HIGHER-ORDER FILTERS Cacade of Biquad Filter Follow the Leader Feedbac Filter (FLF) ELEN 6 (ESS) Than for ome of the material to David Hernandez Garduño
CASCADE FILTER DESIGN N H ( ) Π H ( ) H ( ) H ( ) H i i i N ( ) V i ω 0, Q, G ω 0, Q, G ω 0 N G,Q, N N I the order in which ection are connected important? Mathematically i irrelevant On one hand, to avoid lo of dynamic and to avoid ignal clipping in the high-q ection ue: Q < Q < Q N On the other to minimize noie, ue Q > Q > Q N What i the optimal ordering to yield the bet S/N?
H i ( ) H ( ) n i Π t i a i i i a i ωoi Q ( ) i a ω oi oi How do we aign pole-zero pair? n i even Propoition. Minimize the maximum value of d i where d G log, G i and m i are the maximum and minimum gain in the ω ω ω. Then determine the optimal i i m frequency range of i interet, L equence of the biquad ection, uch that again the maximum number d i i minimized. The final tep i to aign the gain contant ( i ) of the biquad. To yield optimal dynamic range i.e., maxvoi ( jω) maxvo n H ( jω) maxv ( jω), Propoition. To jointly optimize ignal gain and noie gain. Noie by itelf i minimized by aigning the larget poible gain to the firt-tage, then the econd with the larget gain among the other tage and o on. out
Realization of High Order Tranfer Function (N>) Cacade of nd order ection (one t order ection if N i odd) Leapfrog Follow-The-Leader Cacade FLF Leap-Frog Senitivity High Medium Low Eay to Tune Medium Eay Difficult 4
Primary Reonator Bloc -F -F V in BP BP BP V out It provide compenatory internal interaction between the different filter ection through coupling the biquad building bloc. F i incorporated in BP. 5
-F n -F -F. V a. in V o. V. V T () T ( T n () ) B B n V n B 6 B o Follow-the-Leader Feedbac Filter Topology H n ( ) H ( ) V V V V n in out in n a Π Tj ( ) j n F Π Tj ( ) j n Bo j j a n F Π Tj ( ) j [ ] B Π T ( ) V out
Deign Quetion: How do we obtain the feedbac coefficient F and F? How do we determine the pecification for each biquadratic ection? Q ω o Gain 7
Deign Procedure Start with the Lowpa equivalent ytem. -F -F V in K 0 K K K V out However Elliptic Filter need finite zero in their lowpa equivalent tranfer function. 8
Implementation of Finite Zero by the Summation Technique -F -F V in K 0 V 0 K V V V K K B 0 B B B 9 V out
EXAMPLE To deign a 6th order bandpa elliptic filter uing the Follow-the-Leader (FLF) architecture. The pecification are: Specification Value Order 6 Paband Paband Ripple Attenuation.9MHz to.mhz 0.dB 0dB at 0.6MHz 0
Deign Procedure Let for now K K K and () Applying Maon rule, the complete tranfer function i given by: () T ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 F T F T B T B T B T B K V V H in out ( ) ( ) ( ) ( ) ( ) 0 0 F F B B B B K
Deign Procedure From Matlab or Fieta, we can obtain the lowpa prototype tranfer function of the deired 6 th Order Elliptic Filter: () Equating the denominator of equation () and (), we obtain the following et of equation from which we can olve for, F, and F. (4) Alo from equation () and () (5) 0 0 0 0 ) ( a a a b b b b b a m H ( ) 0 a F F a F a 0 0 0 b a K m
Deign Procedure To obtain the ummation coefficient, we equate the numerator of equation () and (). If H() i a bandpa: B 0 b 0. Then, we obtain the following et of equation from which we can determine B, B and B. B 0 b B b 0 B B ( B B B ) b0 b (6)
Deigning for Maximum Dynamic Range We need to ditribute the gain of each ection T(), i.e. K, K and K uch that we maximize the Dynamic Range. The maximum dynamic range will be obtained if the ignal pectra at the output of all ection have equal maxima, i.e. 4 V out,max V,max V,max V,max V 0,max
