COURSE OUTLINE. Introduction Signals and Noise: 3) Analysis and Simulation Filtering Sensors and associated electronics. Sensors, Signals and Noise

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1 Sensors, Signals and Noise 1 COURSE OUTLINE Introduction Signals and Noise: 3) Analysis and Simulation Filtering Sensors and associated electronics

2 Noise Analysis and Simulation White Noise Band-Limited White Noise (Wide-Band Noise) Basic Parameters of Wide-Band Noise Foundations of White-Noise Filtering Generation and Simulation of Any Noise by a Poisson Process APPENDIX: Noise Power Transients

3 3 White Noise

4 White Noise (stationary) 4 IDEAL «white» noise is a concet extraolated from Johnson noise and shot noise defined by its essential feature: no autocorrelation at any time distance τ, no matter how small R "" τ = S & ' δ τ and therefore constant sectrum R nn S n area S b τ S " f = S & In reality such a noise does not exist: it would have divergent ower n f REAL «white» noise has Very small width of autocorrelation, shorter than the minimum time interval of interest in the actual case and therefore aroximated to zero Very wide band with constant sectral density S b, wider than the maximum frequency of interest in the actual case and therefore aroximated to infinite

5 White Noise (non-stationary) 5 Also in non-stationary cases the IDEAL «white» noise is defined by the essential characteristic feature: no correlation at any finite time distance τ, no matter how small, but the noise intensity is no more constant, it varies with time t that is the autocorrelation function is δ-like, but has time-deendent area S b (t) R t, t+ τ = S t δ τ nn ( ) ( ) ( ) b R nn area S b (t) τ

6 Filtering white noise is simle 6 n(t) T s t 1 t t 3 t k t N For clarity, let s consider a discrete case: linear filtering in digital signal rocessing: Samle n 1 at t 1 and multily by a weight w 1, samle n at t = t 1 +T s and multily by a weight w and sum and so on... The filtered noise n f is n 1 = w 3 n 3 + w 5 n = : 9;3w 9 n 9 and its mean square value is n 1 5 = w 3 5 n w5 5 n w 3 n 3 ' w 5 n 5 + w 3 n 3 ' w < n < + = = w 3 5 n w 5 5 n w 3 w 5 n 3 n 5 + w 3 w < n 3 n < +

7 Filtering white noise is simle 7 n 5 1 = w n 3 + w5 n w 3 n 3 ' w 5 n 5 + w 3 n 3 ' w < n < + = = w 3 5 n w 5 5 n w 3 w 5 n 3 n 5 + w 3 w < n 3 n < +... If noise at interval T s is not correlated, then all rectangular terms vanish n 3 n 5 = n 3 n 5 = = 0 and the result is simly a sum of squares, even in case of non-stationary noise If the noise is stationary there is a further simlification n 1 5 = w 3 5 n w 5 5 n = = : 9;3 w 5 9 n 5 9 n 3 5 = n 5 5 = n < 5 = = n 5 n 1 5 = n 5 (w w5 5 + ) = = n 5 : 9;3w 5 9 we will see later that also with continuous filtering white noise brings similar simlification

8 8 Band-Limited White Noise or Wide-Band Noise

9 Band-limited white noise (wide-band noise) 9 Real white noise = white noise with band limited at high frequency The limit may be inherent in the noise source or due to low-ass filtering enforced by the circuitry. Anyway, in all real cases there is such a limit A frequent tyical case is the Lorentzian sectrum: band limited by a simle ole with time constant T, ole frequency f = 1/ πt Rnn τ τ T Sn = n e ( f ) ( ) = 1+ S B ( π ft ) R nn (τ) S n (f) n S B τ f

10 10 Basic Parameters of Wide-Band Noise

11 Simlified descrition of wide-band noise 11 The true R nn (τ) and S n (f) can be aroximated retaining the noise main features: a) equal mean square n and b) equal sectral density S b T n R nn (τ) n n τ in time: R nn (τ) triangular arox, half-width T n a) equal msq noise : R "" 0 = n b) equal sectral density: [area of R nn (τ)] = S b, ( i.e. n 5 T " = S & ) Correlation width = τ = T " S n (f) S b in frequency: S n (f) rectang arox, half-width f n a) equal msq noise : [area of S n (f)] = n ( i.e. S & f " = n ) f n f b) equal sectral density: S " 0 = S & Noise bandwidth: f = f " Note that τ ' f = 1 which is consistent with S " f = F[R "" τ ]

12 Simlified descrition of Lorentzian sectrum 1 Time Rnn τ T ( τ ) = n e ( τ) τ S = R d = n T B nn n T n = T τ -T n Frequency T n S n ( f ) = 1+ S B ( π ft ) 1 = n( ) = B T n S f df S S B f n = 1 T - f n f n f Note that f n f, namely f n 1 π = = f 4T

13 13 Foundations of White-Noise Filtering

14 Noise filtering clarified by the Poisson ulse model 14 Noise is a random suerosition of elementary ulses The elementary ulse tye (i.e. ulse waveform and its F-transform) defines the noise tye (i.e. autocorrelation shae and sectrum shae) The assage through a linear constant-arameter filter modifies the elementary ulse tye The ulse modification causes a corresonding modification of the noise Noise filtering can thus be understood, studied and evaluated by understanding, studying and evaluating the filtering of the elementary ulses

