COURSE OUTLINE. Introduction Signals and Noise Filtering: OPF2 Optimum Filtering2 Sensors and associated electronics. Sensors, Signals and Noise 1

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1 ensors, ignals and Noise 1 OURE OUTINE Introduction ignals and Noise Filtering: OPF Optimum Filtering ensors and associated electronics ignal Recovery, 017/018 Optimum filter

2 Optimum Filtering for High-Impedae ensors High impedae sensors and low-noise preamplifiers Noise whitening filter Matched filter Optimum filtering for measuring the charge of pulse signals Optimum Filtering with Finite Readout Time Practical approximations of the optimum filtering Appendix 1 : Noise whitening filter in case of finite resistae ignal Recovery, 017/018 Optimum filter

3 3 High Impedae ensors and ow-noise Preamplifiers ignal Recovery, 017/018 Optimum filter

4 High-Impedae ensors 4 et us consider sensors that are seen by the circuits connected to their terminals as generators of current signals with high internal impedae (typically a small capacitae with a high resistae R in parallel) Typical examples are: p-i-n jution photodiodes and other photodetectors (Ds, vacuum tube photodiodes, etc.); piezoelectric Force ensors in quartz or other piezoelectric ceramic materials They have internal noise sources (e.g. shot current noise of a jution reverse current) modeled by a current noise generator in parallel to the signal generator ensor ensor Equivalent ircuit ensor Equivalent ircuit approximation with very high R I R id I i ensor ignal ensor Noise ignal Recovery, 017/018 Optimum filter

5 High-Impedae ensors and ow-noise Preamplifiers 5 va va ia ia R ia ia ia RiA Preamplifier equivalent circuit Approximation with very high R ia R ia true physical resistae between the input terminals (NOT the dynamic input resistae modified by the feedback in the amplifier; e.g. not the low dynamic resistae of the virtual ground of an operational amplifier) Besides shot noise of bias currents, the ia iludes Johnson resistor noise of R ia ir 4kT R i The current noise directly faces the sensor current signal I if R ia is small the ir is overwhelming (e.g. with R ia 50 Ω it is )* 18 p A Hz ) and other components of )3 are much lower (about 1 p A Hz or lower ) olusion: for low-noise operation of high-impedae sensors, it is mandatory to employ a preamplifier with high input resistae R ia ignal Recovery, 017/018 Optimum filter

6 High-Impedae ensors and ow-noise Preamplifiers 6 Equivalent circuit of high-impedae ensor and Preamplifier (approximation valid for very high sensor resistae R 5 ) I δ(t) v i RiA Voltage pulse ( ) 1( t) y t urrent pulse to be measured At the preamplifier output: + ia total capacitae load v va voltage noise generator (wideband white spectrum) i id + ia current noise generator (wideband white spectrum) The voltage noise spectrum n has two components, it is NOT white n ω i ( ω) V + The voltage signal is a step with amplitude / ignal Recovery, 017/018 Optimum filter

7 Noise pectrum at Preamplifier Output 7 rossing of the component defines ω Noise-orner angular frequey log n v ω i ω T 1/ω Noise-orner time constant i v v n ω i ( ω) V + T 1 v ω i T and ω are fundamental parameters of the optimum filter: we will see that T rules the duration of the filter weighting and ω the filter bandlimit ω ω i logω We define the Noise orner resistae R v i so that T R with 8 a few nv Hz and ) ranging from a few 0,1 to 0,01 pa Hz R ranges from tens to hundreds of kohms with from 0,1 pf to a few pf T ranges from a few nanoseconds to some hundreds of nanoseconds ignal Recovery, 017/018 Optimum filter

8 8 Noise whitening filter ignal Recovery, 017/018 Optimum filter

9 Noise-Whitening Filter 9 The noise spectrum has a pole at ω p 0 a zero at ω z ω 1/T log n 1 ω T i n( ω) V 1 V + V ω V ω T ω T n V ω T ω T The noise whitening filter H nw must v cael the pole with a zero at ω 0 cael the zero with a pole at ω ω 1/T ω i ω logω H nw ( ω) ω T 1 +ω It is a simple high-pass filter H nw ( ω) jωrww 1 + jωr w w with R w w T T w R w 1 log H nw ω ω T 1 T H nw ω T 1 +ω T logω ignal Recovery, 017/018 Optimum filter

10 Action of the Noise-Whitening Filter 10 imple high-pass R filter w Hnw ( s) R w srww 1 + sr w w with R T w w I v i RiA ( ) ω ( ) ( ) n Noise ω Whitening Filter y t z ( t) ( ) H nw H ω n nw v it makes white the noise at its output and changes the signal into a short exponential pulse with time-constant T y t t NW Filter t 1 exp T ( ) 1( ) z ( ) ( ) t t st H 1 + st 1 Y ( s) ( ) Z s T s aels pole at s0 and replaces it by pole at s - 1/T 1 st ignal Recovery, 017/018 Optimum filter

