Sensors, Signals and Noise

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1 Sensors, Signals and Noise COURSE OUTLINE Introduction Signals and Noise Filtering: LPF3 Switched-Parameter Averaging Filters Sensors and associated electronics Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20

2 Switched-Parameter Averaging Filters Discrete Time Integrator (DTI) Discrete Time Integrator versus Gated Integrator Boxcar Integrator (BI) Ratemeter Integrator (RI) Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 2

3 Discrete-Time Integrator DI Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 3

4 Discrete-Time Integrator DI It is the discrete-time equivalent of a continuous gated integrator with gate = N T s Signal s x N P P P P P P P k T s 2 Noise n x Integratorr w() Weighting outline Samples taken with sampling frequency f s =/T s i.e. at intervals T s within Input: DC-signal s x and wide-band noise n x (autocorrelation width 2T n << T s ) Every sample is multiplied by P and summed, up to a total N = / T s samples Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 4

5 The output signal is The output noise is Discrete-Time Integrator With white noise, the GI gives ; we show now that the DI gives (that is, the DC gain is G = N P) and The noise samples are not correlated 0 and the noise is stationary Therefore By summing N samples the signal is increased by N and the rms noise by The SNR is thus improved by the factor Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 5

6 Discrete-Time Averager An averager is simply a discrete-time integrator with sampling weight P adjusted to give unity DC gain, that is G = N P = and therefore output signal equal to input The output noise is reduced to which corresponds to the enhancement of the S/N Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 6

Discrete-Time Exponential Averager It is the discrete-time equivalent of an RC integrator Signal s x T s Noise n x Averager w() k 2 0 Weighting outline Samples are taken with sampling frequency f s

7 Discrete-Time Exponential Averager It is the discrete-time equivalent of an RC integrator Signal s x T s Noise n x Averager w() k 2 0 Weighting outline Samples are taken with sampling frequency f s =/T s i.e. at intervals T s Input: DC-signal s x and wide-band noise n x (autocorrelation width 2T n << T s ) The sample weight slowly decays with the sample «age»: with ( r) << Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 7

8 Discrete-Time Exponential Averager Output signal Output mean square noise (i.e, DC gain ) The noise samples are not correlated ( 0 for k j ) and the noise is stationary ( ) Therefore The SNR is thus improved to S sy Psx S r N y n r N x r y nxp 2 r But the attenuation ratio r is very close to unity ( r) << hence ( + r) 2 and therefore 2 Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 8

9 Discrete-Time Integrator versus GI Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 9

10 Discrete-Time Integrator vs. GI w G () w D () P P P P P T s GI Gated Integrator in DI Discrete Time Integrator in INPUT: DC signal s x and wide-band noise S b (bandwidth 2f n >> f s, correlation width 2T n << T s ) with rms value 2 2 S/N enhancement by GI S/N enhancement by DI or equivalently with unity DC gain and Noise reduction by GI Noise reduction by DI Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 0

11 Discrete-Time Integrator vs. GI The improvement factor is for the DI, increasing with the number N of samples taken 2 for the GI, constant for a given QUESTION : is it possible to attain with a DI better S/N improvement than a GI just by increasing the number N (i.e. by using very fast sampling electronics)? ANSWER: NO!! In fact, since for having 2 it must be 2 in these conditions the samples are no more uncorrelated the improvement factor is no more given by There is still an improvement factor, but it must be evaluated taking into account the correlation between the noise samples. It is anyway (S/N) DI (S/N) GI with (S/N) DI (S/N) GI as N is increased, as we can demonstrate in time domain and in frequency domain Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20

12 Discrete-Time Integrator vs. GI (time domain) GI Gated Integrator (normalized to unity DC gain G=) DI Discrete-time Integrator (normalized to unity DC gain G=) 2 2 τ τ k ww τ ZOOM around τ = 0 k ww τ k ww 0 T G T s k ww N T T 0 S G τ τ Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 2

13 Discrete-Time Integrator vs. GI (time domain) GI Gated Integrator (with G=) DI Discrete-time Integrator (with G=) k ww 0 T G T s k ww T T 0 S G τ τ 2 nyd TS ( sum of Rxx samples at 0; TS ; 2 TS ;...) T G 2 nyg ( areaof Rxx ) T G 2 nyd ( area of the scaloid that approximatesrx x ) T G The scaloid area is greater than the R xx area, therefore n n n T n yd yg x TG with n n as T 2 2 yd yg S 0 Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 3

14 Discrete-Time Integrator vs. GI (frequency domain) TIME-Domain Weighting FREQUENCY-Domain Weighting GI with G= Free running sampler T s «window» & normalize to G= N sample averager with G= T s Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 4

15 Noise filtering analysis: GI vs. DI (frequency domain) GI with G= S x (f) 2 2 f Bandwidth approx S b 2 true S x (f) f n f The figure illustrates how the output noise is reduced and S/N is enhanced by increasing the sampling frequency f s (for a given averaging time ) f n Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 5

16 Noise filtering analysis: GI vs. DI a) As long as : the noise samples are uncorrelated each line of is identical to of the GI (with same DC gain G=) a high number of lines of falls within the noise bandwidth 2 the output noise of the DI is times that of the GI With good approximation it is 2 and it is confirmed that for uncorrelated samples the S/N increases as b) When f s becomes comparable to or higher the previous result is no more valid. the output noise must be computed with the actual noise spectrum The figure shows that is always higher than and attains it for lim Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 6