15 Low-ass filtering of White noise 15 Shot current white noise: V a S J f qi R R JJ τ qiδ(τ) Diode current: elementary short ulses with rate = I q, T O 100s qh t qδ t q T O t Current in R: elementary exonential ulses with rate = I q, T 1 = RC q R C q ' f t T 1 t f t = 3 1(t)e Z[ ]\ X W X F f = 3 3^_5`1W X e.g. with R= 100kΩ and C = 10F we have T f = 1μs

16 Low-ass filtering of White noise: time domain view 16 Inut noise (current in the diode): δ-like autocorrelation (width 100s) R ii R JJ τ qiδ(τ) for τ >> 100s qi T O R C T f = R C = 1μs τ To comare msq values of noise before and after filtering comare the central values of autocorrelation functions Outut noise (current in R): autocorrelation function R uu R bb τ = qi ' k 11 (τ) k 11 τ = 1 T 1 e d e W X qi T 1 τ

17 Low-ass filtering of White noise: frequency domain view 17 Inut noise (diode current): sectral density S i constant (bandwidth f i 10 GHz) S J f = S & for f << 10 GHz qi S i (f) f i 10 GHz R T f = R C = 1μs C To comare msq values of noise before and after filtering comare the areas of inut and outut sectral densities f Outut noise (current in R): sectral density function S u (f) S b f = qi ' F(f) 5 F(f) 5 = (πft 1 ) 5 qi S u (f) Pole frequency f k = 1 πt 1 Noise bandwidth f " = ` 5 f k 50kHz f

18 18 Generation and Simulation of Any Noise by a Poisson Process

19 Generation of any noise from Poisson rocess 19 Poisson rocess (δ-ulses) Constant-Parameter Linear filter Outut Noise x(t) h(t) (δ-resonse) y(t) ulse rate (/s) Q ulse area White Inut Noise R = Sx ( f ) = Q xx ( τ) Q δ( τ) Filter autocorrelation ( τ) () ( τ) khh = h t h t + dt Tranfer Function H(f) = F[h(t)] Outut Noise R = yy ( τ) Q khh( τ) S y ( f ) = Q H f ( )

20 Generation of any noise from Poisson rocess 0 For roducing a given S y (f) (i.e. a given R yy (τ)= F -1 [S y (f)] ) the filter must have H(f) = S y (f) normalized to 1 at f=0 k hh (τ) = R yy (τ) normalized to unit area Examle: band-limited white noise H ( ω) = + 1 jωt Sy = SB with f = 1+ f f πt ( ) 1 T ht () = e 1() t T t x(t) Poisson inut t x(t) R RC=T C y(t) ( ) S ( f ) = Q H f = Q y y(t) Noise outut t 1+ 1 ( f f )

21 Aendix: Noise Power Transients 1 Q: how does the noise ower rise when a noise source is switched on? A: the Poisson model clarifies!

22 Rise-time of the Noise ower Noise modeling by Poisson rocess also shows how noise rises after switch-on t= 0 Poisson rocess Q δ( t) ulses ulses/s t Contact closed at t=0 Linear filter t= 0 Actual noise n y t S h(t) H( ω) α α Noise switch-on at t=0 modeled by closing at t=0 the switch at the filter inut. The mean square outut at time t is comuted by Cambell s theorem but integrating only over the interval where outut ulses occur, i.e. for 0 < α < t, n t = Q t h ( α) dα ( ) y 0

23 Rise-time of real white noise 3 The rise of the outut noise intensity can be observed by comuting the root-mean square noise (rms) versus time t (normalized to unit sectral density of the inut Poisson noise) ( ) ny t t σ y ( t) = = h ( α) dα Q 0 Let s consider real White Noise with band-limit due to a simle ole with time constant T The noise is modeled by ulses The rise of the rms noise is 1 T ht () = e 1() t T t σ y 1 1 t = e dα = 1 e 0 T T ( ) α t t T T

24 Rise-time of real white noise 4 σ ( ) Wide-band «white» Noise The time constant T is short and the intensity (rms) rises swiftly, reaching in a few T the steady value Moderately Wide-band Noise the time constant T is longer and the intensity rise is slower. In the first art y 1 1 t = e dα = 1 e 0 T T t = T α t t T T σ = y 1 T for t? and the intensity rises aroximately as t T σ y ( t) t 1 T = 1 e T t T

25 Rise-time of random-walk noise 5 «Random-Walk» Noise» denotes noise with sectral density 1/ω. is generated by integration of white noise (e.g. shot current noise S i = qi in caacitor C à voltage noise S v = qi/ C ω ) is modeled by ste elementary ulses h(t) = 1(t) The variance rises as t t t σ t = h ( α) dα = dα = t y ( ) 0 0 At long time t à the variance is divergent In the frequency domain this corresonds to the ower over a band extended down to very low frequency f à 0 σ y 1 lim H ( f ) df = lim df f 0 0 i = fi fi fi f

26 Rise-time of random-walk noise 6 Poisson rocess (δ-ulses) Integrator h(t)=1(t) S Random-walk noise y(t) S-u S-down t Integrator outut y(t) σ ( ) y t = t