Noise What is NOISE A definition Any unwanted

Noise • What is NOISE? A definition: Any unwanted signal obscuring signal to be observed two main origins • EXTRINSIC NOISE examples. . . pickup from external sources unwanted feedback RF interference from system or elsewhere, power supply fluctuations ground currents small voltage differences => currents can couple into system may be hard to distinguish from genuine signals but AVOIDABLE Assembly & connections, especially to ground, are important • INTRINSIC NOISE Fundamental property of detector or amplifying electronics Can’t be eliminated but can be MINIMISED g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 1 14/09/2021

Origins of noise in amplifying systems • 1. Thermal noise Quantum-statistical phenomenon Charge carriers in constant thermal motion macroscopic fluctuations in electrical state of system • 2. Shot noise Random fluctuations in DC current flow originates in quantisation of charge non-continuous current • 3. 1/f noise Characteristic of many physical systems least well understood noise source commonly associated with interface states in MOS electronics g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 2 14/09/2021

Thermal noise • Einstein (1906) , Johnson, Nyquist (1928) Mean voltage <v> = 0 Variance <v 2> = 4 k. T. R. ∆f s(v) = √<v 2> = 1. 3 10 -10 (R. ∆f)1/2 e. g. R = 1 MΩ ∆f = 1 Hz gaussian distribution ∆f = observing bandwidth of fluctuations in v volts at 300 K s(v) = 0. 13µV Noise power = 4 k. T. ∆f independent of R & q independent of f - WHITE • Circuit representations Noise generator + noiseless resistance R • Spectral densities mean square noise voltage or current per unit frequency interval w. V(f) = 4 k. TR (voltage) g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ w. I(f) = 4 k. T/R (current) 3 14/09/2021

Shot noise • Poisson fluctuations of charge carrier number eg arrival of charges at electrode in system - induce charges on electrode quantised in amplitude and time • Examples electrons/holes crossing potential barrier in diode or transistor electron flow in vacuum tube < in 2> = 2 q. I. ∆f WHITE I = DC current (NB notation e = q) gaussian distribution of fluctuations in i g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 4 14/09/2021

1/f noise • White noise sources frequently dominate in many real systems however frequency dependent noise is also common • 1/f noise is a generic term for a wide range of phenomena, possibly not always related Power spectral density w(f) = Af/fn n ~ 0. 8 -1. 5 typical • Most important for MOS FET devices, often dominates but can also arise in other circumstances e. g. dielectrics, … p. MOS transistor noise spectrum g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 5 14/09/2021

Campbell’s theorem • Most amplifying systems designed to be linear S(t) = S 1(t) + S 2(t) +S 3(t) + … • Impulse response h(t) = response to d • Transfer function H(w) = vout(w)/vin(w) = ∫-∞∞ h(t). e-jw t dt ie impulse response h(t) and transfer function H(w) are Fourier pair • In a linear system, if random impulses occur at rate n average response <v> = n ∫-∞tobs h(t)dt variance s 2 = n ∫-∞tobs [h 2(t)]dt so s 2 = n ∫-∞∞ h 2(t) dt g. hall@ic. ac. uk i. e. sum all pulses preceding time, tobs, of observation = n ∫-∞∞|H(w)|2 df www. hep. ph. ic. ac. uk/~hallg/ 6 14/09/2021

Amplifier systems for spectroscopy • typical application - precise measurements of x-ray or gamma-ray energies • pre-amplifier first stage of amplification • main amplifier - adds gain and provides bandwidth limiting ADC - analogue to digital conversion - signal amplitude to binary number • fast amplifier and logic start ADC ("gate") and flag interesting "events" to DAQ system - most signals arrive randomly in time. Other logic required to maximise chance of "good" event, eg second detector g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 7 14/09/2021

"Rules" of low noise amplifier systems • Combine uncorrelated noise sources in quadrature e 2 tot = e 12 + e 22 + e 32 + … + in 2 R 2 +. . . follows from Campbell's theorem consider as combinations of gaussian distributions • First stage of amplifier dominates noise originates at input transistor is most important - defines noise in most cases • Noise is independent of amplifier gain or input impedance so noise can be referred to input • In real systems both are approximations - but normally good ones so often sufficient to focus on input device g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 8 14/09/2021

Amplifiers - dominance of input stage • Amplifier systems (and amplifiers!) usually consist of several stages impractical to put all gain at one location - power, heating, material, size, . . . • Calculate signal and noise at output Sout = G 1. G 2. G 3 Sin for 3 stage system, but can easily extend to N (eout)2= G 12. G 22. G 32 e 12 + G 22. G 32 e 22 + G 32 e 32 (eout /Sout) 2 = (e 12 + e 22/G 12 + e 32 /G 12. G 22)/Sin 2 • Desirable to maximise gain at input stage eg stage 1 boosts signal enough for transmission down cable and should be large enough that environmental noise is not significant g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 9 14/09/2021

Amplifiers - location of noise sources • Normal to partition noise sources not fundamental to calculations but can simplify! • Parallel noise sources appear as currents at input detector leakage current bias resistors feedback resistor Series noise sources appear as voltage at input … transistor gate noise device series resistance microstrip or transistor load resistor in amplifier. . . g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 10 14/09/2021

