08 LPF 01 - Intranet DEIBhome.deib.polimi.it/rech/Download/08_LPF_01.pdf · 2020-02-14 · SignalRecovery, 2019/2020 –LPF-1 Ivan Rech Low-pass filters LPF 5 To understand and to

Signal Recovery, 2019/2020 – LPF-1 Ivan Rech

Sensors, Signals and Noise 1

COURSE OUTLINE

• Introduction

• Signals and Noise

• Filtering: LPF1 Constant-Parameter Low Pass Filters

• Sensors and associated electronics


Constant-Parameter Low-Pass Filters 2

• Low-Pass Filters as Basic Elements for Signal and Noise Filtering

• RC Integrator

• Mobile-Mean Low-Pass Filter

• Bandwidth and Correlation Time of Low-Pass Filters


Low-Pass Filters as Basic Elementsfor Signal and Noise Filtering

3


Filtering signals and noise 4

SIGNALS AND NOISE• Signals carry information, but are accompanied by noise• The noise often is non-negligible and can degrade or even obscure the information• Filtering is intended to improve the recovering of the information• Filtering must exploit at best the differences between signal and noise, taking well

into account what kind of information is to be recovered. For instance: in case of a pulse-signal, is it just the amplitude or is it the complete waveform?

LOW-PASS FILTERS• We deal first with «low-pass filters» (LPF), so called because of their action in

the frequency domain. The filtering weight is concentrated in a relatively narrow frequency band from zero to a limit frequency; above the band-limit it falls to negligible value.

• Correspondingly, in the time domain the weighting function has relatively wide time-width (as well as its autocorrelation).

• The action of the filter as seen in time-domain is to produce approximately a time-average (i.e. a weighted average) of the input over a finite time interval, delimited by the width of the weighting function


Low-pass filters LPF 5

To understand and to be able to deal with LPF is very important because:

a) LPF are a basic element of filtering and a foundation for gaining a better insight on all other kinds of filters and better exploit them.For instance, a high-pass filter (HPF) can be obtained by subtracting from a given input the output of an LPF that receives the same input. In various real HPF, the physical structure of the HPF actually implements this scheme.

b) LPF are employed in real cases of filtering for information recoveryFor instance, in many cases a wide-band noise accompanies signals that have significant frequency components in a relatively narrow frequency band around f=0 . These are not only the cases of DC and slowly varying signals, but also cases where the just the amplitude of a pulse signal (having fairly long pulse-duration and known pulse shape) must be measured (and not the complete waveform)


RC-integrator

6


t

TR

|H(f)|2

f

0.5

1𝑇# t

RC integrator (constant-parameter LPF) 7

R

δ-response

ℎ 𝑡 =1𝑇#1(𝑡)𝑒 *+,

-.

Cx(t)

y(t)

Transfer function

𝑯 𝒇 =𝟏

𝟏 + 𝒋𝟐𝝅𝒇𝑻𝒇

𝐻 𝑓 9 =1

1 + (2𝜋𝑓𝑇#)9

Step-responsewith

risetime 10-90%TR = 2.2 Tf ≈ 1/3fp

3 dB frequency (Pole) 𝑓< = ⁄1 2𝜋𝑇#

1

1

fT RC=


RC integrator: viewpoints on |H(f)|2 8

LIN-LIN:Linear vertical scale

Linear horizontal scale

Log f

|H|2

f

LIN-LOG:Linear vertical scale

Logarithmic horizontal scale

LOG-LOG (Bode plot):Logarithmic vertical scale

Logarithmic horizontal scale

f=0

|H|2 =0

−∞

Log(|H|2)

|H|2 =0

−∞

f=0−∞

Log f

|H|2

f=0

|H|2 =0


RC integrator (constant-parameter LPF) 9

R

Cx(t)

y(t)Weighting function

in time𝒘𝒎 𝜶 = 𝒉 𝒕𝒎 − 𝜶

tm

1𝑇#

α

Output: can be seen as an average over a time interval ≈ 2Tf preceding tm

Weighting function in frequency

𝑊F(𝑓) 9 = 𝐻 𝑓 9

𝑾𝒎(𝒇) 𝟐 =𝟏

𝟏 + (𝟐𝝅𝒇𝑻𝒇)𝟐f

Output: can be seen as a selection of the lower frequency components up to ≈ fp

1

𝑊F(𝑓) 9

fp


RC integrator: filtering wide-band noise:Time-domain analysis 10

𝑘FFI 𝜏 =12𝑇#

𝑒+ K-.