Maximizing Dynamic Range To mae V,max V out,max where K0 K0 K0q (7) V q V out,max prior to caling prior to caling We alo need to adjut the ummation coefficient to eep the overall gain: If we aume a flat pectrum for the Bi Bi B q i,max V out, max Max H ( ω) input, i.e. Vin(ω) (8) (9) (0) 5 V,max Max H( ω)
6 Gain Value Determination Where () To obtain K, K and K : () 0 0 ) ( ) ( ) ( a a a K V V H in,,. ) ( ) ( 4 i for H Max H Max K i i i ω ω ω ω ω ω
7 Feedbac Coefficient The feedbac coefficient need to be readjuted to eep the ame loop gain: () The ummation coefficient alo need to be readjuted again: (4) K K F F F K K K F F F q B B B q K B B B q K K B B B
Summary of Deign Procedure Obtain from Matlab or Fieta the lowpa prototype for the deired filter. From equation (4), (5) and (6), obtain K 0, the feedbac and the ummation coefficient. To maximize dynamic range, obtain q uing equation (8). Recalculate K 0 uing equation (7). Calculate the gain of each ection, i.e. K, K and K uing equation (). Recalculate the feedbac and ummation coefficient uing equation () and (4). Finally, apply a lowpa-to-bandpa tranformation to obtain the deired bandpa filter pecification: ω 0 π Q Q0 where Q i the quality factor of the overall filter and Q 0 i that required for each biquad ection. Note: A Matlab program wa written to automate the deign procedure for an arbitrary filter pecification of order N. f L f U 8
Summary of Reult For the required pecification, the following value were obtained: Feedbac Coefficient F 0.657640 F 0.7545 Feedforward Coefficient B0 0 B 0.6979 B -0.657 B.984 Gain for the input and each biquad tage Center frequency and Q 0 of each biquad tage K0 0.604488 K.49 K.65 K f 0.9975MHz Q 0 5.7 9
Simulation Reult Sytem Level The complete filter wa imulated in Cadence at a ytem-level. The reult are hown below: Ripple 0.dB 0 Magnitude Repone Phae Repone
Tranitor Level Implementation To implement each biquadratic ection, a twointegrator loop biquad OTA-C filter wa ued. Advantage with repect to Active-RC: Eay Tunability by changing the bia current of the OTA. (Active-RC need the ue of varactor). Lower Power Conumption and Smaller Area. Diadvantage with repect to Active-RC: Smaller Dynamic Range Poorer Linearity
Tranitor Level Implementation of each Biquad Section Vin OUT gm gm gm - R C - gm4 OUT - OUT R C - OUT Vout H ( ) g C g m4 C m g m C g C m
Thi image cannot currently be diplayed. Deign of Lole Integrator The lole integrator wa deigned to have unity gain at f 0.9975MHz. H ( ω) g m ω C The following pecification are needed if a 5% variation in Q i allowed: g m µa 76.5 C 0 pf V Exce Phae : φ E Q Q Q a.5 0 rad 0.086 DC Q Gain : AV 60 55. 58dB Q Q a
Deign of Lole Integrator Due to the relatively high DC gain required for the OTA, a folded-cacode topology wa ued: VDD M4 M M M0 Vbp M Ibia Vin M M Vin- Vbn Vout M6 M7 M9 M8 M M4 M5 VSS Tranconductance DC Gain GBW Bia Current Power Conumption Active Area 76.5µA/V 64.dB 6MHz 40µA 79µW 77µm 4
Simulation Reult of the Lole Integrator The exce phae without any compenation wa.7. Paive exce phae compenation wa ued R55Ω. Magnitude Repone Phae Repone 5
Deign of Biquadratic Bandpa Filter To reue the deigned OTA: g m g m 76.5 µ A V C C 0 pf C g m 4 ω0. 87 µ A Q 0 V Tranconductance g m depend on K i g K m i m4 For demontration purpoe, g m g m4, i.e. K g 6
