Lecture #9 Continuou time filter Oliver Faut December 5, 2006 Content Review. Motivation......................................... 2 2 Filter pecification 2 2. Low pa.......................................... 2 2.2 High pa.......................................... 3 2.3 Band pa.......................................... 3 2.4 Band top.......................................... 4 3 Butterworth filter cookbook 4 3. Bode plot - Gain...................................... 4 3.2 Bode plot - Phae..................................... 5 3.3 Butterworth filter...................................... 5 3.4 Introduction......................................... 6 3.5 Butterworth filter in the Laplace domain......................... 6 3.6 Butterworth filter coefficient............................... 7 3.7 LP filter example...................................... 8 4 Summary 9 Reference 0 Review Lat lecture introduced the Laplace tranform for CT ytem: If a man make continuou effort, we can ave him. Johann Wolfgang von Goethe A LTI Linear Time Invariant) ytem i completely decribed by the impule repone The Laplace tranform of the impule repone i the tranfer function The tranfer function can be obtained by dividing the Laplace tranform of the output ignal by the Laplace tranform of the input ignal
A ytem i intable if the tranfer function ha pole on the right ide of the pole zero diagram The Laplace tranform can be ued to olve differential and integral equation Evaluating the tranfer function on the imaginary axi yield the frequency repone In turn the frequency repone allow to analye the frequency characteritic of the ytem. Motivation Continuou time filter are important ytem: To control the frequency characteritic of a ignal with a pecial i in may cae deirable. Such ytem might: Block pecific frequencie which might caue damage to another ytem or which might interfere in a later tage of proceing. Amplify pecific frequencie. Some pectral component might have to be amplified in order to obtain a flat pectrum. Delay pecific frequencie. The uperpoition of multiple delayed ignal might reveal more information. 2 Filter pecification In ignal proceing, filter ytem have the following purpoe: Frequency blocking Gain adjutment Phae adjutment The filter pecification determine the requirement for a particular filter. LTI ytem are completely defined by their tranfer function, therefore the filter pecification target the tranfer function or the frequency repone. 2. Low pa A low pa filter pae lower frequencie, but attenuate higher frequencie. The filter pecification follow a: Hω) + D p D p 0.5 D 0 ω p ω ω where: D p paband ripple 2
D topband ripple ω p End of the paband ω Start of the topband 2.2 High pa A high pa filter block low frequencie but pae high frequencie. The filter pecification follow a: Hω) + D p D p 0.5 D 0 ω ω p ω where: D p paband ripple D topband ripple ω p End of the paband ω Start of the topband 2.3 Band pa A band pa filter pae a band of frequencie but block all the ret of the pectrum. The filter pecification follow a: Hω) + D p D p 0.5 D 0 ω ω p ω p2 ω 2 ω where: D p paband ripple D topband ripple ω p Start of the paband and ω p2 end of the paband ω End of topband and ω 2 tart of topband 2 3
2.4 Band top A band top filter block a certain band of frequencie but pae all the ret of the pectrum. The filter pecification follow a: Hω) + D p D p 0.5 D 0 ω p ω ω 2 ω p2 ω where: D p paband ripple D topband ripple ω p End of paband and ω p2 tart of paband 2 ω Start of the topband and ω 2 end of the topband 3 Butterworth filter cookbook The frequency repone of the Butterworth filter ha the following propertie: The frequency repone i maximally flat i.e. ha no ripple) in the paband, and roll off toward zero in the topband. The logarithmic frequency repone lope off linearly toward negative infinity. The filter order determine how teep the lope i. The general rule i 20dB time filter order per decade. For example a econd order filter lope off with 40dB per decade and a third order filter lope off with 60dB. 3. Bode plot - Gain The following figure how the filter gain in db over the logarithmic frequency. 4
3.2 Bode plot - Phae The following figure how the filter phae over the logarithmic frequency. 3.3 Butterworth filter The magnitude of the frequency repone of an nth order lowpa filter can be defined mathematically a: G n ω) = H n ω) = + ω/ωc ) 2n ) where G i the gain of the filter, H i the tranfer function, n i the order of the filter and i the cutoff frequency 3 db frequency) []. 5
