Signal Processing for TPCs in High Energy Physics ( Part I) Beijing, 9-10 January 2008

1. Signal Processing for TPCs in High Energy Physics(Part I)Beijing, 9-10 January 2008 • Outline • Introduction to signal processing in HEP • Detector signal processing model • Electronic signal processing • Preamplifier and Shaper • Analogue to Digital Conversion • Digital signal processing Luciano Musa - CERN

2. Signal Processing in High Energy Physics Introduction • Signal Processing is a way of converting an obscure signal into useful information • Signal processing includes signal formation due to a particle passage within a detector, signal amplification, signal shaping (filtering) and readout • The basic goal is to extract the desired and pertinent information from the obscuring factors (e.g. noise, pile-up) • The two quantities of greatest importance to be extracted from detector signals are: • amplitude: energy, nature of the particle, localization • time of occurrence: localization, nature of the particle

3. Signal Processing in High Energy Physics Means of Detection • Each detection method has to extract some energy from the particle to be detected • Nearly all detection methods (Cerenkov and Transition Radiation Detector being an exception) make use of ionization or excitation • Charged particles: ionization and excitation is produced directly by the interaction of the particle electromagnetic field with the electrons of the detection medium • A typical particle energy (today’s experiments) is of the order of few 100MeV to GeV, while the energy loss can be below the MeV level. This is an example of a nondestructive method for detection of charged particles. • All neutral particles must first undergo some process that transfers all or part of their energy to charged particles. The detection method is destructive

4. Signal Processing in High Energy Physics Detection of Ionization (1/2) • In most ionization detectors the total ionization charge is collected using an externally applied electrical field • Sometimes an amplification process by avalanche formation in a high electrical field is used. Examples of detectors are: • Proportional chamber (MWPC, GEM, mMegas); • Time Projection Chamber • Liquid-argon chamber; • Semiconductor detector. • All of them provide a certain amount of charge onto an output electrode • The electrode represents a certain capacitance • For signal-processing point of view these detectors are capacitive sources, i.e. their output impedance is dominated by the capacitance

5. Signal Processing in High Energy Physics Detection of Ionization (2/2) • This common feature of all detectors for particle physics allows a rather unified approach to signal processing • Despite of common features among various detectors used in high-energy physics, great differences exist among them • The typical charge at the detector output can differ by six orders of magnitude • The output capacitances can differ by the same factor • Signal dynamics • Pulse repetition rate

6. x V’A(P) QA,el = -q = -q d V’A x V’A(P) QA,ion = q = q d V’A dQA,el q dx = - i = d dt dt   Signal Induced by a Moving Charge Example I Parallel Plate Ion Chamber Anode (A) Applying Green’s Theorem i E d Vb P(x) x A constant induced current flows in the external circuit Cathode (C)

7. Signal Induced by a Moving Charge Cylindrical Proportional Chamber Example II Avalanche region (amplification) Charged particle primary ionization cathode + + + + + + + + + + + - - - - - - - - - - - - - gas anode Electron cloude electron– ion pair E ≠ 0 Ion cloude i(t) = i0 / (1+t/t0)

8. series white noise series 1/f noise e2W=a e2f=c/|f| i2f=d·f i2W=b If s(t) = d(t) A(Q/(Cd+Ci)) parallel white noise parallel f noise Detector Signal Processing Model noiseless preamplifier signal processor A Q ·s(t) Cd Ci noise power spectral density Ax[a+b/w(Cd+Ci)2] The detector is modeled as a current source, delivering a current pulse with time profile s(t) and charge Q, proportional to the energy released, across the parallel combination of the detector capacitance Cd and the preamplifier input capacitance Ci.

9. Electronic Signal Processing Signal Processor F(f) U(f) h(f) F(f) U(f) f f0 f Noise floor f0 Improved Signal/Noise Ratio f f0 Example of signal filtering - the figure shows a “typical” case of noise filtering In particle physics, the detector signals have very often a very large frequency spectrum The filter (shaper) provides a limitation in bandwidth, and the output signal shape is different with respect to the input signal shape.

10. Electronic Signal Processing F(f) U(f) h(f) f(t) u(t) f f Noise floor f0 Improved Signal/Noise Ratio f f0 • The output signal shape is determined, for each application, by the following parameters: • Input signal shape (characteristic of detector) • Filter (amplifier-shaper) characteristic • The output signal shape is chosen such to satisfy the application requirements: • Time measurement • Amplitude measurement • Pile-up reduction • Optimized Signal-to-noise ratio

11. A noiseless Signal Processing for Charge Measurement OPTIMUM PROCESSOR signal processor T(s) f(t) Cd+Ci n(w) i(t) System Transfer function AT(s) / (s  (Cd+Ci)) System Impulse response function vs(t) = ʆ-1AT(s) / (s  (Cd+Ci)) Output root mean square noise [vN2]1/2 Optimum Processor maximize r = {Q A/(Cd+Ci) MAX ʆ-1(I(s) T(s) / s) } / [vN2]1/2

