# System on a Chip. Prof. Dr. Michael Kraft

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1 System on a Chip Prof. Dr. Michael Kraft

2 Lecture 4: Filters Filters General Theory Continuous Time Filters

3 Background Filters are used to separate signals in the frequency domain, e.g. remove noise, tune to a radio station, etc 5 types of filter T(jw) Low pass High pass Band pass Band stop/reject All pass T(jw) T(jw) T(jw) T(jw) w w w w w

4 Ideal Filter Brick-wall LP filter Brick-wall BP filter T(jw) T(jw) 1 1 w p Passband Stopband w w p1 Lower Stopband Passband w p2 Brick-wall filters do not exist in reality Real filters can approximate brick-wall filters as close as required by the filter specification Upper Stopband w

5 Real LP Filter T(jw) [db] 0 A max maximum deviation from 0dB; Bandpass ripple Stopband Transitionband Passband A min minimum attenuation w p w s w p /w s : Measure for filter sharpness; filter selectivity w Parameters required for filter synthesis: A max, w p, A min, w s

6 Filter Types Using Biquads T ( s) K a2s b s a1s a b s b 1 0 0

7 Biquadratic LP Transfer Function Magnitude Response Phase Response Low-pass biquad TF T ( s) K s 2 0 w 2 0 ( w / Q) s w Diagrams normalized to w 0 = K = 1 - Asymptotic fall is -40 db/dec

8 Biquad Block Diagram T( s) V V LP IN ( s) ( s) K s 2 0 w 2 0 ( w / Q) s w 2 0 (K either pos. or neg.) -1/Q V in K V HP -w 0 /s -w 0 /s V BP V LP Universal Active Filter: realizes LP, HP, and BP

9 Tow-Thomas Biquad Realization R 2 V in R 3 w 02 = 1/R 3 R 4 C 1 C C 1 R 1 C 2 R 4 R 5 R 5 V BP - + V LP - + V LP Q = Sqrt(R 12 C 1 /R 2 R 4 C 2 ) K = -R 2 /R 3 (R 5 arbitrarily chosen) V LP : Inverting LP Filter V LP : Non-inverting LP Filter

10 N-th Order Filter ) ( ) )( ( ) ( ) )( ( ) ( N M p s p s p s z s z s z s K s T Number of poles determines order zeros are obviously placed in stopband for stability: M N; N-M zeros at w = for stability: Re{pi} < 0 no general optimisation algorithms known ) ( ) )( ( 1 ) ( 2 1 N p s p s p s K s T Special Case: all zeros at w = ; all pole filter

11 Imag Axis Example: 5-th Order Filter Poles: P[1..5] = 1.0e+002 *[ i i i i ] Pole-zero map Passband Zeros: Z[1..4] = 1.0e+003 *[ i i i i] Real Axis T( s) s s s s s s s

12 Phase (deg); Magnitude (db) T(S) Example: 5-th Order Filter T( s) s s s s s s s Bode Diagrams Step Response From: U(1) Z 1/2 Z 3/4 Amplitude To: Y(1) Frequency (rad/sec) Time (sec.)

13 Butterworth LP Filter - Make T(jw) so that: 1 2 T ( jw) 2N w 1 w0 - N: Filter order - All pole filter w 0 : T(jw) has dropped by 3 db Normalized Butterworth Polynomials: For w 0 =1: N Denominator of T(s) 1 (s+1) 2 (s s+1) 3 (s+1)(s 2 +s+1) 4 (s s+1)(s s+1) 5 (s+1)(s s+1)(s s+1)

14 BW-LP Frequency Response T(jw) (db) Frequency (rad/sec) maximally flat in passband i.e. the first 2N-1 derivatives of T(jw) are 0 at w=0 T(jw) monotonically falling not steepest roll-off

15 BW-LP Design w 0 p 1 a /10 10 max 1 2 N w a/db a min N amin / log amax / w p 2log ws a max wp ws w/rad/s Design a filter so that in the passband T(jw) has fallen not more than by a max and in the stopband the minimum attenuation is a min Find w 0 and N

16 BW Pole Locations Poles located on a circle around the origin k = 90 (2k + N - 1)/N k = 1,2,,2N If N is odd, then there is a pole at = 0, if N is even there are poles at = 90 /N Poles are separated by = 180 /N

17 Chebychev LP Filter Make T(jw) so that: 2 1 T( jw) e ( w) C N 1 C N ( w) cos( N cos ( w)) for w 1 1 ( w) cosh( N cosh ( w)) for w 1 C N N: Filter order All pole filter Normalized for w 0 = 1 e: design parameter; determines ripple Chebychev Polynomials; Denominator of T(s): N e ; (0.5 db ripple) e ; (1 db ripple) 1 (s+2.863) (s+1.965) 2 (s s+1.516) (s s+1.103) 3 (s+0.626)(s s+1.142) (s+0.494)(s s+0.994) 4 (s s+1.064)(s s+0.356) (s s+0.987)(s s+0.279) 5 (s+0.362)(s s+1.036)(s s+0.477) (s+0.289)(s s+0.988)(s s+0.429)