3.4 Introduction The ubequent argument follow Gordon E. Carlon [2, page 326]. generating the Butterworth filter: Two criteria are ued in The filter gain at the cutoff frequency i 2. Thi i equivalent to a power gain at ω = of 2 or 3dB. Therefore, w c i alo referred a the 3dB cutoff frequency. The amplitude repone mut be maximally flat at ω = 0. That i Hω) mut have a many derivative equal to zero at ω = 0. The Butterworth amplitude repone ha 2n derivative equal to zero at ω = 0 and thu become flatter a the order increae. 3.4. Tranfer function Uing the two criteria we have jut dicued, we obtain the tranfer function a with ω 2 = 2 the tranfer function follow a For a firt order filter follow: H n ω) 2 = + ω 2 /ωc 2 ) n = ω 2n c ωc 2n + ω 2n 2) H n ) 2 = ω 2n c ω 2n c 2n 3) H ) 2 = ω2 c 2 = + ) ) 4) And for the econd order filter follow H 2 ) 2 = ω4 c ω 4 c 4 = ω 4 c 4 k= p k ) 5) where p k = e j0.25π+0.5πk) 6) 3.5 Butterworth filter in the Laplace domain The following two pole zero diagram how the pole location for Butterworth filter of dimenion n = and n = 2 repectively. Pole Unued pole Im) Im) Re) Re) 6
The following two Pole-Zero diagram how the pole location for Butterworth filter of dimenion n = 3 and n = 4 repectively. Im) Re) Im) Re) We elect the pole in the left half plane to be the pole of the Butterworth filter. The tranfer function for the firt order filter follow a H ) = + 7) And for the econd order filter H 2 ) = e j0.75π ) e j.25π ) = ωc 2 e j2π e jπ e j0.25π + e j0.25π ) + 2 = ωc 2 + 2 co0.25π) + 2 = ωc 2 + 2 + 2 = }{{} a 0 + The coefficient for the econd order Butterworth filter are: ) 2 8) }{{} 2 + }{{} ωc a a 2 a 0 = a = 2 a 2 = 9) 3.6 Butterworth filter coefficient The following table detail the filter coefficient up to an order of n = 6. Note, a 0 = a n =. n a a 2 a 3 a 4 a 5 2 2 3 2.000000 2.000000 4 2.6326 3.4424 2.6326 5 3.236068 5.236068 5.236068 3.236068 6 3.863703 7.46402 9.4620 7.46402 3.863703 7
3.7 LP filter example The following example wa preented by Richard Baraniuk [3, Butterworth Filter]. Quetion: Deign a Butterworth filter with a paband gain between and 0.89 - db gain) for 0 < ω < 0 and a topband not to exceed 0.036-30 db gain) for ω 20. Anwer tructure:. Determine the filter order n 2. Find the 3dB point, 3. Find the normalied tranfer function 4. Find the final tranfer function 3.7. Firt tep The firt tep i to determine n. To do thi, we mut olve for n uing the paband and topband criteria. We begin by finding the equation for the gain in the paband in db, [ ) ] 2n Ĝ p =20 log [Hω) H ω)] = 0 log + ωp 0) and for the topband in db, [ Ĝ =20 log [Hω) H ω)] = 0 log + ω ) 2n ] ) thee equation can alo take the form ωx ) 2n = 0 Ĝ/0 2) In thi form, we may divide the paband equation by the topband equation to get rid of the. From there, we can olve for n to get ) 0 Ĝ/0 log 0 Ĝp/0 n = ) 3) ω 2 log By plugging in, we find n = 5.9569 However, ince n mut be an integer, we round thi up to n = 6 3.7.2 Second tep The next tep i to find. We can do thi by ubtituting n = 6 into the equation for the paband and topband and olving for. Thi yield =.99 for the paband equation and =.2478 for the topband equation. The difference in thee olution i a reult of n needing to be an integer. If we chooe the olution from the paband equation, the paband will meet ω p 8
it requirement exactly, and the topband will urpa it requirement. If we chooe the olution from the topband equation intead, the topband requirement will be met exactly, while we will exceed the paband requirement. Therefore, we may chooe either value or any value in between. For thi example, we will chooe =.2478. 3.7.3 Third tep Now, we can find the normalized tranfer function. Since we know thi to be a ixth-order Butterworth, we can determine from the table that H) = 6 + 3.863703 5 + 7.46402 4 + 9.4620 3...... +7.46402 2 + 3.863703 + 4) 3.7.4 Lat tep Finally, we can determine the tranfer function a H) =.24786... +9.4620... +3.863703 ) 6 + 3.863703.24783.2478.24785 ) 3 + 7.46402 ) 5 + 7.46402.24782.24784 ) 2... ) 4... ) + 5) Rather than multiplying thi out and factoring, we will leave it in thi form for readability, ince the number can get quite large otherwie. 4 Summary Thi lecture introduced filter deign: Filter deign begin with filter pecification. In the implet cae they are concerned with the frequency repone of the filter: Low pa: A low pa filter pae lower frequencie, but attenuate higher frequencie. High pa: A high pa filter block low frequencie but pae high frequencie. Band pa: A band pa filter pae a band of frequencie but block all the ret of the pectrum. Band top: A band pa filter block a band of frequencie but block all the ret of the pectrum. Butterworth filter deign cookbook Jutification of the Butterworth filter characteritic in time domain Jutification of the Butterworth filter characteritic in the Laplace domain Butterworth filter example 9
Evaluate the filter order n Evaluate of the cutoff frequency State the normalied tranfer function State the final tranfer function Reference [] Mankind. Wikipedia, the free encyclopedia. Homepage: http://en.wikipedia.org/wiki/main_page. [2] Gordon E. Carlon. Signal and Linear Sytem Analyi. John Wiley & Son, econd edition, 998. ISBN: 0-47-2465-6. [3] Connexion. Connexion i an environment for collaboratively developing, freely haring, and rapidly publihing cholarly content on the Web. Homepage: http://cnx.rice.edu/. 0