Deign of Biquadratic Bandpa Filter Due to the relatively mall tranconductance required, ource degeneration wa ued. Alo, a PMOS differential pair wa more uitable. VDD M9 M8 MA MB M4 M5 R M6 Vbp M7 Vin M M Vin- Vout Vbn Ibia M0 M M4 M M VSS Tranconductance DC Gain GBW Bia Current Power Conumption Active Area.87µA/V 6.5dB 6.MHz 4µA 77.µW 80µm 7
Simulation Reult of the Biquadratic Bandpa Filter Frequency Repone Step Repone 8
Summation Node To complete the tranitor-level deign, we need two ummation node: -F -F V in K 0 V 0 K V V V K K B 0 B B B 9 V out
Summation Node The ummation node can be implemented with OTA in the following configuration: Vin K0 gm0 V B gm0 OUT OUT - - V - F gm0 V - B gm0 OUT OUT V - OUT F gm0 - gm0 V OUT B gm0 - gm0 OUT V0 - OUT Vout Summation Node for the Feedbac Path Summation Node at the Output If g m0 i choen large enough, the output reitance of each OTA doe not need to be very high. Exce Phae of OTA can be a concern. 0
Summation Node Due to the deired low exce phae introduced by the OTA, it i more convenient to ue a imple differential pair. VDD M M Vout Ibia Vin M M Vin- M9 M VSS Tranconductance Bia Current Power Conumption Active Area 00µA/V 40µA 64µW 69µm
Simulation Reult of the Complete FLF Filter (Tranitor v. Sytem Level) Ripple ~0.dB Magnitude Repone Phae Repone
Simulation Reult of the Complete FLF Filter (Tranitor v. Sytem Level) Tranient Repone to a Sinewave Step Repone
Summary of Reult Specification Paband Paband Ripple Attenuation Power Conumption Value.9MHz to.mhz ~0.dB 40dB at 0.6MHz 8.5mW Active Area,44µm Total Area ~,549µm 4
Problem to be conidered Voltage Swing: The allowable input voltage wing i only 00mV. A mall voltage wing i expected, ince the OTA have a mall linear range limited by ±VDSAT of the input tranitor (in cae no linearization technique i ued, uch a ource degeneration or other). Neverthele, 00mV i too mall and i baically becaue the OTA with g m 76.5µA/V ue input tranitor with a mall VDSAT and no linearization technique i being ued. I need to redeign thee OTA to increae the linear range. Bia Networ: To deign the bia networ for the folded-cacode OTA capable of effectively tracing change of VT due to proce variation. Senitivity and Tunability: To characterize the complete filter in term of enitivity and tunability. Layout 5
Another tructure for higher-order filter that can be obtained from the FLF i the invere FLF by uing a tranpoed ignal-flow graph. -F -F n V n a -F. V n- V n- T () T ( T n () ) V o B. B n V in B B o -F n -F V in B n -F V V o V n- V n- T T ( n () T () ) a V n B IFLF B 6 B o
Reference [] Sedra, Bracett. Filter Theory and Deign: Active and Paive. Matrix Serie in Circuit and Sytem. pp. 589-659. [] Deliyanni, Sun, Fidler. Continuou-Time Active Filter Deign. CRC Pre 999. pp. 5-80. [] G. Hurtig, III. The Primary Reonator Bloc Technique of Filter Synthei Proc. Int. Filter Sympoium, p.84, 97. [4] Barbargire. Explicit Deign of General High-Order FLF OTA- C Filter. Electronic Letter. 5th Augut 999, Vol. 5, No. 6, pp. 89-90. [5] Jie Wu, Ezz I. El-Mary. Synthei of Follow-the-Leader Feedbac Log-Domain Filter. IEEE 998. 0-780-5008-/98. pp. 8-84. 7
Reference (cont.) [6] G. S. Mochytz, Linear Integrated Networ-Deign, Van Notrand, Reinhold, New Yor 975 [7] R. Schaumann, M.S. Ghaui, and K.R. Laer, Deign of Analog Filter: Paive, Active RC and Switched Capacitor, Prentice Hall, Englewood Cliff, 990. [8] A.S. Sedra, and P.O. Bracett, Filter Theory and Deign: Active and Paive, Matrix, Portland 978. [9] W.M. Snelgrove, and A.S. Sedra, Optimization of Dynamic Range in Cacade Active Filter, Proc. IEEE ISCAS, pp. 5-55, 978. 8