Amplifiers - reference to input assume: signal source and associated impedance noise sources amplifier with gain & input impedance • noise at output Eno 2 = A 2 {e 2 n. Z 2 in/(Zin + Rs)2 + i 2 n. R 2 s. Z 2 in/(Zin + Rs)2 } • transfer function K = Vout/Vin = A. Vsig. Zin /(Zin + Rs)Vsig = AZin/(Zin + Rs) • noise at input Eni 2= Eno 2/K 2 => E 2 ni = e 2 n + i 2 n. R 2 s no Zin or A dependence easy to show analogous result I 2 ni = i 2 n + e 2 n/R 2 s choice is for convenience in most detector systems, there is a current signal source and a parallel capacitance • then the spectral distribution of noise at the input is affected I 2 ni = i 2 n + e 2 nw 2 C 2 g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ no longer white 11 14/09/2021

Preamplifier types • Current sensitive - common for photodiode signals vout ≈ -iin. Rf • signals follow input current, ie fast response but not lowest noise • Charge sensitive amplifier Ideally, simple integrator with Cf but need means to discharge capacitor - large Rf • Simple integrator vout ≈ -Q/Cf • with feedback resistor Rf vout(t) ≈ -(Q/Cf)exp(-t/t) g. hall@ic. ac. uk t = Rf Cf www. hep. ph. ic. ac. uk/~hallg/ 12 14/09/2021

Signal processing • Knowing Noise sources Amplifier components and basic design How to achieve “best” signal to noise? • Possible constraints power consumption layout of system (space, cables, …) signal rate eg. signal "pileup" vs E resolution • Two methods Pulse shaping time invariant filter Pulse sampling time variant filter g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 13 14/09/2021

Noise after pulse shaping • General result is ENC 2 = aen 2 C 2/t + b in 2 t + g. C 2 a, b depend on pulse shape calculate in t or f g : 1/f - can be computed in f only a minimum noise can be achieved with a given shaping time constant chosen depending on magnitudes of noise sources • Useful point of comparison: CR-RC bandpass filter only 36% worse than theoretical optimal filter g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 14 14/09/2021

Equivalent Noise Charge • ENC = signal which produces output amplitude equal to r. m. s. noise desirable to measure in absolute units - e, coul, ke. V(Si), . . • Noise must be compared with known signal to calibrate system • How to inject an impulse of known size? Qtest = Ctest. Vtest = Ne measure Vout for known Qtest g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 15 14/09/2021

Some numerical values • An approximate numerical value • using CR-RC filter, ignoring 1/f noise ie I = 1 n. A t = 1µs ENCp ≈ 100 e Rs = 10Ω g. hall@ic. ac. uk C = 10 p. F t = 1µs www. hep. ph. ic. ac. uk/~hallg/ ENCs ≈ 24 e 16 14/09/2021

Time variant filters • Alternative to pulse shaping filters based on summation • Sample & hold method Vout initially switches S 0 B 1 B 2 open, S 1 S 2 closed switch S 0 is Reset Vout = output from charge sensitive preamplifier open S 1 : preserves Vout on C 1 after time ∆t open S 2 : preserves Vout on C 2 then, close B 1 and B 2: output A = V 1 - V 2 reset preamp later • need to know when signal will arrive! Switched capacitor easy to implement convenient for MOS technology g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 17 14/09/2021

Integrating Analogue to Digital Converter (ADC) • Integrate signal during application of gate - another time variant filter convert charge to digital number • = convolution of pulse shape with gate so w(t) = h(t) * ggate(t) (ignoring t reflection) • Tgate << t Tgate >> t w(t) = h(t) Tgate = 5 t new, wider weighting function can change filtering and increase or decrease noise g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 18 14/09/2021

Digitisation noise • Eventually need to convert signal to a number quantisation (rounding) of number = noise source the more precise the digitisation, the smaller the noise • After digitisation all that is known is that signal was between -∆/2 and ∆/2 <x> = ∫x. p(x). dx/∫p(x). dx s 2 = <x 2> = ∫x 2. p(x). dx /∫p(x). dx = ∫-∆/2∆/2 dx = [x] ∫x 2. p(x). dx = ∫-∆/2∆/2 x 2. dx = [x 3/3] so -∆/2 =∆ -∆/2 = 2∆3/24 s 2 = ∆2/12 • ie statistical noise which is proportional to digitisation unit g. hall@ic. ac. uk www. hep. ph. ic. ac. uk/~hallg/ 19 14/09/2021

Time measurements and noise • When did signal cross threshold ? noise causes “jitter” Dt = snoise/(d. V/dt) • compromise between bandwidth (increased d. V/dt) noise (decreased bandwidth) • limits systems where preamplifier pulse used to generate trigger eg x-ray detection • typical preamp response V = Vmax(1 -e-t/trise) so ∆t ≈ snoisetrise/ Vmax g. hall@ic. ac. uk t << t www. hep. ph. ic. ac. uk/~hallg/ 20 14/09/2021