# Analog Filter Design

The Wolfram Language provides a comprehensive set of methods for designing analog filters.
Analog Filter Transfer Functions
 BiquadraticFilterModel biquadratic filter model ButterworthFilterModel Butterworth filter model Chebyshev1FilterModel Chebyshev type 1 filter model Chebyshev2FilterModel Chebyshev type 2 filter model EllipticFilterModel elliptic filter model BesselFilterModel Bessel filter model
Analog filter design methods.
Each one of the classic filters is defined by a particular choice of the function , where defines the order of the filter.
Here is the Chebyshev polynomial of the first kind of order and is the Chebyshev rational function. The Bessel filter is another popular analog filter with a formulation in terms of rational polynomials.
 TransferFunctionModel transfer function of the analog filter TransferFunctionExpand expanded transfer function TransferFunctionFactor factored transfer function
Representation of analog filters.
A second-order Butterworth filter model with cutoff frequency at :
Plot the magnitude of the transfer function of an order 5 Butterworth filter:
Transfer functions of selected orders of lowpass analog filters:
Plot the magnitude of the transfer functions of selected orders of lowpass analog filters:
Poles and Zeros of Analog Filters
 TransferFunctionPoles extract poles of analog filters TransferFunctionZeros extract zeros of analog filters
Poles and zeros of analog filters.
Show poles of a Butterworth filter:
Poles and zeros of an elliptic filter:
Frequency Response of Analog Filters
Squared frequency response of a second-order Butterworth filter:
Plot the magnitude squared of the frequency response:
Bode plot of the filter:
Bode plots of the selected orders of lowpass analog filters:
Output Response
Symbolic step response of a Butterworth filter:
Step response of the filter computed numerically:
Plot the step response:
Sinusoidal response of a Butterworth filter as a function of frequency:
Filter out the high-frequency sinusoid:
Filter out the low-frequency sinusoid:
Different Types of Analog Filters

### Create Filters of Different Types

Create a lowpass Butterworth filter with edge frequencies at 500Hz and 1000Hz and attenuations of 1dB and 20dB for passband and stopband, respectively:
Create a highpass Butterworth filter with edge frequencies at 500Hz and 1000Hz and attenuations of 1dB and 20dB for passband and stopband, respectively:
Create a bandpass Butterworth filter with edge frequencies at 500Hz and 1000Hz and attenuations of 1dB and 20dB for passband and stopband, respectively:
Create a bandstop Butterworth filter with edge frequencies at 500Hz and 1500Hz and attenuations of 1dB and 20dB for passband and stopband, respectively:

### Convert between Filter Types

 TransferFunctionTransform transform a transfer function
Convert between different filter types.
Convert a lowpass analog prototype to highpass, bandpass, and bandstop: