Analog and IIR Filter Types
http://www.nuhertz.com/filter/
Basic Filters
Gaussian Filters
The Gaussian Filter is the filter type that results in the most gradual pass band roll-off and the
lowest group delay. The step response of the Gaussian filter NEVER overshoots the steady state
value. As the name states, the Gaussian Filter is derived from the same basic equations used to
derive the Gaussian Distribution. The significant characteristic of the Gaussian Filter is that the step
response contains no overshoot at all.
Filter Solutions normalizes the Gaussian filter such that the prototype high frequency
attenuation matches the Butterworth filter. The pass band attenuation of the Gaussian filter increases
with the order of the filter when this normalization is applied. However, Filter Solutions allows the
user the option of selecting the desired pass band attenuation in dB's. 3dB attenuation is a popular
choice for some.
Gaussian Transitional Filters
It is occasionally desirable to transition from a Gaussian frequency response to a stepper roll
off response at a user defined attenuation point. Filter Solutions provides 3, 6, 9, 12, and 15 dB
Transitional Filters. Pass band attenuation is always set to 3.01 dB for Gaussian Transitional filters.
Gaussian Low Pass filter, 100Hz Pass Band Frequency
Gaussian With -3.01dB Pass Band Attenuation
Gaussian With 6dB Transition
Gaussian With 15dB Transition
Gaussian Step Response
Bessel Filters
The Bessel Filter's distinguishing characteristic is the near constant group delay throughout the
pass band of the low pass filter.
Filter Solutions normalizes the Bessel filter such that the prototype high frequency attenuation
matches the Butterworth filter. The pass band attenuation of the Bessel filter increases with the
order of the filter when this normalization is applied. However, Filter Solutions allows the user the
option of selecting the desired pass band attenuation in dB's. 3dB attenuation is a popular choice
for some.
Bessel filters may be modified for equiripple group delay, stop bands, or both. Filters with
equiripple group delays are frequently referred to as Linear Phase filters, which is also the
terminology used by Filter Solutions. The equiripple group delay had added efficiency in that the
group delay remains flat farther into the stop band. See the description of Delay Filters for more
on the equiripple group delay.
Bessel Low Pass filter, 10KHz Pass Band Frequency
Linear Phase With Equiripple Group Delay, Period=2.0
Linear Phase With Equiripple Group Delay, Period=2.6
Bessel Modified With Stop Band
Linear Phase With Stop Band and Equiripple Group Delay
Bessel Low Pass Filter Step Response
Butterworth Filters
The Butterworth Filter is the filter type that results in the flattest pass band and contains a
moderate group delay.
A standard Butterworth Filter's pass band attenuation is -3.01dB. However, Filter Solutions
allows the user the option of selecting any pass band attenuation in dB's that will define the filters
cut off frequency.
Filter Solutions also offers the user the option of placing user-defined zeros in the stop band.
Such a filter with stop band zeros is no long a true Butterworth Filter, but is still in the Maximally
Flat filter family.
Butterworth Low Pass filter, 10KHz Pass Band Frequency
Butterworth High Pass filter, 10KHz Pass Band Frequency
Butterworth Band Pass filter, 10KHz Center Frequency, 10KHz Pass Band Width
Butterworth Band Stop filter, 10KHz Center Frequency, 10KHz Pass Band Width
Butterworth Low Pass Step Response
Chebyshev Type I Filters
The Chebyshev Type I Filter is the filter type that results in the sharpest pass band cut off and
contains the largest group delay. The most notable feature of this filter is the ripple in the pass band
magnitude.
A standard Chebyshev Type I Filter's pass band attenuation is defined to be the same value as
the pass band ripple amplitude. However, Filter Solutions allows the user the option of selecting any
pass band attenuation in dB's that will define the filters cut off frequency.
Filter Solutions also offers the user the option of placing user-defined zeros in the stop band.
Chebyshev Type I Low Pass filter, 1MHz Pass Band Frequency
Chebyshev Type I High Pass filter, 1MHz Pass Band Frequency
Chebyshev Type I Band Pass filter, 1MHz Center Frequency, 1MHz Pass Band Width
Chebyshev Type I Band Stop filter, 1MHz Center Frequency, 1MHz Pass Band Width
Chebyshev Type I Low Pass Step Response
Custom Filters
Filter Solutions supports filter design from user entered poles and zeros, biquads, or
polynomial with the use of the Custom windows. Transfer function pages in Filter Solutions may
export transfer functions to the Custom windows where you may add, delete, or modify the filter
transfer function. It is possible to create filters composed a combination of active and passive filters,
or filters composed of Elliptic and Butterworth filters.
Each Transfer Function window contains a "Vec" button to display textual vectors. Each
Vectors windows has a "Send to Custom" button to transfer all displayed vectors to the Custom
window. The format of the vectors (Poly, Biquad, Root) will match the selected format of the
Custom Window.
The pole/zero plot windows may also be used to add, delete, or modify pole or zero locations
prior generating the transfer function.
Custom Polynomials
Custom Biquads
Custom Roots
Advanced Filters
Chebyshev Type II Filters
The Chebyshev Type II Filter, also known as the Inverse Chebyshev Filter, contains a
Butterworth style, or maximally flat, pass band, a moderate group delay, and an equiripple stop
band.
Like the Butterworth Filter, the pass band attenuation of the Chebyshev Type II Filter is
defined to be -3.01 dB. However, Filter Solutions allows the user the option of selecting any pass
band attenuation in dB's that will define the filters cut off frequency.
Below are examples of 5th order Chebyshev Type II low pass, high pass, band pass and band
stop filters and the low pass step response. The stop band ratio is 1.2 in all cases shown. Compare
the stop band attenuation and the group delay to that of the Hourglass and Elliptic Filters.