18 Magnitude (db) Magnitude (db) CC-LP Frequency Response Properties: Ripples in Bandpass between w = 0 and w = 1/(1+e 2 ) 0.5 H(j1) = 1/(1+e 2 ) 0.5 for all N H(0) = 1 for N odd = 1/(1+e 2 ) 0.5 for N even steeper roll-off than Butterworth Implementation: see Butterworth example N = 5; e = 0.5 db e = 3 db Bode Diagrams N = 6; e = 0.5 db Frequency (rad/sec) e = 3 db 1 10 Bode Diagrams 1 10 Frequency (rad/sec)

19 Imag Axis Chebychev Pole Locations Minor axis: b a = sinh(1/n sinh -1 (1/e)) a Major axis: b = cosh(1/n cosh -1 (1/e)) N = 6; e = 1 db Real Axis Poles located on an ellipse around the origin; narrow ellipse means poles closer to imag. axis larger ripples. Wider ellipse small ripples; approaches Butterworth filter s k = -sinh(1/n sinh -1 (1/ e sin((2k-1)p/2n) w k = -cosh(1/n sinh -1 (1/ e cos((2k-1)p/2n)

20 Motivation Switched Capacitor Filters Pro: Accurate transfer-functions Pro: High linearity, good noise performance Con: Limited in speed Clock rate must be greater than twice the signal frequency Con: Requires anti-aliasing filters Continuous-time filters Con: Moderate transfer-function accuracy (requires tuning circuitry) Con: Moderate linearity Pro: High-speed Pro: Good noise performance Required building blocks: Integrators, summers and gain stages Allow to realise any rational function, hence any integrated continuoustime filter Any rational transfer function with real-valued coefficients may be factored into first- and second-order terms

21 First Order Filter block diagram of a first-order continuous-time filter first-order continuous-time filter requires one integrator, one summer, and up to three gain elements In general: One integrator is required for each pole in an analog filter

22 Second Order Filter block diagram of a second-order continuous-time filter Two integrators are required to realise the two poles For stability: w 0 /Q must be positive One integrator must have feedback around it, hence the integrator is lossy A large feedback coefficient w 0 /Q results in a very lossy integrator, hence the Q is low Q<1/2: both poles are real; Q>1/2: poles are complex-conjugate pairs

23 G m -C Integrators Use a transconductor (or OPA) to build an integrator: i o = G m v i Output current is linearly related to input voltage Output impedance is ideally infinite OTA (operational transconductance amplifier) has a high G m value but is not usually linear

24 Multiple Input G m -C Integrators

25 Example What Gm is needed for an integrator having a unity gain frequency of w ti = 20 MHz when C=2 pf? Or equivalently: G m =1/3.98kW This is related to the unity gain frequency by:

26 Fully Differential Integrators Use a single capacitor between differential outputs Requires some sort of common-mode feedback to set output common-mode voltage Needs some extra caps for compensating common mode feedback loop

27 Fully Differential Integrators Use two grounded capacitors Still requires common-mode feedback but compensation caps for common-mode feedback can be the same grounded capacitors

28 Fully Differential Integrators Integrated capacitors have top and bottom plate parasitic capacitances To maintain symmetry, usually 2 parallel caps used as shown above Note that parasitic capacitance affects time-constant and cause non-linearity

29 G m -C Opamp Integrator Use an extra Opamp to improve linearity and noise performance Also known as a Miller Integrator The gain of extra Opamp reduces the effect of parasitic capacitances Cross coupling of output wires to maintain positive integration coefficient

30 G m -C Opamp Integrator Advantages Effect of parasitic caps reduced by opamp gain more accurate time-constant and better linearity Less sensitive to noise since transconductor output is low impedance (due to opamp feedback) cell drives virtual Gnd output-impedance of G m cell can be lower and smaller voltage swing needed Disadvantages Lower operating speed because it now relies on feedback Larger power dissipation Larger silicon area

31 First Order Filter General first-order transfer-function: Built with a single integrator and two feed-ins branches w 0 sets the pole frequency

32 First Order Filter Can show that the transfer function is given by (using a current equation at the output node): Equating with the block diagram transfer function:

33 Fully-Differential First-Order Filter Same equations as single-ended case but cap sizes doubled Can realize k 1 <0 by cross-coupling wires at C x

34 Example Find fully-diff values when dc gain = 0.5, a pole at 20 MHz and a zero at 40MHz. Assume C A =2pF K 1 =0.25, k 0 =2p 10 7, w 0 =4p 10 7 So:

35 Second Order Filter Block diagram: see lecture on switched capacitor circuits Modified to have positive integrators

36 Differential Second Order Filter (Biquad)

37 Differential Second Order Filter Transfer function: (Biquad) Note that there is a restriction on the high-frequency gain coefficient k 2 as in the first-order case Note that G m3 sets the damping of this biquad G m1 and G m2 form two integrators with unity-gain frequencies of w 0 /s

38 Example Find values for a bandpass filter with a centre frequency of 20 MHz, a Q value of 5, and a centre frequency gain of 1 Assume C A = C B = 2 pf where G =1 Is the gain at the center frequency