11 ignal at the output of the Noise-Whitening Filter 11 I s v i RiA st y H nw ( t ) z ( t) 1 + st NW Filter Input (current) Preamp Output (voltage) NW Filter Output (voltage) δ pulse tep pulse Exponential pulse s ( ) δ ( ) I t t s y t t T ( ) 1( ) z ( t) ( t) ( ) Y ( s) Z ( s) I s 1 s t 1 exp T T 1 st The matched filter has to be tailored to the signal at the whitening filter output, i.e. it must have weighting fution exponential with time constant T ignal Recovery, 017/018 Optimum filter

12 1 Matched Filter ignal Recovery, 017/018 Optimum filter

13 The Matched Filter completes the Optimum Filtering 13 OPTIMUM FITER I R v i Ri n ( ω) Noise Whitening Filter H nw v Matched Filter w m The case with finite load resistae R is treated in Appendix 1, showing that the whitening filter is different but the output signal produced is the same as with R à t z ( t) 1( t) exp T Therefore, the matched filter is the same in the two cases and gives the same result w M 1 t 1( t) exp T T zs( α) wm( α) dα T 1 T o w m( ) d N opt ( ) v v v wm α dα η α α ignal Recovery, 017/018 Optimum filter

14 Optimum Filtering 14 At the output of the optimum filter (i.e. of the matched filter) we have ignal Noise /N 1 α 1 s z ( α) w ( α) dα exp dα o 0 s m 0 T T 1 α no v kww( 0) v exp dα T T T so 1 T η o n v o 0 v The optimum filter theory specifies what is the best /N physically obtainable and the kind of filter required for attaining it The optimum filter can be implemented in reality, but the filter design turns out to be quite complex Furthermore, the optimum filter takes infinite time (after the signal onset) to complete its action, which is not acceptable in practice It is possible, however, to consider and evaluate fairly simple filters that approximate the optimum filter and closely approach its performae ignal Recovery, 017/018 Optimum filter

15 Optimized Noise harge 15? We want measure the signal charge, so we consider the noise in terms of charge q > We have T η o q q ( ) v no T v no? Recalling T we can express q > in terms of the main parameters R v i of the sensor and preamplifier q no v i q? > is the charge pulse for which /N 1. It represents the minimum measurable pulse min and the minimum differee between two distinguishable pulses. The very small size of q? > is better appreciated if expressed in number of electron charges (NB electron charge e 1, ) q v no i 1/ 1/ Neo 78 [ pf] v nv Hz i pa Hz e e ignal Recovery, 017/018 Optimum filter

16 Optimized Noise harge Examples 16 It is instructive to evaluate and compare typical values of the optimized noise charge in typical cases of PIN photodiodes coupled to high-impedae preamplifiers. a) PIN with wide sensitive area ( 1 mm diameter ) and discrete-component preamplifier with moderately low-noise. Fairly high capacitae of PIN & connections ( A 10pF) D? and not-so-low preamp noise (typical 8 4 nv Hz D? ; ) 1 pa Hz ) set a fairly high limit electrons N eo 1800 b) PIN with reduced sensitive area ( 100 μm diameter) and discrete-component preamplifier with low-noise. Reduced capacitae of PIN & connections ( A 1pF) D? and reduced preamp noise (typical 8 nv Hz D? ; ) 0,1 pa Hz ) bring a strong reduction electrons N eo 10 c) PIN with small sensitive area ( 0 μm diameter ) integrated with a very low-noise preamplifier. apacitae minimization by monolithic integration ( A 0,1pF ) D? and very low preamp noise ( 8 nv Hz D? ; ) 0,01 pa Hz ) bring further progress electrons N eo 10 ignal Recovery, 017/018 Optimum filter

17 17 Practical approximations of the optimum filtering ignal Recovery, 017/018 Optimum filter

18 R integrator Approximation of the Matched Filter 18 The whitening filter is simple and easily and exactly implemented. For completing the optimum filter it is sufficient to find out how to approximate the matched filter. The features of the matched filter weighting fution observed in time and in frequey point out that it is a low-pass filter A simple R integrator (single-pole low-pass filter) can be an approximation of the matched filter. With R T its δ-response h F (t) is identical to the weighting fution w M (t) of the matched filter. The R weighting w F (t) has the same shape as w M (t) of the matched filter, but it s not fully correct because it is reversed in time! 1 T J 1 t hf ( t) 1( t) exp TF T F δ-response with T F T t ( ) ( ) w t h t t F F P weighting fution for readout at t P t t P Noise filtering is equal to the matched filter, sie it is unaffected by time-inversion; the output is white noise with band-limit set by a simple pole with time-constant T F. ignal filtering is different from the matched filter, sie it is modified by time-inversion ignal Recovery, 017/018 Optimum filter

19 R integrator Approximation of the Matched Filter 19 ensor output current Preamplifier output Whitening filter output δ-response of R integrator Output of R integrator A A 1 T J ( ) δ ( ) I t t s y t t ( ) 1( ) t z ( t) 1( t) exp T 1 t hf ( t) 1( t) exp TF T F t t t with T F T t t uf ( t) z * hf exp T T t Weighting fution of R integrator ( ) ( ) w t h t t F F P t peaking time t P T t ignal Recovery, 017/018 Optimum filter

