Sensors, Signals and Noise COURSE OUTLINE Introduction Signals and Noise Filtering Noise Sensors and associated electronics Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 1
Processing Noise with Linear Filters Mathematical Foundations Filtering Stationary Noise Filtering White Noise Filtering Noise with Constant-Parameter Filters and for those who want to gain a better insight Appendix: Input-Output Crosscorrelation and Autocorrelation with Stationary Noise and Constant-Parameter Filters Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 2
Mathematical Foundations of Noise Processing by Linear Filters Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 3
Noise filtering Input noise x(α) w t (α) Output noise y(t) characterized by, characterized by, The output autocorrelation can be obtained in terms of the input autocorrelation and of the filter weighting function :,, Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 4
Noise filtering The output autocorrelation,, 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,, and in particular the mean square noise at time t 1 is,, 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. Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 5
Filtering Stationary Noise Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 6
Filtering Stationary Noise In case of stationary noise the input autocorrelation depends only on the time interval γ, The output autocorrelation is correspondingly simplified, 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! Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 7
Filtering Stationary Noise, Denoting by k 12w ( γ) the crosscorrelation of the weighting functions w 1 ( α) and w 2 ( α) We can write, For the mean square noise we must consider the autocorrelation k 11w (α) of w 1 ( α), Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 8
Filtering Stationary Noise 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 By the Parseval theorem extension and recalling that the output mean square noise can be computed also in the frequency domain Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 9
Filtering Stationary Noise The mean square output of a filter that receives stationary noise can be computed in the time domain as in the frequency domain as and in case of white noise, i.e. with it is simply 0 Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 10
Filtering White Noise Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 11
Filtering White NON-Stationary noise The fact that a White NON-Stationary noise has δ-like autocorrelation, brings simplification to the equation of the output autocorrelation,, and to the equation of the output mean square value, the equation is conceptually similar to that for in discrete time filtering, with samples x taken at clocked times α i and multiplied by weights w i and summed Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 12
Filtering White Stationary noise The fact that White Stationary noise has constant intensity (power), further simplifies the equation of the output autocorrelation, and of the output mean square value 0 0 the equation is similar to that for discrete time filtering of stationary white input noise By Parseval theorem we have also Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 13
Filtering Noise with Constant-Parameter Filters Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 14
About CONSTANT-PARAMETER filters 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 therefore have 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. NB1: this is absolutely true in principle, but it is limited in practical implementation by the limitations of linear behaviour (e.g. the finite dynamical range of real circuits). 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. NB2: this is absolutely true in principle, but it is limited in practical implementation by the limitations of linear behaviour and by the presence of noise (i.e. higher noise can be associated to the restored signal) Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 15
CONSTANT-PARAMETER filters with NON-stationary input noise The output autocorrelation is,,, That is,,, Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 16
CONSTANT-PARAMETER filters with Stationary input noise Starting from,, and taking into account that: the stationary input autocorrelation depends only on the interval the output autocorrelation is also stationary and depends only on the interval τ we can obtain (detailed equations available in Appendix) and therefore Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 17
CONSTANT-PARAMETER filters with Stationary input noise From the output autocorrelation we obtain for the output mean square value and by Parseval s theorem 0 In the case of white input noise 0 and therefore Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 18
Appendix: output-input cross-correlation and output autocorrelation with constant parameter filters and stationary noise Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 19
Appendix: output-input cross-correlation with constant parameter filters and stationary noise yx Let us see first the output-input crosscorrelation R yx (τ), R tt yt xt x ht xt d x xt ht d 1 2 1 2 1 2 2 1 xx, R t h t d R t h t d 2 1 xx 2 1 and setting t2 d d,, yx 1 2 yx 1 1 xx 1 2 xx xx t t 2 1 R t t R t t R h t t d R h d R h d we see that yx R R h xx Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 20
Appendix: input-output cross-correlation with constant parameter filters and stationary noise xy Let us see now the input-output crosscorrelation R xy (τ), R t t x t y t x t x h t d x t x h t d 1 2 1 2 1 2 1 2 xx, R t h t d R t h t d 1 2 xx 1 2 and setting ; ; d d t 1 2 1 t t,, R t t R t t R h t t d xy 1 2 xy 1 1 xx 2 1 xx R h d We see that xy R R h xx Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 21
Appendix: output auto-correlation with constant parameter filters and stationary noise yy, R t t y t y t y t x h t d y t x h t d 1 2 1 2 1 2 1 2 yx, R t h t d R t h t d 1 2 yx 1 2 d d and setting ; ; t 1 2 1,, yy 1 2 yy 1 1 yx 2 1 t t R t t R t t R h t t d Ryx h d Ryx h Rxx h h Rxx h h and finally R R k yy xx hh Sergio Cova SENSORS SIGNALS AND NOISE SSN04b FILTERING NOISE rv 2017/01/25 22