|
|
| Line 1: |
Line 1: |
| {{Refimprove|date=October 2008}}
| | The author is known as Wilber Pegues. For years she's been working as a travel agent. My husband doesn't like it the way I do but what I truly like performing is caving but I don't have the time lately. Some time in the past she chose to live in Alaska and her parents live nearby.<br><br>Stop by my site - online psychics ([http://xovibe.com/members/liamboyes/activity/314711/ xovibe.com the full report]) |
| {{More footnotes|date=October 2008}}
| |
| {{Linear analog electronic filter|filter2=hide|filter3=hide}}
| |
| | |
| In [[electronics]] and [[signal processing]], a '''Bessel filter''' is a type of [[linear filter]] with a maximally flat [[group delay]] (maximally linear [[phase response]]). Bessel filters are often used in [[audio crossover]] systems. Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband.
| |
| | |
| The filter's name is a reference to [[Friedrich Bessel]], a German mathematician (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called '''Bessel–Thomson filters''' in recognition of W. E. Thomson, who worked out how to apply [[Bessel functions]] to filter design.<ref>Thomson, W.E., "Delay Networks having Maximally Flat Frequency Characteristics", ''Proceedings of the Institution of Electrical Engineers'', Part III, November 1949, Vol. 96, No. 44, pp. 487–490.</ref>
| |
| | |
| The Bessel filter is very similar to the [[Gaussian filter]], and tends towards the same shape as filter order increases.<ref>http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l3.pdf</ref><ref>http://www.dsprelated.com/showmessage/130958/1.php</ref> The Bessel filter has better [[shaping factor]], flatter [[phase delay]], and flatter [[group delay]] than a Gaussian of the same order, though the Gaussian has lower time delay.<ref>Design and Analysis of Analog Filters: A Signal Processing Perspective By Larry D. Paarmann p 238 http://books.google.com/books?id=l7oC-LJwyegC</ref>
| |
| | |
| == The transfer function ==
| |
| | |
| [[File:Bessel4 GainDelay.png|right|thumb|A plot of the gain and group delay for a fourth-order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency.]]
| |
| | |
| A Bessel [[low-pass filter]] is characterized by its [[transfer function]]:<ref name=bianchi>
| |
| {{cite book
| |
| | title = Electronic filter simulation & design
| |
| | author = Giovanni Bianchi and Roberto Sorrentino
| |
| | publisher = McGraw–Hill Professional
| |
| | year = 2007
| |
| | isbn = 978-0-07-149467-0
| |
| | pages = 31–43
| |
| | url = http://books.google.com/books?id=5S3LCIxnYCcC&pg=PT53&dq=Bessel+filter+polynomial&lr=&as_brr=3&ei=gyeWSvTbIpmwNPyaqNcH#v=onepage&q=Bessel%20filter%20polynomial&f=false
| |
| }}</ref>
| |
| | |
| :<math>H(s) = \frac{\theta_n(0)}{\theta_n(s/\omega_0)}\,</math>
| |
| | |
| where <math>\theta_n(s)</math> is a reverse [[Bessel polynomial]] from which the filter gets its name and <math>\omega_0</math> is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of <math>1 / \omega_0</math>.
| |
| | |
| ==Bessel polynomials==
| |
| | |
| [[File:Bessel 3rd-order poles.svg|right|thumb|The roots of the third-order Bessel polynomial are the [[pole-zero plot|poles]] of filter transfer function in the [[s plane|''s'' plane]], here plotted as crosses.]]
