COURSE OUTLINE. Introduction Signals and Noise Filtering: LPF3 Switched-Parameter Averaging Filters Sensors and associated electronics

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1 ensors, ignals and Noise COURE OUTLINE Introduction ignals and Noise Filtering: LPF3 witched-parameter Averaging Filters ensors and associated electronics

2 witched-parameter Averaging Filters 2 Discrete Time Integrator (DTI) Discrete Time Integrator versus Gated Integrator Boxcar Integrator (BI) Ratemeter Integrator (RI)

3 3 Discrete-Time Integrator DI

4 Discrete-Time Integrator DI 4 It is the discrete-time equivalent of a continuous gated integrator with gate = N T s ignal s x P P P P P P P N 2 k T s Noise n x Integratorr w() Weighting outline amples 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

5 Discrete-Time Integrator 5 With white noise, the GI gives N T B ; we show now that the DI gives N N The output signal is s y = N % Ps x (that is, the DC gain is G = N P) The output noise is n ) =. -/0P % n,- and n 2 ) = P 2 (n 2,0 + n 2,2 + n,0 n,2 + ) = P 2 ( n 2,0 + n 2,2 + + n,0 n,2 + ) The noise samples are not correlated n,0 n,2 = n,2 n,8 = = 0 and the noise is stationary n,0 2 = n,2 2 = = n, 2 Therefore n y 2 = N % P 2 n x 2 By summing N samples the signal is increased by N and the rms noise by N The NR is thus improved by the factor N = s y = N % Ps x = N % N y n 2 y N % P 2 n 2 N x x

6 Discrete-Time Averager 6 An averager is simply a discrete-time integrator with sampling weight P adjusted to give unity DC gain, that is G = N P = P = N and therefore output signal equal to input s y = s x The output noise is reduced to n 2 ) = N % P 2 n 2, = 0. % n, 2 n y 2 = n x 2 which corresponds to the enhancement of the /N N ) = N % N N,

7 Discrete-Time Exponential Averager 7 It is the discrete-time equivalent of an RC integrator ignal s x T s Noise n x Averager w() k 2 0 Weighting outline amples 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 slowlydecays with the sample «age»: w - = Pr - with ( r) <<

8 Discrete-Time Exponential Averager 8 Output signal s ) = s, % P % G -/H r - = s, % P 0 0IJ Output mean square noise (i.e, DC gain G = P 0 0IJ ) n ) 2 = P 2 ( n,h 2 + r 2 % n, r 2- % n, r - r L % n,- n,l + ) The noise samples are not correlated ( n,- n,l = 0 for k j ) and the noise is stationary ( n,h 2 = n,0 2 = = n, 2 ) Therefore The NR is thus improved to n ) 2 = n, 2 % P 2 + r r 2- + = n, 2 % P 2 % sy Psx + r = = = N y n r x y nxp N r 2 r But the attenuation ratio r is very close to unity ( r) << hence ( + r) 2 and therefore 2 N y N x r 0 0IJ M

9 9 Discrete-Time Integrator versus GI

10 Discrete-Time Integrator vs. GI 0 w G () w D () P P P P P T s GI Gated Integrator in DI Discrete-Time Integrator in N = T B T R = f R T B INPUT: DC signal s x and wide-band noise b (bandwidth 2f n >> f s, correlation width 2T n << T s ) with rms value n 2, = T 2f V = T 2T V With unity DC gain s y = s x Noise reduction by GI n 2 )B = n 2, Z W X 2W Y Noise reduction by DI n 2 )[ = n 2, \ N

11 Discrete-Time Integrator vs. GI The improvement factor is N for the DI, increasing with the number N of samples taken 2T n for the GI, constant for a given T B QUETION : is it possible to attain with a DI better /N improvement than a GI just by increasing the number N (i.e. by using very fast sampling electronics)? ANWER: NO!! In fact, since N = T B T R for having N > T B 2T V it must be in these conditions the samples are no more uncorrelated T R < 2T V the improvement factor is no more given by N There is still an improvement factor, but it must be evaluated taking into account the correlation between the noise samples. It is anyway (/N) DI (/N) GI with (/N) DI à (/N) GI as N is increased, as we can demonstrate in time domain and in frequency domain

12 Discrete-Time Integrator vs. GI (time domain) 2 GI Gated Integrator (normalized to unity DC gain G=) DI Discrete-time Integrator (normalized to unity DC gain G=) n, 2 R,, n, 2 R,, 2T V 2T V T B T B k ww T B τ τ N ZOOM around τ = 0 k ww τ N = T B T R τ k ww ( 0) = T G R,, n, 2 R,, T s k ww ( ) T = = N T 0 G τ τ

13 Discrete-Time Integrator vs. GI (time domain) 3 GI Gated Integrator (with G=) DI Discrete-time Integrator (with G=) k ww ( 0) = T G R,, n, 2 T s k ww ( ) T T 0 = G τ τ 2 nyd = T ( sum of Rxx samples at τ = 0; ± T ; ± 2 T ;...) T G 2 nyg = ( area of 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 0

14 Discrete-Time Integrator vs. GI (frequency domain) 4 TIME-Domain Weighting FREQUENCY-Domain Weighting GI with G= T B W B Free-running sampler T R T R T R T R T s f R «window» & normalize to G= N N = T R N-sample averager with G= N T s N N N N W [ N = T s f R = T R

