedia