Deigning of Analog Filter. Aliaing and recontruction filter are analog filter, therefore we need to undertand the deign of analog filter firt before going into the deign of digital filter. Further the concet in the deign of analog filter can be ued for deign of digital filter later on. Deign of analog Butterworth, Chebyhev, Ellitic Filter.
Analog Lowa Filter Secification H ( j a Stoband +δ δ Paband δ Stoband Tranition band
Analog Lowa Filter Secification In Paband 0, we require - δ H a ( j + δ, In toband <, we require H a ( j δ, <
Terminology ued for Analog Lowa Filter Secification δ δ - aband edge frequency - toband edge frequency - eak rile value in the aband - eak rile value in the toband Peak aband rile : α -0log Minimum toband attenuation : α 0 (-δ -0log db. 0 ( δ db.
Analog Lowa Filter Normalized Secification H ( j a + ε A Stoband Paband Tranition band Stoband
Terminology ued for Analog Lowa Filter Normalized Secification - - Maximum value of Paband rile denoted by. + ε (minimum value of aband magnitude. Maximum aband edge frequency toband edge frequency the magnitude in aband toband magnitude A..
Terminology ued for Analog Lowa Filter Normalized Secification Additional Parameter defined a : - ( Tranition ratio k For lowa filter k <. ( Dicrimination arameter k Uually, k <<. A ε.
Butterworth Aroximation Magnitude - quared reone of N - th order analog lowa Butterworth filter : - H a ( j + ( ( Firt N -derivative of above reone at are zero. ( i.e. maximally flat magnitude at 0. c N 0.
Butterworth Aroximation Gain in db of analog lowa Butterworth filter : - G( 0log 0 H ( j ( Gain at 0 : - G(0 0 a 0log 0 ( 0log 0 ( + ( c N ( c Gain at c : - G( 0-0log -0 log ( + ( + ( ( -3.003-3dB. i known a 3 db cutoff frequency. c 0 0 N c c N 0log 0 (
Butterworth Aroximation with cutoff frequency normalized to. 0 db Magnitude -3dB Higher N c Frequency
Ste in Lowa Butterworth Filter Deign. Ste. Determine the two arameter :- ( N, the filter order. c (, 3db cutoff frequency Ste. Determine the Tranfer Function H a of the Butterworth Lowa Filter uing the above two arameter. (
Determining the two arameter Given the ecification : aband edge frequency toband edge frequency minimum aband magnitude maximum toband rile -,,. A c and N : - + ε,
N Determining order of filter and H H From thee two equation ( ( cutoff frequency. From Magnitude - quared reone of order analog lowa Butterworth filter : - a a j j 0 + + ( ( / / log0[( A / ε ]. log ( / c c N we can olve for N and log log N 0 0 A (/ k (/ k N - th + ε...eqn.4.34a... Eqn. 4.34b c...eqn.4.35.
Butterworth Aroximation N mut be rounded u to the next highet integer. Ue thi rounded N to determine c via either of the eqn. 4.34a or eqn.4.34b If you ue eqn. 4.35a the aband edge ecification i atified exactly while the toband edge ecification i exceeded.
Tranfer Function of the Lowa Butterworth Filter Deign. order N. olynomialof the Butterworth known a i ( D radiu circle of aced on a are equally on the - lane ( H ( H N ole of...4.37..., where ( ( ( H N c a a ] / ( [ 0 a e i Eqn N l e d D C N l N j c l l N l N N l l l N N N c c Π + +
Examle 4.6. Deign Butterworth Lowa Filter. Given filter ecification : - (db cutoff with minimum value of ( frequency at khz.i.e. aband edge frequency ikhz, Stoband edge frequency i 5kHz, magnitude in aband db. with minimum toband attenuation 40 db. 0 db - db -40 db 000Hz. 5000Hz.
From eqn. 4.34a : -0log i.e.0log0 (, ε 0.5895 + ε From eqn. 4.34b : -0log H ( j 40 db. i.e.0log 0 From eqn. 4.3 : - From eqn. 4.3: - From eqn. 4.35: - Roundu N ( A 4. 40, A k k N 0 0 log log A ε 0 0 H a a ( j 0,000. 5000 000 (/ k (/ k db. 96.5334 5 3.8
+ ( 000 / ( 000 / 8log log 000 / Determining the 3dB Cutoff Frequency. From Eqn. 4.34a : - and ubtituting c 0 0 8 ( 000 / ( 000 / 0 000 /0 c c c 8 H a 000, N 4, ε 0.5895 into above eqn. + 0.5895 0.5895 c c -0.07335-0.07335 ( j log log 0 0 + ( (0.5895, / (0.5895-0.07335 8 000*0 c 0.07335 N, + ε 84 Rad/ec.
Determining the Tranfer Function of Filter( i.e.ole.. 4 4 4.84 (.84 (.84 (.84 ( 84 ( ( ( ( ( 84 ( ( H ( ( -H 4.36 : From eqn. 84 84 84 84 84 84 84 84 4. N 84,, - 4.37 : From eqn. 8] / [ 8] 9 / [ 8] 7 / [ 8] 5 / [ 4 4 3 4 a a 8] / [ 8] / 7 (4 [ 4 8] 9 / [ 8] / 5 (4 [ 3 8] 7 / [ 8] / 3 (4 [ 8] 5 / [ 8] / (4 [ ] / ( [ j j j j l N l N j j j j j j j j c N l N j c l e e e e e e e e e e e e N l e c Π + + + + + H(- H(
Gain Reone of Butterworth Lowa Filter N4, 3dB cutoff frequency84hz.. 0-0 -0 Gain, db -30-40 -50-60 0 000 000 3000 4000 5000 6000 Frequency, Hz
Other Tye of Analog Lowa Filter Chebyhev Aroximation Tye Chebyhev Aroximation Tye Ellitic Aroximation Linear-Phae Aroximation Beel Filter
Other Analog Lowa, Higha, Banda and Bandto Thee are derived from imle frequency tranformation of the rototye lowa filter a follow:- Tye of Tranformation Lowa Higha Banda Bandto Tranformation ( ( + u u l + l l u l u Band edge frequencie of new filter l l,, u u
Deigning of IIR Digital Filter. Our objective i to realize the tranfer function H(z of IIR Digital Filter by aroximating the given frequency reone ecification. The ytem deigned hould be table and caual. H(z hould be a table real rational function. Deign of Digital Filter i the roce of deriving the tranfer function H(z.
Digital Filter Secification. Magnitude reone ecification. Phae (delay reone ecification. Deign of Digital Filter i the roce of deriving the tranfer function H(z through either aroximating to one of or both of the above given ecification. In ome ituation, deign can be via unit imule or te reone. We retrict to deign via magnitude reone ecification only.
Ideal Lowa Filter H ( e j Stoband Paband Stoband c c (cutoff frequency
Ideal Higha Filter H ( e j Paband Stoband Paband c c (cutoff frequency
Ideal Banda Filter H ( e j Paband Stoband Paband Stoband c c c c (cutoff frequency
Ideal Bandto Filter H ( e j Stoband Paband Stoband Paband c c c c (cutoff frequency
Digital Filter Secification The above ideal filter are not realizable becaue their imule reone are noncaual and of infinite length. To overcome thi, in ractice we tend to relax on the ideal har ecification and accet ome form of tolerance. In addition we allow ome form of tranition between the aband and the toband.
Practical Lowa Filter H ( e j Stoband +δ δ Paband δ Stoband Tranition band
Digital Filter Secification From the reviou lide : - In Paband, 0, we require H(e j, with an error ± δ, i.e.-δ In Stoband, with an error δ, i.e. H(e j H(e δ, j + δ,., we require. H(e j 0,
Terminology ued for Digital Lowa Filter Secification δ δ - aband edge frequency - toband edge frequency - eak rile vale in the aband - eak rile vale in the toband Peak aband rile : Minimum toband attenuation : Lo Function A( α -0log -0log 0 α 0 H ( e (-δ db. j -0log db. 0 ( δ db.
Digital Lowa Filter Normalized Secification H ( e a j + ε A Stoband Paband Tranition band Stoband
Terminology ued for Digital Lowa Filter Normalized Secification - - Maximum value of Paband rile denoted by. + ε (minimum value of aband magnitude. Maximum aband edge frequency toband edge frequency the magnitude in aband toband magnitude A..
Digital Filter Secification In ractice, aband edge frequency F toband edge frequency F the amling eriod T (Sec. The amling frequency F (Hz and T are given. T Hz. (Hz, Need to convert F F F T T F F F F T T & F F T, F T, to normalized angular edgeband frequencie : -
IIR Filter Deign Aroache Convert the digital filter ecification into analog rototye lowa filter ecification. Determine the analog lowa filter tranfer function H(. Tranform H( into the deired digital tranfer function H(z.
IIR Filter Deign Aroache Aly maing from -domain to z- domain. Proertie of the analog frequency reone are reerved. Imaginary axi in the -lane be maed onto the unit circle of the z-lane. A table analog tranfer function be maed into a table digital tranfer function.
IIR Digital Filter Deign Bilinear Tranformation Aly maing to analog tranfer function H( uing bilinear tranformation : - T being the te ize in numerical integration. The digital tranfer function : - H(z For imlicity we can take T ( z ( z T + ( ( + H(, z z, ( z T (+ z.
Bilinear Tranformation. / tan(, ng T Again taki.9.8... / tan( / tan( / co( / jin( ( ( ( ( e z,, r 0, When - axi to unit circle: Maingthe j. re z, ( ( / / / / / / j j σ σ + + + + Eqn T j T T e e e e e e T e e T j j j z z T j j j j j j j j
Bilinear Tranformation Maing. j Im(z σ - Re(z Maing of -lane into the z-lane
Maing of T Bilinear Tranformation angular analog frequencie to angular digital frequencie : - tan( /. For T, tan( / Maing i highly non-linear, oitive(negative imaginary axi in -lane maed into uer(lower half of unit circle in z-lane.
Ste in Deign of IIR Lowa Digital Filter Uing Bilinear Tranformation Ste in deign : - ( Normalizing to angular edgeband frequency ( & if ec given in Hz. ( Prewra * ( & to determine analog equivalent ( &. (3 Deign the equivalent analog lowa filter H a (. (4 Ue bilinear tranformation to deign H(z from H a (. * Nonlinear maing introduce ditortion in frequency axi known a frequency wraing. need to rewra the digital frequency to the analog frequency.
Examle IIR Lowa Butterworth Digital Filter Deign a lowa Butterworth digital filter with F eak aband rile minimum toband attenuation ( Normalizing to the angular bandedge frequncie : - F F 50Hz, F F ( Prewraing T T tan( / tan( / F (50 0.5, 000 (550 0.55, 000 & 550Hz, Samling frequency F α 0.5dB, α 5dB. the digital edge frequencie : - tan(0.5/ 0.4436 tan(0.55/.708496 T khz.,
Examle IIR Lowa Butterworth (3 Deigning the equavilent α 0log 0log + ε + ε Since α 5dB. 0log 0log A 0 0 0 0 0 + ε (/ A 5dB. 0 5/ 0 0 0.05 0.5/ 0 ε 0.085 A ( A 5. 0 3.6777 + ε 0.5 0.085 The invere tranition ratio 0.75 Digital Filter 0.5 0.05 5.63435 analog filter : - k.866809 The invere dicrimination ratio k A ε 5.84979
Examle IIR Lowa Butterworth Digital Filter The filter order N Chooe N 3, the next nearet interger number. Determine 3rd - order normalized lowa Butterworth tranfer function for H c an.4995( ( ( + ( c log log via : - H 0 a 0 (j 0.58848. + + (/ k (/ k /., + ε Denormalizing we have Ha ( H an ( 0.58848 (4 Uing bilinear tranform : - H(z H an ( 0.58848.6586997 + ( c N z + z. c i :
Deign of Higha, Banda & Bandto IIR Digital Filter. Firt Aroach ( Normalize all the critical bandedge frequencie if given in Hz. ( Prewar the ecified digital frequency ecification of the deired digital filter G(z to arrive at the frequency ecification of an analog filter H( of ame tye. (3 Convert the frequency ecification of H( into thoe of a rototye analog lowa filter Hl( uing aroriate frequency frequency tranformation. (4 Deign the analog lowa filter Hl( uing the analog filter deign. (5 Convert the tranfer function Hl( into H( uing the invere of the frequency tranformation ued in te (3. (6 Tranform the tranfer function H( uing the bilinear tranformation to arrive at the deired IIR digital filter tranfer function G(z.
Examle Deign of a Banda IIR Digital Filter Deired ecification of a banda IIR digital filter : - (a aband edge frequencie (b toband edge frequencie (c aband rile of db. f f 450Hz, 300Hz, f f 650Hz. 750Hz. (d minimum toband attenuation of 40dB. (e amling frequnecy F T khz.
Examle Deign of a Banda IIR Digital Filter Deign Ste. Normalization of critical bandedge frequencie : F F T F F F F T T F F T (450 0.45 000 (650 0.65 000 (300 0.3 000 (750 0.75 000 -
Examle Deign of a Banda IIR Digital Filter Deign Ste. Prewar the digital bandedge frequencie : - tan( tan( tan( tan( 0.45 tan( 0.8540807, 0.65 tan(.63857, 0.3 tan( 0.509554, 0.75 tan(.44356.
Examle Deign of a Banda IIR Digital Filter Deign Ste 3 Convertion of frequency ecification of analog banda filter to aroriate ecification of rototye analog lowa filter : - (Referring to frequency tranformation table of formula. ( On the imaginary axi : - + ( + (
( 0 The Examle Deign of a Banda IIR Digital Filter edge frequencie exhibit geometric ymmetry with reect to Making value of New adjuted 0.509554 0.63857 0.8540807.63857 will 0.8540807.44356 be adjuted o a 0.393733.44356 0.77777..393733..30035. the two toband 0.577303. 0.
Examle Deign of a Banda IIR Digital Filter Uing the rototye analog lowa filter with normalied aband edge frequency we arrive at the toband edge frequency of the rototye analog lowa filter : - ( ( + ± and from the frequency tranformation formula, ( 0.577303.393733 0.333788.393733 ± ( ± m.36767. 0.577303 0.77777 0.577303 0.77777
Examle Deign of a Banda IIR Digital Filter Deign Ste 4. Deign of rototyeanalog lowa filter. Secification of the rototyeanalog lowa filter i : - aband edge frequency toband edge frequency aband rile db. minimum toband attenuation 40dB. Ue thee information to get N the order of the filter and ( a er analog Butterworth filter deign revioulyor via M - Thi c the 3dB cutoff frequency.36767 will give you the rototye tranfer function H l (. File.
Examle Deign of a Banda IIR Digital Filter Deign Ste 5.(Can be done via M - File lb Aly the lowa - to - banda uing the H H b b ( Deign Ste 6.(can be done via Having obtain H for the analog banda filter H H b ( b ( finally aly the bilinear tranformation to get at ( z frequency tranformation table of H l ( by ubtituting z + z tranformation to H l b (, ( formulae to get + M - File bilinear change back the variable ( H to b (.
Examle of Simlified Deign of a Banda IIR Digital Filter Deired ecification of a banda IIR digital filter : - (a aband edge frequencie f 450Hz, f 650Hz. (b aband rile of db. (c rototye filter order. (d amling frequency F T khz.
Examle of Simlified Deign of a Banda IIR Digital Filter Deign Ste. Normalization of critical bandedge frequencie : F F T F F T (450 0.45 000 (650 0.65 000 -
Examle of Simlified Deign of a Banda IIR Digital Filter Deign Ste. Prewar the digital bandedge frequencie : - tan( tan( 0.45 tan( 0.8540807, 0.65 tan(.63857,
Examle of Simlified Deign of a Banda IIR Digital Filter Ski Deign Ste ( Aly the lowa - to - banda tranformation to the normalized H ( l H ( l H (i.e. ( 0.8540807.63857.393733. (.63857 0.8540807 0.77777. by uing H 3,4 +.393733 + 0.77777 and go to Deign Ste 5. ( the +.393733 0.77777 Since the rototye filter order, b +, + b H ( l frequency tranformation table of by ubtituting 0.77777 +.393733 + 0.77777 formulae. +.393733, 0.77777
Examle of Simlified Deign of a Banda IIR Digital Filter Deign Ste 6. Having obtain H for the analog banda filter H finally aly the bilinear tranformation to get at H By ubtituting H b b ( z ( We get H H ( 0.77777, +.393733 + 0.77777 b b ( z b ( ( change back the variable z z+ z z z into z + b (, z 0.77777( z + +.393733 + 0.77777( + z z + to
Deign of Higha, Banda & Bandto IIR Digital Filter. Second Aroach ( Normalize all the critical bandedge frequencie if given in Hz. ( Prewar the ecified digital frequency ecification of the deired digital filter G(z to arrive at the frequency ecification of an analog filter H( of ame tye. (3 Convert the frequency ecification of H( into thoe of a rototye analog lowa filter Hl( uing aroriate frequency frequency tranformation. (4 Deign the analog lowa filter Hl( uing the analog filter deign. (5 Convert the tranfer function Hl( into the tranfer function Gl(z of an IIR digital filter uing the bilinear tranformation. (6 Tranform the tranfer function Gl(z into the deired IIR digital filter tranfer function G(z uing the aroriate ectral tranformation from table 9. age 509 of Mitra..