15 Noise filtering analysis: GI vs. DI (frequency domain) 5 GI with G= W B 2 x (f) n B 2 = T % T B = n, 2 % 2f V T B = n, 2 % 2T V T B f Bandwidth approx b n, 2 = T % 2f V true x (f) W [ 2 f n f G n B 2 = d, f IG f R W B (f) 2 df f R f R G n [ 2 = d, f IG f n W [ (f) 2 df The figure illustrates how the output noise n D 2 is reduced and /N is enhanced by increasing the sampling frequency f s (for a given averaging time )

16 Noise filtering analysis: GI vs. DI 6 a) As long as f s f n : the noise samples are uncorrelated each line of W 2 [ is identical to W 2 B of the GI (with same DC gain G=) a high number N h of lines of W 2 [ falls within the noise bandwidth 2f V the output noise of the DI is N h times that of the GI n 2 [ = n 2 B % N h With good approximation it is N h 2f V f R and it is confirmed that for uncorrelated samples the /N increases as N n [ 2 = n, 2 % 0 = V k W X i j. b) When f s becomes comparable to f n or higher the previous result is no more valid. the output noise must be computed with the actual noise spectrum G n [ 2 = d, f IG W [ (f) 2 df n B 2 The figure shows that n [ 2 is always higher than n B 2 and attains it for f R lim i j G n [ 2 = n B 2 M

17 7 Boxcar Integrator BI

18 Boxcar Integrator (BI) 8 This simple analog circuit combines two functions:. ample 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 REET between the acquisitions. R C -down -up TA T A T R =T A + t m T F = RC >> Weighting w B () r 2 T r r T r T r 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 r = e I T F. NB: r does NOT depend on the interval T A

19 Boxcar Integrator (BI) 9 R -down -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 (-down); it is in HOLD state when the switch is open (-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 -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) k vvw 0 = k vvxy 0 = 2RC = 2T r r 2 T r r T r T r

20 Boxcar Integrator (BI): /N enhancement 20 The input wide-band noise b with bandwidth 2f n, autocorrelation width 2T n, has mean square value n 2, = T % 2T V The BI output noise is n 2 ) = T % k vvw (0) = T % 2T r = n 2, % T V T r Therefore, since BI has G= the /N enhancement is N y = N x % T F T n The /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

21 Boxcar Integrator (BI): /N enhancement 2 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 /N by the factor 2T n b) Exponential averaging of the samples with attenuation ratio r = e IW X W T B T r which enhances the /N by the factor + r r 2 r = 2T F NB: this factor is INDEPENDENT of the RATE of samples, because the AVERAGE I DONE ON A GIVEN NUMBER OF AMPLE and not on a given time. The /N enhancement is thus confirmed and clarified N y = N x % 2T n % 2T F = N x % T F T n

22 22 Ratemeter Integrator RI

23 Ratemeter Integrator (RI) 23 T =T A + T R = RC >> T for averaging many samples R C -down -up RC-weighting w L () RI-Weighting w R () T A r 2 T x r TA T x t m T x T x By inserting a buffer between 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. There is no HOLD state. The memory loss goes on all the time; the weight reduction from sample to sample is r = e I (}T A ) T F = e I T T F. NB: r DEPENDon the sample RATE! During (with -down) the input is integrated in C During T A (with -up) the input is NOT allowed

24 Ratemeter Integrator (RI) 24 T =T A + T R = RC >> T for averaging many samples R C -down -up RC-weighting w L () RI-Weighting w R () T A r 2 T x r TA T x t m T x T x The DC gain is G < (the RC filter has G=, but it receives just a fraction of the input!) With T R >> T the DC gain G is proportional to the sample rate f = /T G G = w x d W X % IG w h d W X IG W G W = f % T B NB: if the input signal amplitude x is constant but f varies, the output signaly varies. In fact, the circuit is also employed as analog ratemeter: with constant input voltage x it produces a quasi DC output signal proportional to the repetition rate f

25 Ratemeter Integrator (RI): /N enhancement 25 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 /N by the factor 2T n b) Exponential averaging of the samples with attenuation ratio r = e IW W ƒ T T x which enhances the /N by the factor + r r 2 r = 2T R T = 2T R f NB: this factor DEPEND on the sample RATE f because the AVERAGE I 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 /N enhancement thus depends on the sample rate f s N y = N x % 2T n % 2T R T = N x % f T R T n

26 BI and RI: Passive Circuit comparison 26 R R C C RATEMETER INTEGRATOR witch acts as gate on the input source witch is decoupled from the RC passive filter by the voltage buffer The RC integrator is unaffected by, 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 witch acts as gate on the input source witch 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 (-down) to infinite (-up, HOLD state) The sample average is done on a given number of samples, defined by the T F / value

27 BI and RI: Active Circuit comparison 27 R F DC gain G = R F /R i R F 2 C F R i R i C F RATEMETER INTEGRATOR witch acts as gate on the input witch 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 ; 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 witch acts as gate on the input witch is decoupled from the active RC integrator by the buffer action of the OP-AMP virtual ground A second switch 2 is required for switching the time constant T F of the integrator from finite R F C F ( 2 -down) to infinite ( 2 -up, HOLD state) The sample average is done on a given number of samples, defined by the T F / value