Now identified: Noise sources Amplifier components and basic design How to achieve best signal to noise?

Transcription

1 Signal processing Now identified: Noise sources Amplifier components and basic design How to achieve best signal to noise? Possible constraints power consumption ability to provide power & extract heat, material for cooling layout of system (space, cables, ) signal rate eg. signal "pileup" vs E resolution C f Two methods Pulse shaping time invariant filter Pulse sampling time variant filter C det Bandwidth limiting filter g.hall@ic.ac.uk 1

2 Pulse shaping Preamplifier pulse shape is impractical step (exponential) with long duration short, sharp peak C f H(ω) Calculate signal and noise in t and f C det H preamp (ω) H filter (ω ) h(t) using results so far =1/jωC f Signal S out (ω) = S in (ω).h preamp (ω).h filter (ω) = S in (ω).h(ω) S out (t) = S in (t)*h preamp (t)*h filter (t) = S in (t)*h (t) convolution Noise σ = I n H(ω) df = (e n ω C +i n ) H(ω) df = e n C ωh (ω) df + i n H(ω) df σ = e n C [h'(t)] dt + i n [h(t)] dt provided e n & i n are white g.hall@ic.ac.uk

3 Equivalent Noise Charge Noise must be compared with signal - normalisation S out (t) = Qδ(u)*h(t-u)du = h(t) inject unit impulse ie. Q = 1 h(t) contains gain ENC = e n C [h'(t)] dt + i n [h(t)] dt normalised to signal of unit amplitude ENC = signal which produces output amplitude equal to r.m.s. noise desirable to measure in absolute units - e, coul, kev(si),.. How to inject an impulse of known size? V T C test Q test = C test V test = Ne measure V out for known Q test Pulser Transmission line Z = R 0 R 0 amplifier τ << T g.hall@ic.ac.uk 3

4 Example noise calculation Preamplifier + CR-RC bandpass filter = equal τ high & low pass filters h(t) = e 1 (t/τ)e -t /τ = e 1 (at)e -at = 1 at t = τ h'(t) = e 1 a(e -at -at e -at ) = e 1 ae -at (1-at) C f 1 [h(t)] = e a t e -at [h'(t)] = e a e -at (1-at+a t ) 1 = I 0 = 0 e -at dt = [e -at /(-a)] 0 = 1/(a) I 1 = 0 te -at dt = [te -at /(-a)] 0 + (1/a) 0 e -at dt = 1/(4a ) t = I = 0 t e -at dt = [t e -at /(-a)] 0 + (1/a) 0 te -at dt = 1/(4a 3 ) so 0 [h(t)] dt = e a /(4a 3 ) = e - τ/4 0 [h'(t)] dt = e a [1/(a) -a/(4a ) +a /(4a 3 )] = e a/4 = e /(4τ) ENC = e 8 4kTR s C tot τ + eiτ 1/f noise ignored Factor (1/) from requiring frequencies from 0 to g.hall@ic.ac.uk 4

5 Improved time invariant filters Optimal filter = infinite exponential cusp h(t) = exp(- t /τ opt ) gives equal contributions from series and parallel sources, if... τ opt = C tot.r s.r p R s and R p are equivalent noise resistances impractical in real systems R / R / Practical filters - wide range of possibilities! C CR-RC n type n ~1- RC = low pass filter CR= high pass easy to implement R 1 C 1 C 1 CR-RC n n ~5-7 semi-gaussian symmetric pulses active filters based on op-amp configurations R 1 examples R C + - R R(k-1) g.hall@ic.ac.uk 5

6 Noise after pulse shaping General result is ENC = αe n C /τ + β i n τ + γc α,β depend on pulse shape calculate in t or f γ : 1/f - can be computed in f only logenc series 1/f parallel a minimum noise can be achieved with a given shaping time constant chosen depending on magnitudes of noise sources logτ Useful point of comparison: CR-RC bandpass filter ENC = e 4kTR s C tot + eiτ + 4A f C tot 8 τ only 36% worse than theoretical optimal filter g.hall@ic.ac.uk 6