20 Filtering system with R as approx matched filter 0 ensor output current signal Output signal of the filtering system with RI as approx matched filter Overall weighting fution acting on the sensor output s ( ) δ ( ) I t t ( ) ( ) h t u t of ( ) ( ) w t h t t F of of P t t t 0 t P t Note that: The weight is maximum at t 0, i.e. in correspondee to the sensor pulse Weighting is limited to t P on the t > 0 side Weighting is not limited on the t < 0 side (see for comparison the optimum filtering in slide 6) ignal Recovery, 017/018 Optimum filter

21 R integrator Approximation compared to the Optimum Filter 1 The R output signal waveform is ignal peak value (at t T ) t t uf ( t) exp T T 1 sf uf ( T ) e v Noise nf v khh ( 0) T sf 1 T /N η F n e v F omparing the R approximation with the ideal optimum filter system we see that s s 0,736 s e F o o n n F o ηf ηo 0,736 η the /N is lower o e the performae of the filter system with R approximation of matched filter is about 7% worse than the absolute optimum. Note that the loss is due to bad exploitation of the signal the signal is lower the noise is equal ignal Recovery, 017/018 Optimum filter

22 R integrator Approximation compared to Finite-Readout Optimum Filter The filter system with R approximation of the matched filter has finite readout time t R T (the output pulse peak amplitude is measured at t T ). The ideal optimum filter has infinite readout time. A more objective assessment of the R approximation is obtained by taking as referee the optimum filter with the constraint of finite readout time t R T. The optimum filter with the constraint of finite readout time t R gives η ( t ) Ro R o t R η 1 exp T in our case with t R T it is ηro ( T ) ηo 0,99 In colusion ηf ηo 0,736 ηo 0,79 ηro ( T ) e the performae of the RI approximation is about 1% worse than the optimum filter with the constraint of finite readout time t R T This performae is remarkably lower than the optimum, but the R is just a crude approximation: better results can be obtained with more sophisticated filter design ignal Recovery, 017/018 Optimum filter

23 3 Appendix 1 Noise whitening filter: case with finite load resistae ignal Recovery, 017/018 Optimum filter

24 High-Impedae ensor and Preamplifier 4 Equivalent circuit of a high-impedae ensor and Preamplifier configuration with finite load resistae R A urrent pulse I δ(t) v R i RiA ong exponential voltage pulse R total resistae of sensor and load T R time constant of the load At the preamplifier output: The voltage noise spectrum n has two components and is NOT white R n( ω) V + i V + i ω R ω T The output voltage signal is an exponential pulse with long time constant T A R A A ( ) ( t) y t R t 1 exp T ignal Recovery, 017/018 Optimum filter

25 Noise pectrum at Preamplifier Output 5 recalling + n V i R ω ( ) T R v i ω T 1 T T + T T + T n V V T ω T T ω T T T log n T + R v i v + n V i R ω + T T T v The noise spectrum has a pole at a zero at ω 1 T T ω T z n V T z ω T 1 1 T 1 ωz + Tz T T ω ω z logω With high load resistae R, namely with the same as with infinite load resistae R R T T > 10 1 ω ω z T, the zero is practically ignal Recovery, 017/018 Optimum filter

26 Noise pectrum at Preamplifier Output 6 with T ω T n V T T z z ω T z T T T + The noise whitening filter H nw must T and T R log n + R v i R n V + i ω R cael the pole with a zero at ω 1/T 1 1 cael the zero with a pole at ω p Tz T It s a quasi-high-pass R with finite D attenuation R T y R w R w z R w RR w w T z + Rw log v ω ω H nw ω z ω p logω logω 1 H nw ( ω) T jωt T jωt T jωt T jωt z z ignal Recovery, 017/018 Optimum filter

27 Action of the Noise-Whitening Filter 7 uasi-high-pass R filter I R R Hnw ( s) y z w R w v i Ri ( ω ) ( ω) ( ω) n with R Noise Whitening Filter H nw Tz T st st RR w w T z T + Rw y ( t ) z ( t) z and R w H n nw v T a) it makes white the noise at its output and b) it changes the signal into a short exponential pulse with time-constant T. Note that this «post-whitening» signal is the same found with R à t y ( t) 1( t) exp T Y ( s) T 1 st T T NW Filter st st aels pole at s -1/T and replaces it by pole at s -1/T t z ( t) 1( t) exp T 1 Z ( s) T st ignal Recovery, 017/018 Optimum filter

28 ignal at the Noise-Whitening Filter Output 8 v y ( t ) z ( t) T T st st I s R i Ri NW Filter Input (current) Preamp Output (voltage) NW Filter Output (voltage) δ pulse ong exponential pulse hort exponential pulse s ( ) δ ( ) I t t s ( ) Y ( s) Z ( s) I s t y ( t) 1( t) exp T T 1 st T t z ( t) 1( t) exp T T 1 st The matched filter has to be tailored to the signal at the whitening filter output, i.e. it must have weighting fution exponential with time constant T ignal Recovery, 017/018 Optimum filter