39 Example Since w 0 = 2p 20MHz and Q = 5, we find: Since k 0 and k 2 are zero, we have C x = C ma = 0 The transconductance values are:

40 CMOS Tranconductors A large variety of methods Best approach depends on application Two main classifications: triode or active transistor based Triode vs. Active Triode based tends to have better linearity Active tend to have faster speed for the same operating current

41 Triode Tranconductors A large variety of methods Best approach depends on application Two main classifications: triode or active transistor based Triode vs. Active Triode based tends to have better linearity Active tend to have faster speed for the same operating current

42 Triode Tranconductors Recall n-channel triode equation Conditions to remain in triode or equivalently: Above models are only reasonably accurate Higher order terms are not modelled Not nearly as accurate as exponential model in BJTs Use fully-differential architectures to reduce even order distortion terms also improves common mode noise rejection The third order term dominates

43 Fixed Bias Triode Tranconductors Use a small v DS voltage so v 2 DS term goes to zero Drain current is approximately linear with applied v DS. Transistor in triode becomes a linear resistor Resulting in: Can use a triode transistor where a resistor would normally be used resistance value is tunable

44 Fixed Bias Triode Tranconductors [Welland, 1994] Q9 is in the triode region transconductor has a variable transconductance value that can be adjusted by changing the value of V gs9 Moderate linearity

45 Fixed Bias Triode Tranconductors [Kwan, 1991] Alternative approach with lower complexity and p-channel inputs transconductor has a variable transconductance value that can be adjusted by changing the value of V gs9

46 Fixed Bias Triode Tranconductors Circuit can be easily made with multiple scaled output currents Multiple outputs allow filters to be realized using fewer transconductors

47 Biquads Using Multiple Outputs Can make use of multiple outputs to build a biquad filter scale extra outputs to desired ratio Reduces the number of transconductors saves power and die area Above circuit makes use of Miller integrators

48 Varying-Bias Triode Transconductor [Krummenacher, 1988] Linearizes MOSFET differential stage Transistors primarily in triode region

49 Varying-Bias Triode Transconductor gates of Q 3 and Q 4 connected to the differential input (and not to bias voltage) Q 3 and Q 4 undergo varying bias conditions to improve linearity It can be shown that With Note, G m is proportional to square-root of as opposed to linear relation for a BJT transconductor Transconductance can be tuned by changing bias current I i

50 Drain-Source Fixed-Bias Transconductor If v DS is kept constant, then i D varies linearly with v GS Model is too simple, neglecting second order effects such as velocity saturation, mobility degradation Possible implementation using fully differential architecture

51 Drain-Source Fixed-Bias Transconductor Can realize around 50 db linearity (not much better since model is not that accurate) Requires a fully-differential structure to cancel even-order terms V C sets v DS voltage Requires a non-zero common-mode voltage on input Note that the transconductance is proportional to v DS For v DS small the bias current I 1 is also approximately proportional to v DS

52 Alternative: MOSTFET-C Filters Gm-C filters are most commonly used but MOSFET-C have advantages in BiCMOS for low power applications MOSFET-C filters similar to active-rc filters but resistors replaced with MOS transistors in triode Generally slower than Gm-C filters since opamps capable of driving resistive loads required Rely on Miller integrators Two main types 2 transistors or 4 transistors

53 Alternative: MOSTFET-C Filters Gm-C filters are most commonly used but MOSFET-C have advantages in BiCMOS for low power applications MOSFET-C filters similar to active-rc filters but resistors replaced with MOS transistors in triode Knowledge and architecture of active RC filters can be transferred Generally slower than Gm-C filters since Opamps capable of driving resistive loads required Rely on Miller integrators Two main types 2 transistors or 4 transistors

54 Two Transistor Integrators Banu 1983

55 Two Transistor Integrators For resistor integrator can be shown If negative integration is required cross-couple wires For MOSFET-C integrator, assuming transistors are biased in triode region, the small-signal resistance is given by: Therefore, the differential output of the MOSFET-C integrator is: With:

56 General MOSFET-C Biquad Filter Equivalent active RC half circuit v 0 s v i s = C 1 C B s 2 + G 2 C B s + G 1G 3 C A C B s 2 + G 5 C B s + G 3G 4 C A C B

57 Tuning Circuitry Tuning can often be the MOST difficult part of a continuous-time integrated filter design Tuning required for continuous-time integrated filters to account for capacitance and transconductance variations 30 percent timeconstant variations Must account for process, temperature, aging, etc. While absolute tolerances are high, ratio of two like components can be matched to under 1 percent Note that SC filters do not need tuning as their transfer-function accuracy set by ratio of capacitors and a clock-frequency

58 Indirect Tuning Most common method build an extra transconductor and tune it Same control signal is sent to filter s transconductors which are scaled versions of tuned extra Indirect since actual filter s output is not measured

59 Constant Transconductance G m = 1 R ext Can tune Gm to off-chip resistance and rely on capacitor absolute tolerance to be around 10 percent

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