𝑅MM ≅ 𝑆PQ𝛿(𝜏)

τ

kmmw

𝑘FFI(0)

τ

𝑦9 = U+V

V

𝑅MM(𝜏) W 𝑘FFI(𝜏) 𝑑𝜏

The noise is considered wide-band if it has autocorrelation much narrower than the filter weight autocorrelation, that is, if Tn << Tf

We can then approximate 𝑅MM ≅ 𝑆PQ𝛿(𝜏) and obtain

𝒚𝟐 = 𝑺𝒃𝑩 W 𝒌𝒎𝒎𝒘 𝟎 =𝑺𝒃𝑩𝟐𝑻𝒇

𝑥9

2𝑇

FILTER

NOISE


RC integrator: filtering wide-band noise:Frequency-domain analysis

11

𝑊F(𝑓) 9 = 𝐻(𝑓) 9 =

=1

1 + (2𝜋𝑓𝑇#)9

|Wm(f)|2 Noise bandwidth of the filter

𝑓#` =𝜋2𝑓<

SbB

Sn(f)

1

f

f

The noise is considered wide-band if it has spectrum much widerthan the filter weighting spectrum, that is, if its bandlimit fn >> fp

We can then approximate 𝑆M 𝑓 ≅ 𝑆PQ and obtain

𝒚𝟐 = 𝑺𝒃𝑩 W U+V

V

𝑾𝒎 𝒇 𝟐 𝒅𝒇 = 𝑺𝒃𝑩 W 𝒌𝒎𝒎𝒘 𝟎 =𝑺𝒃𝑩𝟐𝑻𝒇

= 𝑺𝒃𝑩 W 𝟐𝒇𝒇𝒏

𝑦9 = U+V

V

𝑆M(𝑓) W 𝑊F(𝑓) 9 𝑑𝑓fp

FILTER

NOISE


RC integrator: noise band-width 12

Noise bandwidth ff n of the filter:

defined with reference to a white noise input Sb as the bandwidth value to be employed for computing simply by a multiplication the output mean square noise

𝒚𝟐 = 𝑺𝒃𝑩 W 𝟐𝒇𝒇𝒏Since it is

𝑦9 = 𝑆PQ W U+V

V

𝑊F 𝑓 9 𝑑𝑓 = 𝑆PQ W 𝑘FFI 0

for any LPF the correct bandwidth limit ffn is

𝑓#` =𝑘FFI(0)

2and in particular for the RC integrator

𝒇𝒇𝒏 =𝟏𝟒𝑻𝒇

=𝝅𝟐𝒇𝒑


RC integrator active filter 13

x(t) y(t)

1𝑇#𝑅e𝑅f

t

− ℎ 𝑡 =𝑅e𝑅f

1𝑇#1(𝑡)𝑒 *+,

-.𝑇# = 𝑅e𝐶e

𝐻 𝑓 9 =𝑅e𝑅f

9 11 + (2𝜋𝑓𝑇#)9

𝑤F 𝛼 = ℎ 𝑡F − 𝛼

𝑊F(𝑓) 9 = 𝐻 𝑓 9

In comparison with the passive RC:• still a constant-parameter filter• same shape of the weighting

• dc gain = jkjl

instead of 1

dc gain 𝑊F(0) = jkjl

TIME DOMAIN

FREQUENCY DOMAIN


Mobile-Mean Low-Pass Filter

14


Mobile-Mean Filter 15

y(t)

• A mobile-mean filter (MMF) produces at any time tm an output y(tm) which is not just the integral of the input x(t) over a time interval Ta that precedes tm , but rather the mean value of the input x(t) over the time interval Ta , that is, the integral over Ta divided by Ta

• In order to obtain this, if we vary the averaging time Ta we must vary inversely the weight 1/Ta (this ensures constant area of wm(α) i.e. constant DC gain).

