# Low-pass filter

A low-pass filter is a filter which lets pass the low frequencies and which attenuates the high frequencies, i.e. frequencies higher than the Frequency cut-off. It could also be called filter cut-high. The low-pass filter is the reverse of the high-pass Filtre and these two combined filters form a Filtre band pass.

The concept of low-pass filter is a mathematical transformation applied to data (a signal). The implementation of a low-pass filter can be done numerically or with electronics components. This transformation has as a function to attenuate the frequencies higher than its Frequency cut-off $f_c$ and this, with an aim of preserving only the low frequencies. The cut-off frequency of the filter is the frequency separating the two ideal operating processes from the filter: passer by or blocking.

## Filter ideal

An ideal low-pass filter has a constant profit in its band-width and a null profit in the cut band. The transition between the two states is instantaneous. Mathematically, it can be carried out by multiplying the signal by a rectangular window in the frequential field or by a convolution with a cardinal sine (sinc) in the temporal field. This type of filter is called “wall of brick” in the jargon of the engineers.

Naturally, an ideal filter is practically not realizable, because a cardinal sine is an infinite function. Thus, the filter should predict the future and to have an infinite knowledge of the past to carry out the convolution and to obtain the desired effect. It is possible to approximate this filter very accurately in a numerical way when one lays out of a signal preregistered (while adding of the zeros at the two ends of the series of samples) or for a periodic signal.

In real-time, the numerical filters can approximate this filter while inserting a voluntary time in the signal, which makes it possible “to know the future of the signal”. This operation creates a dephasing between the exit and the entry and naturally, plus the inserted time is long, plus the filter will approach the ideal filter.

## Analogical low-pass filter

A low-pass filter can be implemented in an analogical way with electronics components. Thus, this kind of filter applies to continuous signals in real-time. The components and the configuration of the circuit will fix the various characteristics of the filter, such as the order, the frequency cut-off and its Diagramme of Bode. The traditional analogical filters are of the first or the second order. There exist several families of analogical filters: Butterworth, Tchebychev, Bessel, elliptic, etc the implementation of the of the same filters family is generally done by using the same configuration of circuit, and those have the same form of Transfer function transfer, but in fact the parameters of this one change, therefore the value of the components of the electrical circuit.

### First order low-pass filter

A first order low-pass filter is characterized by its cut-off frequency $f_c$. The transfer transfer function of the filter is obtained by dénormalisant the low-pass filter standardized in substituent $\ omega_n$ by $\ Omega/\ omega_c$, which gives the following transfer transfer function:

$H \left(J \ Omega\right) = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{K\right\} \left\{1+j \ frac \left\{\ Omega\right\} \left\{\ omega_c\right\}\right\}$

$\ Omega = 2 \ pi f$
$\ omega_c =2 \ pi f_c$

The module and the phase of the transfer transfer function equalize with:
$|H \left(\ Omega\right)| = |\frac\left\{v_o\right\}\left\{v_i\right\}|= \ frac \left\{K\right\} \left\{\ sqrt \left\{1+ \ big \left(\ frac \left\{\ Omega\right\} \left\{\ omega_c\right\} \ big\right) ^2\right\}\right\}$

$\ phi \left(\ Omega\right) = \ arg H \left(J \ Omega\right) = - \ arg \left(1+j \ frac \left\{\ Omega\right\} \left\{\ omega_c\right\}\right) = - \ arctan \left(\ frac \left\{\ Omega\right\} \left\{\ omega_c\right\}\right)$

There are several methods to implement this filter. An active realization and a realization passivate are presented here. To note that K is the profit of the filter.

#### Passive circuit

The manner simplest to produce this filter physically is to use a Circuit RC. As its name indicates it, this circuit consists of a resistance $R$ and a condensing of capacity $C$. These two elements are placed in series with the $v_i$ source of the signal. The output signal $v_o$ is recovered at the boundaries of the condenser. To find the transfer transfer function of this filter, it is necessary to work in the field of Laplace by using the impedances elements. With this technique, the circuit becomes simple a Tension divider, and one obtains:
$H \left(J \ Omega\right) = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{1\right\} \left\{1+jRC \ Omega\right\}$
In this equation, $j$ is a Complex number, the square root of -1, and $\ omega$ is the pulsation of the circuit or radial frequency, expressed in rad/s. As the cut-off frequency of a Circuit RC is:
$f_c = \ frac \left\{1\right\} \left\{2 \ pi RC\right\}$ or $\ omega_c= \ frac \left\{1\right\} \left\{RC\right\}$

Here $\ omega_c$, the pulsation of cut, is also the own pulsation $\ omega_o$ of the circuit, it is also the reverse of the time-constant $\ tau$ of the circuit (raised constant $2 \ pi$). Thus, one indeed obtains the typical transfer transfer function of the first order low-pass filter.

With this transfer transfer function, one can obtain the diagrams of Bode:

$G_ \left\{dB\right\} \left(\ Omega\right) = 20 \ cdot \ log |H \left(\ Omega\right)| = -10 \ cdot \ log \left(1+ \ big \left(\ Omega RC\right) ^2\right)$
• the phase in Radian S:
$\ phi \left(\ Omega\right) = - \ arctan \ big \left(\ Omega RC\right)$
Two ideal situations then are distinguished:
• When $\ Omega \ L \ omega_c$, one a:
$G_ \left\{dB\right\} \ simeq 0$ and $\ phi \ simeq 0$
(the filter is busy)
• When $\ Omega \ gg \ omega_c$, one a:
$G_ \left\{dB\right\} \ sim -20 \ cdot \ log \left(\ frac \left\{\ Omega\right\} \left\{\ omega_c\right\}\right)$ and $\ phi \ simeq 90$
(the signal is then filtered)

It is noticed that for $\ omega= \ omega_c$, one has $G_ \left\{dB\right\}$ = -3  dB.