17 Boxcar Integrator BI Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 7

18 Boxcar Integrator (BI) This simple analog circuit combines two functions:. Sample Acquisition by gated integration 2. Exponential averaging of samples The circuit employed is the same of the Gated Integrator, but with a fundamental difference: the capacitor is NOT RESET between the acquisitions. S R S down S up TA T A C T R =T A + t m T F = RC >> Weighting w B () In T A the C is in HOLD state: nothing changes, no memory loss and no new charge input In the discharge of C (memory loss) reduces the previously stored value by the factor. NB: r does NOT depend on the interval T A Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 8

19 Boxcar Integrator (BI) S R S down S up T A TA C T R =T A + t m T F = RC >> Weighting w B () BI behaves as RC-integrator (RCI) when the switch is closed (S-down); it is in HOLD state when the switch is open (S-up) In fact, the weighting function w B () of the BI is obtained by subdividing w RC () of the RCI it in «slices» of width and placing them over the S-down intervals G= : the DC gain of BI (area of w B ) is unity (like that of RCI): the BI is an averager The autocorrelation functions k wwb of BI and k wwrc of RCI are very different, but have equal central value k ww (0) Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 9

20 Boxcar Integrator (BI): S/N enhancement The input wide-band noise S b with bandwidth 2f n, autocorrelation width 2T n, has mean square value 2 The BI output noise is 0 2 Therefore, since BI has G= the S/N enhancement is The S/N enhancement does NOT depend on the RATE of the samples because it is obtained by averaging over a given number of samples and not over a given time interval. In fact, counting the samples (from the measurement time t m and going backwards) the sample weight is reduced below /00 for sample number > 4.6T F /, irrespective of the sample rate Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 20

21 Boxcar Integrator (BI): S/N enhancement The BI is equivalent to the cascade of two filtering stages a) Acquisition of samples by a GI with same and T F as the BI, which enhances the S/N by the factor 2 b) Exponential averaging of the samples with attenuation ratio which enhances the S/N by the factor 2 2 NB: this factor is INDEPENDENT of the RATE of samples, because the AVERAGE IS DONE ON A GIVEN NUMBER OF SAMPLES and not on a given time. The S/N enhancement is thus confirmed and clarified 2 2 Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 2

22 Ratemeter Integrator RI Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 22

23 Ratemeter Integrator (RI) T S =T A + S T R = RC >> T S for averaging many samples R C S down S up RC weighting w L () RI Weighting w R () By inserting a buffer between S and RC a new exponential averager is obtained, radically different from BI. The integrator is no more a switched-parameter RC filter: it is now a constant-parameter RC filter, unaffected by the switch S. There is no HOLD state. The memory loss goes on all the time; the weight reduction from sample to sample is. NB: r DEPENDS on the sample RATE! During (with S-down) the input is integrated in C During T A (with S-up) the input is NOT allowed T A TA t m Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 23

24 Ratemeter Integrator (RI) T S =T A + S TR = RC >> T S for averaging many samples R C S down S up RC weighting w L () RI Weighting w R () T A TA t m The DC gain is G < (the RC filter has G=, but it receives just a fraction of the input!) With T R >> T S the DC gain G is proportional to the sample rate f S = /T S NB: if the input signal amplitude x S is constant but f S varies, the output signal y S varies. In fact, the circuit is also employed as analog ratemeter: with constant input voltage x S it produces a quasi DC output signal proportional to the repetition rate f S Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 24

25 Ratemeter Integrator (RI): S/N enhancement The RI is equivalent to the cascade of two filtering stages a) Acquisition of samples by a GI with same and T F as the RI, which enhances the S/N by the factor 2 b) Exponential averaging of the samples with attenuation ratio which enhances the S/N by the factor NB: this factor DEPENDS on the sample RATE f S because the AVERAGE IS DONE ON A GIVEN TIME and not on a given number of samples. The weight reduction is below /00 for samples that at the measurement time t m are «older» than 4.6 T R The S/N enhancement thus depends on the sample rate f s 2 2 Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 25

26 BI and RI: Passive Circuit comparison S R S R C C RATEMETER INTEGRATOR Switch S acts as gate on the input source Switch S is decoupled from the RC passive filter by the voltage buffer The RC integrator is unaffected by S, it has constant parameters, it does NOT have a HOLD state The sample average is done on a given time, defined by the RC value BOXCAR INTEGRATOR Switch S acts as gate on the input source Switch S acts also on the RC passive filter (changes the resistor value) The time constant T F of the integrator filter is switched from finite RC (S-down) to infinite (S-up, HOLD state) The sample average is done on a given number of samples, defined by the T F / value Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 26

27 BI and RI: Active Circuit comparison R F DC gain G = R F /R i R F S 2 S C F R i S R i C F RATEMETER INTEGRATOR Switch S acts as gate on the input Switch S is decoupled from the active RC integrator by the buffer action of the OP-AMP virtual ground The R F C F integrator is unaffected by S ; it has constant parameters, it does NOT have a HOLD state The sample average is done on a given time, defined by the R F C F value BOXCAR INTEGRATOR Switch S acts as gate on the input Switch S is decoupled from the active RC integrator by the buffer action of the OP-AMP virtual ground A second switch S 2 is required for switching the time constant T F of the integrator from finite R F C F (S 2 -down) to infinite (S 2 -up, HOLD state) The sample average is done on a given number of samples, defined by the T F / value Sergio Cova SENSORS SIGNALS AND NOISE SSN05c LOW PASS FILTERS 3 LPF3 rv 207/02/20 27

Sensors, Signals and Noise

Sensors, Signals and Noise COURSE OUTLINE Introduction Signals and Noise Filtering: LPF2 Switched-Parameter Filters Sensors and associated electronics Sergio Cova SENSORS SIGNALS AND NOISE SSN05b LOW PASS