12. Electronic Signal Processing R Vout C Vin Low-pass (RC) filter Example RC=0.5 s=jw Integrators-transfer function H(s) = 1/(1+RCs) Impulse rsponse function Step function response |h(s)| t f Log-Log scale

13. Electronic Signal Processing C Vin Vout R High-pass (CR) filter Example RC=0.5 s=jw Differentiator time function H(s) = RCs/(1+RCs) |H(s)| Step function response

14. Electronic Signal Processing Combining one low-pass (RC) and one high-pass (CR) filter : R C 1 Vout Vin C R Example RC=0.5 s=jw CR-RC s-transfer function h(s) = RCs/(1+RCs)2 CR-RC time function |h(s)| Step function response f Log-Log scale

15. Electronic Signal Processing Combining N low-pass (RC) and one high-pass (CR) filter : R C 1 Vout Vin C R N times Example RC=0.5, n=5 s=jw CR-RC4s-transfer function H(s) = RCs/(1+RCs)5 CR-RC4 time function |H(s)| Step function response f Log-Log scale

16. Preamplifier - Shaper O I Preamplifier Shaper u(t) n(t) d(t) • What are the functions of the preamplifier and the shaper (in an ideal world) ? • Preamplifier - An ideal integrator : it detects an input charge burst Q d(t).The output is a voltage step Q/Cf•u-1(t). It has a large signal gain such that the noise of the subsequent stage (shaper) is negligible. • Shaper - A filter with : characteristics fixed to give a predefined output signal shape, and rejection of (input) noise components outside of the useful output signal band.

17. Cf Qi Zi=  Cd Vi V0 Preamplifier - Shaper Active Integrator (“charge-sensitive amplifier”) Inverting Voltage Amplifier dVo / dVi = -A  vo = - A vi Input Impedance =  (no signal current flows into amplifier input) Voltage difference across Cf: vf = (A+1)vi Charge deposited on Cf: Qf = Cfvf = Cf(A+1)vi Qi=Qf (since Zi = ) Effective Input capacitance (A>>1) Gain

18. Cf Qi Zi=  Cd Vi V0 Preamplifier - Shaper Active Integrator (“charge-sensitive amplifier”) Inverting Voltage Amplifier dVo / dVi = -A  vo = - A vi Input Impedance =  (no signal current flows into amplifier input) Qi is the charge flowing into the preamplifier …. but some charge remains on Cdet What fraction of the signal charge is measured? (if Ci >> Cdet) A=103 Cdet = 10pF Ci = 1nF Qi / Qs = 0.99 Cf = 1pF

19. Preamplifier - Shaper Preamplifier Shaper I O t t f f n(t) d(t) Ideal Integrator TRANSFER FUNCTION CR-RC shaper Output signal for an “ideal” input charge t

20. f Preamplifier - Shaper Preamplifier Shaper I O t t f f n(t) d(t) Ideal Integrator TRANSFER FUNCTION CR-RC4 shaper Output signal for an “ideal” input charge t

21. Preamplifier - Shaper Basic scheme of a Preamplifier-Shaper structure Cf n Integrators Diff Vout Cd T0 T0 T0 Semi-Gaussian Shaper Vout(s) = Q/sCf . [sT0/1+ sT0].[A/(1+ sT0)]n Vout(t) = [QAn nn /Cf n!].[t/T0s]n.e-nt/Ts Peaking time Ts = nT0 Vout(peak) = QAn nn /Cf n!en Vout (normalized to 1 ) vs. n Vout peak vs. n

22. Preamplifier - Shaper O I Preamplifier Shaper z(t) d(t) Non Ideal Integrator CR_RC shaper u(t) T1= 10  R  C

23. Preamplifier - Shaper O I Preamplifier Shaper z(t) d(t) Non Ideal Integrator CR_RC shaper pz cancellation u(t) Pole-zero cancellation

24. Preamplifier - Shaper Basic scheme of a Preamplifier-Shaper structure with p-z cancellation Cf Diff N Integrators Rp Vout Cd Tp T0 T0 Semi-Gaussian Shaper By adjusting Tp such that Tp = Tf, we obtain the same shape as with a perfect integrator Ts = nT0

25. Analogue vs. Digital Filters Advantages of Digital Filters Finite-duration impulse responses are achievable Time-varying filters (even self-adaptive) are realized without any special component by simply programming a different set of numbers in the filter Certain realization problems, such as negative element values, and practical problems, such as inconveniently large components at low frequencies, do not arise Programmability Greater accuracy is achieved No sensitive to environmental conditions (e.g. temperature, supply voltages, etc.) Disadvantages of Digital Filters Power Limited by A/D speed and resolution Real-time Digital Filters

