COURSE OUTLINE. Introduction Signals and Noise Filtering Noise Sensors and associated electronics. Sensors, Signals and Noise 1

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1 Sensors, Signals and Noise 1 COURSE OUTLINE Introduction Signals and Noise Filtering Noise Sensors and associated electronics

2 Processing Noise with Linear Filters 2 Mathematical Foundations Filtering Stationary Noise Filtering White Noise Filtering Noise with Constant-Parameter Filters Appendix: Input-Output Crosscorrelation and Autocorrelation with Stationary Noise and Constant-Parameter Filters

3 3 Mathematical Foundations of Noise Processing by Linear Filters

4 Noise filtering 4 Input noise x(α) w t (α) Output noise y(t) characterized by R "" α, α + γ = x α x(α + γ) characterized by R ++ t, t + τ = y t y(t + τ) The output autocorrelationcan be obtained in terms of the input autocorrelation and of the filter weightingfunction : R ++ t /,t 0 = y t / y(t 0 ) = = 1 x α w / α dα 4 1 x(β))w 0 (β)dβ = 8 x α x β 4 w / α w 0 β dα dβ = 8 R "" (α, β) w / α w 0 β dα dβ = =

5 Noise filtering The output autocorrelation R ++ t /, t 0 = R ++ t /,t / + τ = y t / y(t / + τ) by setting in evidence the intervals of autocorrelation at the input γ = β α and at the output τ = t 2 t 1 can be expressed as R ++ t /, t / + τ = 8 R "" (α, α + γ) w / α w 0 α + γ dα dγ and in particular the mean square noise at time t 1 is y 0 t / = R ++ t /,t / = 8 R "" (α, α + γ) w / α w / α + γ dα dγ NB: these equations are valid for all cases of noise and linear filtering, that is, also for non-stationary input noise and for time-variant filters.

6 6 Filtering Stationary Noise

7 Filtering Stationary Noise 7 In case of stationary noise the input autocorrelation depends only on the time interval γ R "" α, α + γ = R "" γ The output autocorrelation is correspondingly simplified R ++ t /,t / + τ = 8 R "" (γ) w / α w 0 α + γ dα dγ = 1 R "" (γ) dγ 1 w / α w 0 α + γ dα NB: with stationary input noise: a) a constant parameter filter produces stationary output noise. b) a time-variant filter can produce a non-stationary output noise! =

8 Filtering Stationary Noise 8 R ++ t /, t / + τ = 1 R "" (γ) dγ 1 w / α w 0 α + γ dα Denoting by k 12w ( γ) the crosscorrelation of the weighting functions w 1 ( α) and w 2 ( α) We can write k /0; (γ) = 1 w / α w 0 α + γ dα R ++ t /, t / + τ = 1 R "" γ 4 k /0; (γ) dγ For the mean square noise we must consider the autocorrelation k 11w (α) of w 1 ( α) y 0 t / = R ++ t /,t / = 1 R "" (γ) dγ 1 w / α w / α + γ dα y 0 t / = 1 R "" (γ) 4 k //; (γ) dγ

9 Filtering Stationary Noise 9 With stationary input noise and for any linear filter (i.e. both constant-parameter and time variant filters) the output noise mean square value can be computed y 0 t / = 1 R "" (γ) 4 k //; (γ) dγ By the Parseval theorem extension and recalling that F k //; γ = W / (f) 0 the output mean square noise can be computed also in the frequency domain y 0 t / = 1 S " (f) 4 W / (f) 0 df

10 Filtering Stationary Noise 10 The mean square output of a filter that receives stationary noise can be computed in the time domain as in the frequency domain as y 0 t / = 1 R "" (γ) 4 k //; (γ) dγ y 0 t / = 1 S " (f) 4 W / (f) 0 df and in case of white noise, i.e. with R "" γ = 4 δ(γ) S " f = it is simply y 0 t / = 4 k //; 0 = 1 w / 0 α dα y 0 t / = 1 W / (f) 0 df

11 11 Filtering White Noise

12 Filtering White NON-Stationary noise 12 The fact that a White NON-Stationary noise has δ-like autocorrelation R "" α, α + γ = α δ(γ) brings simplification to the equation of the output autocorrelation R ++ t /, t / + τ = 8 α δ(γ) 4 w / α w 0 α + γ dα dγ R ++ t /, t / + τ = 1 α 4 w / α w 0 α dα and to the equation of the output mean square value y 0 t / = R ++ t /, t / = 1 α 4 w / 0 (α)dα the equation is conceptually similar to that for y 0 in discrete-time filtering, with samples x taken at clocked times α i and multiplied by weights w i and summed y 0 t / = x 0 (α D ) 4 w D 0 E DFG

