Sensors, Signals and Noise

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1 Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1

2 Noise Descripion Noise Waveforms and Samples Saisics of Noise Samples and Probabiliy Disribuion (PD) Complee Descripion of Noise wih Probabiliy Disribuions Basic Descripion of Noise wih he 2 order Momens of PD Auocorrelaion Funcion of Noise Power Specrum of Noise and for hose who rus only analyical demonsraions APPENDIX: Exchanging he order of Time-Averaging and Ensemble-Averaging in he definiion of Power Specrum rv 2017/02/08 2

3 Se-Up of Sensor Measuremens SIGNAL NOISE Signal Noise PREAMPLIFIER SENSOR FILTERING (or FRONT-END) METER Noise of Sensor SIGNIFICANT Noise of Preamp or Fron-end circuis SIGNIFICANT Noise of Filering circuis NEGLIGIBLE (hopefully!) Noise of Meer circuis NEGLIGIBLE (hopefully!) rv 2017/02/08 3

4 Noise Waveforms and Samples rv 2017/02/08 4

5 Noise waveforms 50μs/div) Whie Noise specrum S = consan Random-Walk Noise specrum Flicker Noise specrum rv 2017/02/08 5

6 Noise Waveform Ensemble Se of idenical noise sources (many idenical amplifiers or resisors or oher) x() waveforms of noise x( 1 ) ampliude sample a ime 1 on each waveform rv 2017/02/08 6

7 Saisics of Noise Samples and Probabiliy Disribuion (PD) rv 2017/02/08 7

8 Classifying he Ampliude of Noise Samples x() 1 The ampliude x( 1 ) of he noise waveform a ime 1 is compared o a scale of discree values x k spaced by consan inerval and is classified a he neares value x k of he scale A high number N of noise waveform is sampled and measured of which is he number of sample waveforms classified a x k is called saisical frequency of he ampliude x k rv 2017/02/08 8

9 Noise Sample Saisics and Probabiliy x() N values x( 1 ) measured (in unis Δx) in N waveforms ΔN 0 in he cenral Δx (around x=0) ΔN 1 in he firs Δx (cenered in x 1 = Δx)... ΔN k in he k-h Δx (cenered in x k = kδx) saisical frequency of x k is if hen hence = if hen 1 ΔN p Hisogram of measured xvalues x p(x) Probabiliy Densiy of x values x rv 2017/02/08 9

10 Saionary and Non-saionary Noise STATIONARY noise : he probabiliy densiy is consan in ime p = p(x) NON-STATIONARY noise : he probabiliy densiy varies in ime p= p(x, ) BEWARE!! he probabiliy densiy p alone does no give a complee descripion of he noise, in fac differen cases can have equal probabiliy densiy p rv 2017/02/08 10

11 Noise Waveforms and Sample Saisics x() Case A : oupus of a se of noisy amplifiers, saionary noise x wih prob. densiy p A (x) 1 2 τ Values x( 1 ) and x( 2 ) measured on a sample waveform a differen 1 and 2 are random values wih equal probabiliy densiy p A (x) and hey are in pracice idenical for ulra-shor inerval τ somewha differen for shor inerval τ differen and independen for longer inerval τ rv 2017/02/08 11

12 Noise Waveforms and Sample Saisics x() Case B : oupu of a se of low noise amplifier, wih random baseline offse x wih prob. densiy p B (x) τ 1 2 Values x( 1 ) and x( 2 ) measured on a sample waveform a differen 1 and 2 : hey are random values wih probabiliy densiy p B (x); hey are equal for any inerval τ, shor or long Case B is differen from A, bu i can have equal probabiliy densiy p B (x) = p A (x) rv 2017/02/08 12

13 Complee Descripion of Noise wih Probabiliy Disribuions rv 2017/02/08 13

14 Full Descripion of Noise x() τ For a proper descripion of he noise he marginal probabiliy p m (x, )dx of having a value x a ime is NOT sufficien The join probabiliy p j (x 1, x 2, 1, 2 )dx 1 dx 2 of having a value x 1 a ime 1 and a value x 2 a ime 2 mus also be considered rv 2017/02/08 14