The MMF is a constant-parameter filter: this is pointed out by the weighting function, which is the same for any readout time tm

α

𝑇mMobile-Mean Filter

with averaging time Ta(constant-parameter)

1𝑇m

x(t)weighting wm(α)

tm


The Mobile-Mean Filter is a constant-parameter filter 16

ACTIVEINTEGRATOR

ANALOGTRANSMISSION LINE

WITH DELAY Ta

+

AMPLIFIERA = -1

+

x(t) y

branch (b)

branch (a)

Ta

tm

(b) weighting wb(α)

Total weighting w(α) = wb(α) - wa(α)

α

α

α

(a) weighting wa(α)

tm


tm

1𝑇#

α

Mobile-mean filter versus RC-integrator 17

R

Cx(t)

y(t)

The mobile-mean filter produces an output y(tm) that is exactly the mean value of the input x over the time interval Ta preceding tm . When Ta is changed, the area of wm(α) is kept constant, similarly to the case of the RC integrator when Tf is varied (the weight is reduced; the dc gain is kept constant)Question: can we use a mobile-mean filter as equivalent to a given RC integrator for evaluating the result obtained by processing a signal with low-frequency content in presence of wide-band noise?Answer: yes, the time Ta of the mobile-mean filter can be adjusted to produce equal output rms noise of the given RC integrator.

α

𝑇mMobile-mean filter

(constant-parameterwith averaging time Ta)

1𝑇m

x(t)

weighting wm(α)

weighting wm(α)


kmmw

𝑘FFI(0)

1𝑇#

α

12𝑇#

Mobile-mean filter equivalent to RC-integrator 18

Signal: the filters have equal DC gain (unity) and produce equal output with DC signal in.Noise: for wide-band input noise the output noise is computed as

𝑦9 = 𝑆PQ W 𝑘FFI 0 = 𝑆PQ W U+V

V

𝑤F9(𝛼) 𝑑𝛼

therefore, for having equal output rms noise it must be

𝑇m = 2𝑇#

α

1𝑇m

𝑇mwm(α)

kmmw

ττ

1𝑇m

RC integrator Mobile-mean filter−𝑇m 𝑇m


Mobile-mean filter equivalent to RC-integrator 19

The weighting functions of the two filters in frequency domain plotted with the same scales clearly illustrate the equivalence

RC integrator with RC = Tf

𝑾𝒇 𝒇 = 𝑯𝒇 𝒇 =𝟏

𝟏 + (𝟐𝝅𝒇𝑻𝒇)𝟐

Mobile-mean filter with averaging interval 𝑇m = 2𝑇#

(and unity gain)

𝑾𝒂 𝒇 = 𝑯𝒂 𝒇 =𝐬𝐢𝐧𝟐𝝅𝒇𝑻𝒇𝟐𝝅𝒇𝑻𝒇

f

f

1𝑇m=

12𝑇#

= 𝝅𝒇𝒑

𝑓< =1

2𝜋 𝑇#

𝑊# 𝑓

𝑊m 𝑓


Band-Width and Correlation Time of Low-Pass filters

20


«Rectangular» approximations of real filters 21

• The noise bandwidth ffn of a low-pass filter is currently employed forevaluating the output noise of low-pass filters in frequency-domaincomputations.

• A REAL filter that implements such a «rectangular weighting» in frequencyDOES NOT EXIST: it would be a non-causal system, with δ-response that beginsbefore the δ-pulse.

• A REAL filter that implements such a «rectangular weighting» in timeEXISTS: it is the mobile-mean filter with averaging time Ta = Tfn .

• There are, however, practical limitations to the implementation of mobile-mean filters, mainly due to the impractical features and limited performance ofthe real analog transmission lines with long delay, namely delay longer than afew tens of nanoseconds.


Other constant-parameter LPF 22

x

𝑧9 = 𝑆PQ W 𝑘ss 0 = 𝑆PQ W ∫+VV ℎ 𝑡 9 𝑑𝑡 = 𝑆PQ W ∫+V

V ,-.u

9𝑒+ uvw.𝑑𝑡

which integrated by parts gives

𝑧9 = 𝑆PQ W14𝑇#

Since 𝑧9 = 𝑆PQ 2𝑓 , the noise bandwidth 𝑓 is

𝒇𝒏 =𝟏𝟖𝑻𝒇

For LPF filters with real poles, it is often easier to compute the noise bandwidthin time-domain rather than in frequency-domain, because it implies simple integrals (of exponentials and powers of t). Example: cascade of two identical RC cells

ℎ 𝑡 =𝑡𝑇#9

𝑒+ ,-.

𝑇# = 𝑅𝐶

Documents

08 LPF 01 - Intranet DEIBhome.deib.polimi.it/rech/Download/08_LPF_01.pdf · 2020-02-14 · SignalRecovery, 2019/2020 –LPF-1 Ivan Rech Low-pass filters LPF 5 To understand and to