| |
| | |
| The transfer function of the Bessel filter is a [[rational function]] whose denominator is a reverse [[Bessel polynomial]], such as the following:
| |
| | |
| :<math>n=1; \quad s+1</math>
| |
| :<math>n=2; \quad s^2+3s+3</math>
| |
| :<math>n=3; \quad s^3+6s^2+15s+15</math>
| |
| :<math>n=4; \quad s^4+10s^3+45s^2+105s+105</math>
| |
| :<math>n=5; \quad s^5+15s^4+105s^3+420s^2+945s+945</math>
| |
| | |
| The reverse Bessel polynomials are given by:<ref name=bianchi/>
| |
| | |
| :<math>\theta_n(s)=\sum_{k=0}^n a_ks^k,</math>
| |
| | |
| where
| |
| | |
| :<math>a_k=\frac{(2n-k)!}{2^{n-k}k!(n-k)!} \quad k=0,1,\ldots,n.</math>
| |
| | |
| == Example ==
| |
| | |
| [[File:Bessel 3rd-order gain.svg|right|thumb|Gain plot of the third-order Bessel filter, versus normalized frequency]]
| |
| | |
| [[File:Bessel 3rd-order delay.svg|right|thumb|Group delay plot of the third-order Bessel filter, illustrating flat unit delay in the passband]]
| |
| | |
| The transfer function for a third-order (three-pole) Bessel [[low-pass filter]], normalized to have unit group delay, is
| |
| | |
| :<math>H(s)=\frac{15}{s^3+6s^2+15s+15}.</math>
| |
| | |
| The roots of the denominator polynomial, the filter's poles, include a real pole at {{nowrap|''s'' {{=}} −2.3222}}, and a [[complex conjugate|complex-conjugate pair]] of poles at {{nowrap|''s'' {{=}} −1.8389 ± ''j''1.7544}}, plotted above. The numerator 15 is chosen to give a gain of 1 at [[direct current|DC]] (at ''s'' = 0).
| |
| | |
| The gain is then
| |
| | |
| :<math>G(\omega) = |H(j\omega)| = \frac{15}{\sqrt{\omega^6+6\omega^4+45\omega^2+225}}. \, </math>
| |
| | |
| The phase is
| |
| | |
| :<math>\phi(\omega)=-\arg(H(j\omega))=
| |
| -\arctan\left(\frac{15\omega-\omega^3}{15-6\omega^2}\right). \, </math>
| |
| | |
| The [[group delay]] is
| |
| | |
| :<math>D(\omega)=-\frac{d\phi}{d\omega} =
| |
| \frac{6 \omega^4+ 45 \omega^2+225}{\omega^6+6\omega^4+45\omega^2+225}. \,
| |
| </math>
| |
| | |
| The [[Taylor series]] expansion of the group delay is
| |
| | |
| :<math>D(\omega) = 1-\frac{\omega^6}{225}+\frac{\omega^8}{1125}+\cdots.</math>
| |
| | |
| Note that the two terms in ''ω''<sup>2</sup> and ''ω''<sup>4</sup> are zero, resulting in a very flat group delay at {{nowrap|''ω'' {{=}} 0}}. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at {{nowrap|''ω'' {{=}} 0}} and a second specifies that the gain be zero at {{nowrap|''ω'' {{=}} ∞}}, leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order ''n'': the first {{nowrap|''n'' − 1}} terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at {{nowrap|''ω'' {{=}} 0}}.
| |
| | |
| == See also ==
| |
| | |
| * [[Butterworth filter]]
| |
| * [[Comb filter]]
| |
| * [[Chebyshev filter]]
| |
| * [[Elliptic filter]]
| |
| * [[Bessel function]]
| |
| * [[Group delay and phase delay]]
| |
| | |
| ==References==
| |
| | |
| <references/>
| |
| | |
| == External links ==
| |
| *http://www.filter-solutions.com/bessel.html
| |
| *http://www.rane.com/note147.html
| |
| *http://www.crbond.com/papers/bsf.pdf
| |
| *http://www-k.ext.ti.com/SRVS/Data/ti/KnowledgeBases/analog/document/faqs/bes.htm
| |
| *http://www.source-code.biz/dsp/java/ (Java source code to compute bessel filter poles)
| |
| | |
| [[Category:Linear filters]]
| |
| [[Category:Network synthesis filters]]
| |
| [[Category:Electronic design]]
| |
The author is known as Wilber Pegues. For years she's been working as a travel agent. My husband doesn't like it the way I do but what I truly like performing is caving but I don't have the time lately. Some time in the past she chose to live in Alaska and her parents live nearby.
Stop by my site - online psychics (xovibe.com the full report)