15 Noise Descripion wih Probabiliy Disribuions x() τ 1 2 A full descripion of he noise is obained by knowing: The marginal probabiliy densiy p m (x) = p m (x; 1 ) for every insan 1. For saionary noise p m does NOT depend on ime 1 : p m = p m (x) The join probabiliy densiy p j (x 1, x 2 ) = p j (x 1, x 2 ; 1, 2 ) = p j (x 1, x 2 ; 1, 1 + τ) for every couple of insans 1 and 2 = 1 + τ. For saionary noise p j depends only on he ime inerval τ, NOT on he ime posiion 1 rv 2017/02/08 15

16 Noe: Time-Average and Ensemble-Average x() Average of x over he ime lim 2 Average of x over he ensemble rv 2017/02/08 16

17 Basic Descripion of Noise wih 2 nd order Momens of Probabiliy Disribuion rv 2017/02/08 17

18 NOTE: Momens of Probabiliy Disribuions NB: for clariy, we call here he wo saisical variables x and y insead of x 1 and x 2 Momens of a marginal p(x) = Momens of a join p(x,y) =, he m n (and m jk ) give informaion on he feaures of he disribuions as he order (n or j+k) increases, he informaion is increasingly of deail Le s consider a descripion of noise limied o he 2 order momens, i.e. Mean square value (or variance) = Mean produc value (or covariance of x and y) =, NB: i is obviously m o =m oo = 1 he oal probabiliy is normalized o 1 = = 0 he mean value of noise is zero rv 2017/02/08 18

19 Noise Descripion wih 2 order Momens x() τ for every insan 1 he mean square value (or variance) For saionary noise does NOT depend on ime 1 for every couple 1 and 2 = 1 + τ he meanproduc For saionary noise i depends only on he ime inerval τ, NOT on he ime posiion 1 rv 2017/02/08 19

20 Auocorrelaion Funcion of Noise rv 2017/02/08 20

21 Noise Descripion wih he Auocorrelaion Funcion =, =, is called Auocorrelaion Funcion of he noise is always a funcion of he inerval τ beween he wo insans 1 and 2 is also a funcion of 1 only for non-saionary noise NOTE THAT: for a noise x he auocorrelaion R xx (τ) is an ensemble-average, for a signal x he auocorrelaion funcion K xx (τ ) is a ime-average The noise mean square value is called NOISE POWER i is he auocorrelaion wih τ = 0, 0 for saionary noise i is consan a any 0 rv 2017/02/08 21

22 Power Specrum of Noise rv 2017/02/08 22

23 Noise Descripion wih he Power Specrum Noise has power-ype waveforms (divergen energy ) which have saisical variaions from waveform o waveform of he ensemble. By averaging over he ensemble of he auocorrelaions of he noise waveforms, he conceps of power and power specrum inroduced for he signals can be exended o he noise lim lim lim = = lim lim Therefore, he Power Specrum of he noise is defined as rv 2017/02/08 23 = lim and he noise power is

24 Noise Descripion wih he Power Specrum By averaging over he ensemble we can exend o he noise also he second definiion of Power Specrum inroduced for he signals = = lim Flim, ]= lim, ] The Power Specrum of he noise can be direcly defined as S x f XT lim T 2 T The noise power is 0 f 2 rv 2017/02/08 24

25 Bilaeral and Unilaeral Specral Power Densiy The mahemaical specral densiy S x (f) defined over - < f <, is a bilaeral specral densiy S xb (f) aenion is called on his fac by he second subscrip B The noise power compued wih he bilaeral densiy S xb is Since S xb (f) is symmerical S xb ( f) = S xb (+f), i is 2 2 A unilaeral «physical» specral densiy S xu (f) 2 (f) is usually employed in engineering asks for making compuaions only in he posiive frequency range The noise power compued wih wih he unilaeral densiy S xu is rv 2017/02/08 25

26 Power Specrum of Non-Saionary Noise = resuls from he double average, firs over he ime hen over he ensemble I can be shown ha he order of averaging can be exchanged (see laer) =, The power specrum hus is relaed o he ensemble auocorrelaion funcion =, ] For non saionary noise S x (f) can be defined wih reference o he ime average of he ensemble auocorrelaion funcion of he noise. For saionary noise here is no need of ime-averaging: i is simply, and = rv 2017/02/08 26

27 APPENDIX : he order of Time-Averaging and Ensemble-Averaging can be exchanged in he definiion of he Noise Power Specrum Le s verify ha, In fac: lim lim lim, 2, rv 2017/02/08 27

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