Chebyshev Type II Low Pass filter, 100KHz Pass Band Frequency
Chebyshev Type II High Pass filter, 100KHz Pass Band Frequency
Chebyshev Type II Band Pass filter, 100KHz Center Frequency, 100KHz Pass Band Width
Chebyshev Type II Band Stop filter, 100KHz Center Frequency, 100KHz Pass Band Width
Chebyshev Type II Low Pass Step Response
Hourglass Filters
The Hourglass filter's distinguishing trait is that the reflection zero frequencies are exactly the
inverse of the transmission zero frequencies. The Hourglass Filter is similar to the Chebyshev Type
II Filter, but has a sharper cut off, higher group delay, and greater stop band attenuation. The pass
band also contains a slight equiripple characteristic, which makes it a special case of the Elliptic
filter. The distinguishing feature of the Hourglass filter is that the reflection zeros are the reciprocal
of the transmission zeros.
Like the Chebyshev Type II Filter, the pass band attenuation of the Chebyshev Type II Filter is
defined to be -3.01 dB. However, Filter Solutions allows the user the option of selecting any pass
band attenuation in dB's that will define the filters cut off frequency.
The Hourglass Filter was first derived by Dr. Byron Bennett of Montana State University,
and is documented in IEEE Transactions on Circuits and Systems, December 1988, volume 12 page
1469.
Below are examples of 5th order Hourglass low pass, high pass, band pass and band stop filters
and low pass step response. The stop band ratio is 1.2 in all cases shown. Compare the stop band
attenuation and the group delay to that of the Chebyshev II and Elliptic Filters.
Hourglass Low Pass filter, 100KHz Pass Band Frequency
Hourglass High Pass filter, 100KHz Pass Band Frequency
Hourglass Band Pass filter, 100KHz Center Frequency, 100KHz Pass Band Width
Hourglass Band Stop filter, 100KHz Center Frequency, 100KHz Pass Band Width
Hourglass Band Stop Step Response
Elliptic Filters
The Elliptic Filter contains a Chebyshev Type I style equiripple pass band, an equipped stop
band, a sharp cut off, high group delay, and greatest stop band attenuation.
Like the Chebyshev Type I Filter, the Elliptic pass band attenuation is defined to be the same
value as the pass band ripple amplitude. However, Filter Solutions allows the user the option of
selecting any pass band attenuation in dB's that will define the filters cut off frequency.
Below are examples of 5th order Elliptic low pass, high pass, band pass and band stop filters
and low pass step response. The stop band ratio is 1.2 in all cases shown. Compare the stop band
attenuation and the group delay to that of the Chebyshev II and Hourglass Filters.
Raised Cosine
Elliptic Low Pass filter, 100KHz Cutoff Frequency
Elliptic High Pass filter, 100KHz Cutoff Frequency
Elliptic Band Pass filter, 100KHz Cutoff Frequency, 100KHz Pass Band Width
Elliptic Band Stop filter, 100KHz Cutoff Frequency, 100KHz Pass Band Width
Elliptic Step Response
Matched Filters
The Matched Filter is for use in communications. The distinguishing characteristic of a
Matched Filter is the step response approximating a ramp, and the impulse response approximates a
pulse. The purpose of the Matched Filter is to maximize the signal to noise ratio and to minimize
the probability of undetected errors received from a signal.
The function of a Matched filter is to optimize the signal to noise ratio at the sampling point of
a bit stream. This happens if the filter applied to the bit stream has an impulse response that is the
time inverse of the pulse shape that is being sampled. If the pulse is rectangular, the filter impulse
response must therefore also be a rectangle, and the step response is a ramp.
Filter Solutions and Filter Light allow you to define your Matched Filter by setting the rise
time of the ramp. The proper use of the matched filter is to set the rise time to be equal to the pulse
width of the pulses in a bit stream.
Since ideal continuous and IIR matched filter solutions are not realizable, they must be
approximated. Filter Solutions uses an approximate solution that optimizes the time response of the
filter with the constraint that the transfer function zeros remain on the JW axis.
Specifically, the
integration of the square of the error between the filter impulse response and the ideal impulse
response (a square pulse) is minimized under the mentioned restraint conditions. The purpose of the
JW zeros constraint is to allow the filter to be realized with passive elements.
Below are examples of Matched filters step, impulse, and frequency responses. Below the
frequency response is the Matched filter square wave response when the rise time of the filter is set
to match the pulse width of the square wave.
Matched Filter Step and Impulse Response
Matched Filter Frequency Response
Matched Filter Square Wave Response
Delay Filters
The Delay Filter simulates a transport delay frequency response. The frequency response
magnitude of an ideal transport delay filter is unity for all frequencies, and the frequency response
group delay is equal to the duration of the transport delay for all frequencies.
It is frequently necessary to account for transport delay in controls applications, and it is
occasionally useful to delay a signal for timing purposes. Filter Solutions provides a Pade
approximation of an ideal transport delay frequency response consisting of a series of all pass stages
with a controlled group delay. With an equiripple period of 2.0, the group delay is designed to be
accurate for frequencies up to (N-1) /(T*PI) Hz where N is the order of the filter, and T is the design
transport delay. This value may be adjusted slightly by adjusting the equiripple period. The
same equiripple group delay response is available for modified Bessel Filters.
Filter Solutions allows you to define your Delay Filter by setting the delay time of the filter.
The delay filter implemented by Filter Solutions optimizes the frequency response as opposed to the
time response of the filter.
Below are examples of 8th order, 1 microsecond Delay filters frequency and step responses.
1 uSec Delay Filter Magnitude and Group Delay Response Ripple Period = 2.0
1 uSec Delay Filter Magnitude and Group Delay Response Ripple Period = 2.6
1 uSec Delay Filter Step Response