Electronic
Linear filtering
Fondamental :
Harmonic analysis (or frequential) of a system is its study through its harmonic response , that is to say, its response in sinusoidal permanent regime when subjected to a sinusoidal input which is varied frequency .

Filter order :
Example of a lowpass filter :
Example of a highpass filter :
, with the cutoff frequency to .

Filter order :
Example of a lowpass filter :
where is the natural angular frequency of the filter and the damping coefficient.
is defined as the quality factor.
Example of a highpass filter :
Example of a bandpass filter :
here is also the bandpass filter resonance pulse.
Example of a notch filter (or bandstop) :
Définition : General notations and definitions
We note : (in complex notations)
is the phase shift of the output voltage with respect to the filter input voltage.
is the voltage transfer function of the filter :
And :

is the real gain :

is the gain in decibels ( :
The previous figure gives a Bode diagram of a bandpass filter, that is to say, the graphical representation of the gain in ( ) and the phase shift as a function of the pulse and on a logarithmic scale.
Fondamental : Fundamental filtering 1st order (eg RC series circuit)
Studying an RC series circuit and measuring the output voltage in the open output.
Qualitative study of the nature of flitre :

At low frequencies, the capacitor is equivalent to an open switch (its impedance is infinite).
The current in the circuit is zero. Thus, the voltage at the terminal is also zero and therefore .
At high frequency, the capacitor is equivalent to a zero resistance wire. So, .
The filter studied here is thus a lowpass filter.
Open output, the voltage divider rule gives :
With :
The natural angular frequency of the filter, which is also the cutoff frequency to , that is to say the pulse for which the real gain is ,or, what is equivalent, .
It is the transfer function of a lowpass filter of the first order.
The figure above shows the Bode amplitude of this lowpass filter of the first order.
The following figure shows the Bode phase.
Other filter order :

RC series, output voltage across :
It is a highpass filter.
With .

Circuit RL series, output voltage across :
It is a highpass filter.
With :

Circuit RL series, output voltage across :
It is a lowpass filter.
With :
The preceding figures show the Bode diagram in amplitude and phase of a highpass filter of the first order.
Fondamental : Filter 2nd order (eg the series RLC circuit)
Fondamental : The terminals of C in open output : low  pass filter
For identification :
Whence :
So :
Referring to the study of mechanical oscillators (response in elongation), one obtains :
There are "resonance voltage" across terminals ( "load resonance "), ie corresponding to a maximum voltage across , for the pulse of GBF such that :
And the maximum voltage across at "load resonance" is :
The above formulas become, using the quality factor in place of the damping coefficient ( ) :
And :
Note :
For low amortization ( «small» and «large»), then :
Thus, if , the amplitude at the resonance is times that of the excitation : the resonance is termed "sharp" and may cause destruction of the oscillating system.
The following figures give the Bode diagram in gain and phase of a lowpass filter order. Different numerical values of are used.
Fondamental : The terminals of R open output : band  pass filter
For identification:
Hence (Member to Member ratio) :
So :
Resonant voltage across ("resonance intensity") :
A go "resonance voltage" across (« resonance current »), corresponding to a maximum voltage terminals , for a pulse GBF equal to the oscillation frequency of the circuit (RLC) series :
And the maximum voltage across for the "resonance current" is :
The passband is :
The following figures give the Bode diagram in gain and phase of a band pass filter order. Different numerical values of are used.
Fondamental : The terminals L open output : high  pass filter
For identification, we find the same characteristics as for the lowpass and bandpass filters :
So :
The following figures give the Bode diagram in gain and phase of a highpass filter order. Different numerical values of are used.
Fondamental : The terminals of (C + L) output open: notch or band  stop filter
For identification, we find the same characteristics as for the lowpass, bandpass and highpass filters :
So :
The following figures give the Bode diagram in gain and phase of a notch filter order. Different numerical values of are used.
Simulation : JAVA animations by JJ.Rousseau (University of Le Mans)
Suspension of a vehicle : click HERE
RC circuits, filters, splitters and integrators : click HERE
Passive filters : click HERE
Passive filters (2) : click HERE
Passive filters L, T and Pi : click HERE
Second order passive filter : click HERE
Passive filters T and T bridged : click HERE
Filter double T bridged : click HERE
Filter two ways : click HERE