13 Filtering White Stationary noise 13 The fact that White Stationary noise has constant intensity (power) R "" α, α + γ = R "" γ = 4 δ(γ) further simplifies the equation of the output autocorrelation R ++ t /, t / + τ = 1 w / α w 0 α dα = 4 k /0; (0) and of the output mean square value y 0 t / = 4 k //; 0 = 1 w / 0 α dα the equation is similar to that for discrete-time filtering of stationary white input noise y 0 t / = x 0 E w 0 DFG D By Parseval theorem we have also y 0 t / = 1 W / (f) 0 df

14 14 Filtering Noise with Constant-Parameter Filters

15 About CONSTANT-PARAMETER filters 1 The constant-parameter filters: are completely characterized by the δ-response h(t) in time and by the transfer function H(f) = F[h(t)] in the frequency domain have weighting w m (α) for acquisition at time t m simply related to the δ-response w m α = h(t α) therefore have W m (f) 2 = H(f) 2 They are PERMUTABLE. In a cascade of constant parameter filters, if the order of the various filters in the sequence is changed, the final output does NOT change. They are REVERSIBLE. A constant parameter filter can change the shape of a signal, but it is always possible to find a restoring filter, that is, another constant parameter filter which restores the signal to the original shape.

16 CONSTANT-PARAMETER filters with NON-stationary input noise 16 The output autocorrelation is R ++ t /,t 0 = 8 R "" (α, β) w / α w 0 β dα dβ = 8 R "" α, β 4 h t / α h t 0 β dα dβ = = 1 h t / α dα 1 R "" α, β 4 h t 0 β dβ = That is = R ++ t /,t 0 = R "" α, β h β h(α)

17 CONSTANT-PARAMETER filters with Stationary input noise 17 Starting from R ++ t /,t 0 = R ++ t / + τ = R "" α, β h β h α and taking into account that: the stationary input autocorrelation depends only on the interval γ = β α dβ = dγ dα = dγ the output autocorrelation is also stationary and depends only on the interval τ we can obtain (detailed equations available in Appendix) R ++ τ = R "" γ h γ h γ = R "" γ k TT γ and therefore S + f = S " f 4 H(f) 0

18 CONSTANT-PARAMETER filters with Stationary input noise 18 From the output autocorrelation R ++ τ = R "" γ k TT γ we obtain for the output mean square value y 0 = R ++ 0 = 1 R "" γ k TT γ dγ and by Parseval s theorem y 0 = 1 S " f H(f) 0 df In the case of white input noise R "" γ = δ γ and therefore y 0 = k TT 0 y 0 = 1 H(f) 0 df

19 19 Appendix: output-input cross-correlation and output autocorrelation with constant parameter filters and stationary noise

20 Appendix: output-input cross-correlation with constant parameter filters and stationary noise 20 yx Let us see first the output-input crosscorrelation R yx (τ) (, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R t t = y t x t = x α h t α x t dα = x α x t h t α dα = xx (, ) ( ) ( ) ( ) = R α t h t α dα = R t α h t α dα 2 1 xx 2 1 and setting t2 α = γ dα = dγ τ = (, ) (, ) ( ) ( ) yx 1 2 yx 1 1 xx 1 2 xx ( ) ( ) ( ) ( ) xx t ( ) t 2 1 R t t = R t t + τ = R γ h t t + γ dγ = = R γ h τ + γ dγ = R γ h τ γ dγ we see that R τ = R τ h τ yx ( ) ( ) ( ) xx

21 Appendix: input-output cross-correlation with constant parameter filters and stationary noise 21 xy Let us see now the input-output crosscorrelation R xy (τ) (, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R t t = x t y t = x t x β h t β dβ = x t x β h t β dβ = xx (, ) ( ) ( ) ( ) = R t β h t β dβ = R β t h t β dβ 1 2 xx 1 2 and setting ; ; γ = β dγ dβ t 1 τ = = 2 1 t t (, ) (, ) ( ) ( ) R t t = R t t + τ = R γ h t t γ dγ = xy 1 2 xy 1 1 xx 2 1 xx ( ) ( ) = R γ h τ γ dγ We see that R τ = R τ h τ xy ( ) ( ) ( ) xx

22 Appendix: output auto-correlation with constant parameter filters and stationary noise 22 yy (, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R t t = y t y t = y t x β h t β dβ = y t x β h t β dβ = yx (, ) ( ) ( ) ( ) = R t β h t β dβ = R β t h t β dβ 1 2 yx 1 2 and setting ; ; γ = β dγ dβ t 1 τ = = 2 1 (, ) (, ) ( ) ( ) yy 1 2 yy 1 1 yx 2 1 t t R t t = R t t + τ = R γ h t t γ dγ = ( γ) ( τ γ) γ ( τ) ( τ) ( τ) ( τ) ( τ) = Ryx h d = Ryx h = Rxx h h = ( τ) ( τ) ( τ) = Rxx h h and finally R τ = R τ k τ ( ) ( ) ( ) yy xx hh