A third-order Butterworth filter model with cutoff frequency at :

Express the filter model in terms of variable :

Bode plot of the filter:

A lowpass Butterworth filter using the full specification:

Magnitude response of the filter showing the design specification:

A symbolic lowpass Butterworth filter of order 3 with a cutoff frequency :

Exact computation of the model:

Computation of the model with precision 24:

Use cutoff frequency :

Create a highpass Butterworth filter of order 3:

Bode plot of the filter:

Create a highpass Butterworth filter of order 3 with a cutoff frequency of 10:

Same filter using the full specification:

Create a "Bandpass" filter with passband frequencies and and attenuation of order 3:

Same filter using center frequency and quality factor specification {{ω,q}}:

Same filter using the full specification:

Create a bandstop filter with passband frequencies and and attenuation of order 3:

Same filter using center frequency and quality factor specification {{ω,q}}:

Create a bandstop Butterworth filter using the full specification:

Create a lowpass Butterworth filter:

Filter out high-frequency noise from a sinusoidal signal:

Butterworth filter phase shifts the response by Arg[tf[ω ]], where ω is the frequency of the input sinusoid:

Correct for the phase shift:

Create a highpass Butterworth filter from the lowpass prototype:

Filter out low-frequency sinusoid from the input:

Design a digital FIR lowpass filter using the Butterworth approximation that satisfies the following passband and stopband frequencies and attenuations:

Obtain the equivalent analog frequencies assuming a sampling period of 1:

Compute the analog Butterworth transfer function:

Convert to discrete-time model:

Plot the magnitude:

Create an FIR approximation of a discrete-time Butterworth IIR filter.

Implement a lowpass digital Butterworth filter:

Obtain the desired number of FIR samples from the impulse response of the discrete-time Butterworth filter:

Plot the FIR filter:

Smooth financial data using an FIR approximation of a Butterworth filter:

Filter an image using a discrete-time lowpass Butterworth filter:

Filter an image using a highpass Butterworth filter:

Stopband attenuation increases by a factor of per decade as order increases:

Passband width of "Bandpass" filter decreases with increasing quality factor q:

Phase response of a third-order lowpass Butterworth filter:

Compare phase responses for different filter orders:

Phase response of a "Bandpass" filter for several quality factors:

Extract the order of a Butterworth polynomial:

The order of a Butterworth polynomial for lowpass and highpass is the same as the specified order:

The filter order for bandpass and bandstop is twice the given order:

Show the Butterworth polynomial in the denominator of the transfer function:

Find the poles of a Butterworth filter by solving for the roots of the denominator:

Extract poles using TransferFunctionPoles:

Plot poles of the Butterworth filter:

Implement a lowpass digital Butterworth filter:

Plot poles of the digital Butterworth filter:

Create a highpass filter using a lowpass prototype: