The Laplace transform has shown to allow to analyse the frequency response of circuits based on the differential equations of their capacitive and inductive components. Filter design starts with finding the proper transfer function in order to ampify selected parts of a signal and to damp other ones as a function of their frequency.
Choosing the proper filter structure and deriving the coefficient values is a further topic prensented in the wikibook Signal Processing which deals with the application of signal and systems.
Separating signal from noise or different signals in the same transmission channel basing on their frequency content is best done with a brick-wall filter which shows full transmission in the passband and complete attenuation in the nearby stopbands, with abrupt transitions.
This can be done with the help of the Fourier transform which provides complete information of the frequency content of a given signal. Having calculated a Fourier transform, one can zero out unwanted frequency contents and calculate the inverse Fourier Transform, in order to provide the signal filtered with a brick-wall gauge.
one finds out that the Fourier transform integral, with its infinite bounds, would have to be calculated from the day of the creation of our universe and all the way up to the day of its decay before the integral could have been fully calculated. And only then can the ideal brick-wall filtered signal be delivered.
In more technical terms, the ideal brick-wall filter suffers from an infinite latency.
The analysis of analog circuits shows that their outputs are related to their input by a set of differential equations. The Laplace transform rewrites these differential equations as a set of linear equations of the complex variable . With this, a polynomial function multiplying the Laplace transform of the input signal can be equated to another polynomial function multiplying the Laplace transform of the ouput signal:
Thus, the transfer function of a realizable analog filter can be written as the ratio of two polynomial functions of :
Hence, the problem of analog filter design is to find a pair of polynomial functions which, put together, best approximate the ideal but not realizable brick-wall transfer function.
In the early days of electric signal processing, scientists have come up with filter functions which are still largely used today. The functions they have devised are all of lowpass type. Frequency transformation techniques allow to find polynomials for other filter types such as highpass and bandpass.
The transfer function of an analog filter is the ratio of two polynomial functions of :
The complex plane of
The variable is a complex number which can be written as . The complex plane is a plane with the imaginary axis vertical and the horizontal axis as the real part.
The roots of the transfer function numerator polynom are called the transfer function zeroes. The roots of the transfer function denominator polynom are called the transfer function poles.
The transfer function can be written as a function of its zeroes , its poles and an additional gain factor in the form:
The poles and the zeroes of a transfer function can be drawn in the complex plane. Their position provide information about the frequency response of the system. Indeed, the frequency response is equal to the transfer function taken for , which is along the imaginary axis.
Programs such as Octave or Matlab provide functions which allow to determine the minimal filter order required to fulfill a given specification. This is a good help when choosing from the possible functions.
The following figure shows the result: one needs at least a 13th order Butterworth filter to meet the specifications.
On the graph, one can note that all the amplitude responses go through the same point at -3 dB.
The specification frequencies have been scaled down to fit to the normalized cutoff frequency of 1 Hz. In the script, one might have noted an additional scaling factor of fc = 0.87: this is due to the fact that the corner cutoff amplitude is -0.125 dB and not -3 dB. That value has been adjusted by hand for this example. Again, Octave or Matlab scripts automate this task.