7 Some numerical values An approximate numerical value ENC [e ] 4 R s [kω]c tot [pf] τ[µs] τ[µs] using CR-RC filter, ignoring 1/f noise ie I = 1nA τ = 1µs ENC p 100e R s = 10Ω C = 10pF τ = 1µs ENC s 4e g.hall@ic.ac.uk 7

8 Time variant filters Alternative to pulse shaping filters based on summation Sample & hold method initially switches S 0 B 1 B open, S 1 S closed switch S 0 is Reset V out = output from charge sensitive preamplifier S 0 S 1 V 1 B 1 V out V C B S C 1 A open S 1 : preserves V out on C 1 after time t open S : preserves V out on C then, close B 1 and B : output A = V 1 - V reset preamp later need to know when signal will arrive! V V 1 t 1 t Reset preamp t Switched capacitor easy to implement convenient for MOS technology t g.hall@ic.ac.uk 8

9 Time variant filters Can perform same process with delay line x1 V 1 A = V 1 -V delay t x(-1) V delay line Double Correlated Sampling delay t << RC of preamplifier V 1 t V How to analyse noise performance of time variant systems? g.hall@ic.ac.uk 9

10 Weighting function What output is produced at T m by impulse at time t? consider all t - defines weighting function T m w(t) t 1 t t 3 t t- Tm Time invariant filter w(t) is mirror image of h(t) Time variant filter usually need to think carefully Noise calculation ENC = e n C [w' (t)] dt + i n [w (t)] dt g.hall@ic.ac.uk 10

11 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) * g gate (t) (ignoring t reflection) t = T gate T gate << τ T gate >> τ T gate = 5 w(t) = h(t) new, wider weighting function can can change change filtering and and increase or or decrease noise noise g.hall@ic.ac.uk 11

12 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 - / and / <x> = x.p(x).dx/ p(x).dx σ = <x > = x.p(x).dx / p(x).dx p(x).dx = / - / dx = [x] / - / = 1 p(x) 0 x = 0 σ = / 1 x x.p(x).dx = - / / x.dx = [x 3 /3] - / / = 3 /4 so σ = /1 ie statistical noise which is proportional to digitisation unit g.hall@ic.ac.uk 1

13 Level crossing statistics Binary counting systems have noise! fluctuations cause threshold crossing Rate of zero (level) crossing f Z proportional to spectral density of noise w(f) f Z = General proof is complicated: imagine noise at several distinct frequencies: f 0, f 1, w(f) = δ(f-f 0 ) + δ(f-f 1 ) f w(f)df w(f)df 0 1/ f Z = 0 f δ(f f 0 )df δ(f f 0 )df 0 1/ + 0 f δ(f f 1 )df δ(f f 1 )df 0 1/... = f 0 + f 1 + factor because crossings can occur in both directions g.hall@ic.ac.uk 13

14 Positive level crossing rate If we know level crossing rate f V by measurement or calculation Rate of crossing, in positive direction, of level V is f v = f v Z exp σ because noise has gaussian distribution of amplitudes factor 1/ because one direction so improvement for any other threshold X, compared to threshold Y f X = exp (X Y ) f Y σ g.hall@ic.ac.uk 14

15 Time measurements and noise When did signal cross threshold? noise causes jitter V t = σ noise /(dv/dt) compromise between bandwidth (increased dv/dt) noise (decreased bandwidth) t t limits systems where preamplifier pulse used to generate trigger eg x-ray detection Preamplifier Pulse shaper ADC to DAQ typical preamp response V = V max (1-e -t/τ rise ) so t σ noise τ rise / V max t << τ C Detector Fast amplifier Gate Discriminator Coincidence logic g.hall@ic.ac.uk 15

16 Time and amplitude Time walk due to amplitude variation solutions: constant fraction discriminator not simple circuit but easily possible t walk or zero-crossing discriminator differentiate pulse.0 measure time when output crosses zero invariant pulse shape 1.0 always peaks at same time h(t) h'(t) penalty differentiation also adds noise! 0.0 more high frequencies passed 0 4 t g.hall@ic.ac.uk 16