26. 1 0100 1 0111… 1 1 analog input Output Code 1111 t t 0000 Analogue to Digital Converter Example - 4 bits A/D Converter D0 ADC D1 Analog Digital Output 4-bit D2 Input D3 Full scale amplitude reconstructed signal LSB=Full scale/2N Sampling Clock 62.5 mV for 1V/4bits 16 possible output codes

27. Analogue to Digital Converter Time and Amplitude Resolution • Limiting Factors in the Time and Amplitude resolution • Time: Aperture time and Clock Jitter • Amplitude: Noise floor signal jitter clock

28. Analogue to Digital Converter Accuracy – Speed ENOB 22 Heisenberg 20 1Kohm thermal 18 16 1ps jitter 14 12 10 8 6 4 2 Sampling Sample/s 0 10K 100K 1M 10M 100M 1G 10G 100G

29. The Uniform Sampling Theorem Introduced by Shannon in 1948 (original idea by Nyquist in 1928) It establishes the theoretical maximum sampling interval for complete signal reconstruction The theorem holds (rigorously) only for physically unrealizable band-limited signals Band-limited signals are a good approximation of many signal encountered in practice The Uniform Sampling Theorem

30. |F(w)| p(w) (low pass filter) - wc wc w  0 The Uniform Sampling Theorem Fourier Spectrum of a band-limited function f(t) Theorem f(t) is uniquely determined by its values at uniform time intervals that are 1/2fc seconds apart Cardinal function Nyquist frequency 2fc

31. The z-Transform The Laplace transform reduces constant-coefficient linear differential equations to linear algebraic equations Continuous-Time Domain The z-transform reduces constant-coefficient linear difference equations to linear algebraic equations Discrete-Time Domain • Four tools for the analysis, synthesis, and understanding of linear time-invariant electronic systems, either continuous or discrete time: • Fourier transform • Laplace transform • Hilbert transform • z-transform

32. fs(t) f(t) . . . t  0 T 2T 3T 4T 5T The z-Transform The ideal sampler circuit f(t) fs(t) Fs(s) F(s) T f(t) is a real-life causal signal  f(t) = 0 for t < 0 We consider f(t) as modulating d(t) d(t) is a train of periodic impulse functions of area T

33. The z-Transform The sampling process regarded as a modulation process LAPLACE TRANSFORM Unfortunately Fs(s) contains exponentials, hence is not algebraic in s. Z-transform or

34. Sampled time functions that exponentially decrease with increasing time Oscillating sampled time functions Z=1, constant or increasing functions depending on the multiplicity of the pole Sampled time functions that exponentially increase with increasing time Mapping the s-Plane into the z-Plane Im Im jw s-plane z-plane s 0 1 0 Re Re

35. |F(jw)| w  0 |Fs(jw)| -ws 0 ws w  Frequency-Domain Characteristics Function not band-limited EFFECT OF SAMPLING Aliasing or Foldover

36. The z-Transform |F(jw)| Band-limited function -wc wc w  0 ws  2wc |Fs(jw)| wc -2ws -ws -wc 0 ws 2ws w  |Fs(jw)| ws< 2wc wc -2ws -ws -wc 0 ws 2ws w  -3ws 3ws

37. ANALOG WORLD Readout & Recording antialias filter reconstr. filter S&H A&D DSP D/A input samples f(nT) output samples u(nT) DIGITAL FILTER Difference Equation Key element of a sampled-data system ANALOG WORLD Difference Equation The operation of the filter is described by a difference equation that relates u(nT) as a function of the present input sample f(nT) and any number of past input and output samples Recursion formula

38. 5 50 4 40 Input function Ouput function 3 30 2 20 … 1 10 1 3 5 7 1 2 3 4 2 4 0 0 t/T  t/T  6 -10 -20 Difference Equation Example of a Difference Equation Computation First-order difference equation (m=1, q=3) u(nT) = 3f(nT) – 2f(nT – T) + 6f(nT - 2T) + 2f(nT – 3T) – u(nT-T) fs(t) us(t)

39. + + L0 L0 T T K1 Difference Equation • The recursiveness of the Difference Equation suggest for its implementation: • program a General Purpose Computer • program a Digital Signal Processor • Hardwired Digital Filter Digital Filter Simulation of a First-Order Difference Equation Recursion formula u(nT) Adder f(nT) Multiplier Delay (Register)

40. Lq L2 L1 L0 + + fs(t) Z-1 Z-1 Z-1 Z-1 us(t) -K1 -Km -K2 -Kq Digital Networks Digital filter network values are easily obtained, often by inspection Difference equation System Function Canonic Form Feed-forward path m delays IIR FILTER Feed-back path

41. L0 L1 L2 Lq + + Z-1 Z-1 Z-1 Digital Networks Digital filter network values are easily obtained, often by inspection Difference equation System Function If all Ki’s are 0 Non-recursive fs(t) us(t) FIR FILTER q delays