#### Active circuit

It is also possible to produce a low-pass filter with an active circuit. This option makes it possible to add profit to the output signal, i.e. to obtain an amplitude higher than 0  dB in the band-width. Several configurations make it possible to implement this kind of filter. In the configuration presented here, the cut-off frequency is defined as follows:
$f_c = \left\{1 \ over 2 \ pi R_2 C\right\}$ or $\ omega_ \ mathrm \left\{C\right\} = \ frac \left\{1\right\} \left\{R_2 C\right\}$

By utlisant the properties of the operational amplifier , and the impedances of the elements, the following transfer transfer function is obtained:

$H \left(J \ Omega\right) = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{- R_2\right\} \left\{R_1\right\} \ cdot \ frac \left\{1\right\} \left\{1+jR_2C \ Omega\right\}$

As low frequency, the condenser acts like a Open circuit, which is confirmed by the fact that the term of right-hand side of the preceding equation tends towards 1. The simplified formula thus obtained gives us the profit in the band-width:

$H \left(\ Omega\right) _ \left\{\ Omega \ L \ omega_c\right\} = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{- R_2\right\} \left\{R_1\right\}$

As high frequency, the condensing acts like a Closed circuit and the term of right-hand side tends towards 0, which makes tighten the formula towards zero.

$H \left(\ Omega\right) _ \left\{\ Omega \ gg \ omega_c\right\} = \ frac \left\{v_o\right\} \left\{v_i\right\} \ simeq 0$

With the transfer transfer function, one can show that the attenuation in the rejected band is of 20  dB/décade or of 6  dB per octave as awaited for a filter of order 1.

It is frequent to see a circuit of amplification or by attenuation transformed into low-pass filter by adding a condenser C. This decreases the answer of the high frequency circuit and assistance to decrease the oscillations in the amplifier. For example, an audio amplifier can be an active low-pass filter with a cut-off frequency about 100  Khz to reduce the profit to frequencies which differently would oscillate. This modification of the signal does not deteriorate “useful” information of the signal, because the audio band (audible waveband by the human one) extends until approximately 20  Khz, which is largely included in the band-width of the circuit.

### Low-pass filter of the second order

A low-pass filter of the second order is caractésiré by its frequency of resonance $f_o$ and by the Facteur of quality Q. It is represented by the following transfer transfer function:

$H \left(J \ Omega\right) = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{K \ omega_o^2\right\}$

$\ Omega = 2 \ pi f$
$\ omega_o =2 \ pi f_o$

The module and the phase of the transfer transfer function equalize with:
$|H \left(\ Omega\right)| = |\frac\left\{v_o\right\}\left\{v_i\right\}|= \ frac \left\{K\right\} \left\{\ sqrt \left\{\ frac \left\{\ omega^2\right\} \left\{Q^2 \ omega_o^2\right\} + \left(1 \ frac \left\{\ omega^2\right\} \left\{\ omega_o^2\right\}\right) ^2\right\}\right\}$

$\ phi \left(\ Omega\right) = - \ arctan \left(- \ frac \left\{\ frac \left\{\ Omega\right\} \left\{Q \ omega_o\right\}\right\} \left\{1 - \ frac \left\{\ omega^2\right\} \left\{\ omega_o^2\right\}\right\}\right)$

#### Passive circuit

The manner simplest to produce this filter physically is to use a Circuit RLC. As its name indicates it, this circuit consists of a resistance $R$, a condensing of capacity $C$ and of a Inductance $L$. These three elements are placed in series with the $v_i$ source of the signal. The output signal $v_o$ is recovered at the boundaries of the third and last element, the condensing . To find the transfer transfer function of this filter, it is necessary to work in the field of Laplace by using the impedances elements. With this technique, the circuit becomes simple a Tension divider, and one obtains:

$H \left(J \ Omega\right) = \ frac \left\{v_o\right\} \left\{v_i\right\} = \ frac \left\{\ frac \left\{- 1\right\} \left\{LLC\right\}\right\} \left\{\ omega^2-j \ Omega \ frac \left\{R\right\} \left\{L\right\} - \ frac \left\{1\right\} \left\{LLC\right\}\right\}$

With:

$\ omega_o = \ frac \left\{1\right\} \left\{\ sqrt \left\{LLC\right\}\right\}$

$Q = \ frac \left\{1\right\} \left\{R\right\} \ sqrt \left\{\ frac \left\{L\right\} \left\{C\right\}\right\}$

The module and the phase of this circuit are:

$|H \left(\ Omega\right)| = |\frac\left\{v_o\right\}\left\{v_i\right\}|= \ frac \left\{1\right\} \left\{\ sqrt \left\{R^2 C^2 \left\{\ Omega\right\} ^2 + \ big \left(1 - LLC \left\{\ Omega\right\} ^2\right) ^2\right\}\right\}$

$\ phi \left(\ Omega\right) = - \ arctan \left(- \ frac \left\{RC \ Omega\right\} \left\{1 - LLC \left\{\ Omega\right\} ^2\right\}\right)$

#### Active circuit

Several types of filters exist to produce a second-order active filter. Most popular are structures MFB and VCVS.

### Filter of a higher nature

The filters of a higher nature are generally composed of filters of order 1 and 2 in cascade. The realization of a filter of order 5, for example, is done while placing two filters of order 2 and one filter of order 1. It would be possible to produce directly a filter of order 5, but the difficulty of design would be largely increased by it.

## Numerical low-pass filter

See numerical Filter.

## See too

### External bonds

• Explanation on the filters
• Filters
• low-pass Filter of order 2 liability

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