Sensors, Measurement systems, Signal processing and Inverse

Moving Average Autoregressive (ARMA) filtering. ▷ Errors, noise ... Partition function: F(x) = P(X ≤ x) = ∫ x. ∞ p(x) dx. ▷ Expected value: E{X} = ∫ xp(x) dx.
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Sensors, Measurement systems Signal processing and Inverse problems Ali Mohammad-Djafari Laboratoire des Signaux et Syst`emes, UMR8506 CNRS-SUPELEC-UNIV PARIS SUD 11 SUPELEC, 91192 Gif-sur-Yvette, France http://lss.supelec.free.fr Email: [email protected] http://djafari.free.fr Files: http://djafari.free.fr/Cours/Master_MNE/Cours/Cours_MNE_2014_01.pdf

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Contents 1. Sensors and Measurement systems I I

Direct and indirect measurement sensors Primary sensor characteristics

2. Basic sensors design and their mathematical models I I

R, L, C models and equations Forward model and simulation

3. Basic signal and image processing of the sensors output I I

Averaging, Convolution Fourier Transform, Filtering

4. Indirect measurement and inverse problems I I I

Deconvolution X ray Computed Tomography Ultrasound, Microwave and Eddy current NDT

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Contents 5. Inversion methods I I I

Analytical methods Algebraic methods Regularization

6. Bayesian estimation approach I I

Basics of Bayesian estimation Bayesian inversion

. Multivariate data analysis I I

Principal Component Analysis (PCA) Independent Component Analysis (ICA)

8. Blind sources separation I I

Classical methods Bayesian approach

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1. Sensors and measurement systems Direct and indirect measurement I

Direct measurement: Length, Time, Frequency

I

Indirect measurement: All the other quantities I I I I I I I I I I I

Temperature Sound Vibration Position and Displacement Pressure Force ... Resistivity, Permeability, Permittivity, Magnetic inductance Surface, Volume, Speed, Acceleration ...

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Sensors and measurement systems Different sensors: I I I I I I I I I I I I I I

Fluid Property Sensors Force Sensors Humidity Sensors Mass Air Flow Sensors Photo Optic Sensors Piezo Film Sensors Position Sensors Pressure Sensors Scanners and Systems Temperature Sensors Torque Sensors Traffic Sensors Vibration Sensors Water Resources Monitoring

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Sensors and measurement systems Main Glossary and Nomenclature: I

Sensor: Primary sensing element (example: thermistor which translates changes in temperature to changes to resistance)

I

Transducer: Changes one instrument signal value to another instrument signal value (example: resistance to volts through an electrical circuit)

I

Transmitter: Contains the transducer and produces an amplified, standardized instrument signal (example: A/D conversion and transmission)

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Primary sensor characteristics I

Range: The extreme (min and max) values over which the sensors can make correct measurement over controlled variable.

I

Response time: The amount of time required for a sensor to completely respond to a change in its input.

I

Accuracy (variance): Closeness of the sensor output to indicating the actual value of the measured variable.

I

Precision (bias): The consistency of the sensor output in measuring the same value under the same operating conditions over a period of time.

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Primary sensor characteristics I

Sensitivity: The minimum small change in the controlled variable that the sensor can measure.

I

Dead band: The minimum amount of a change to the process which is required before the sensor responds to the change.

I

Costs: Not simply the purchase cost, but also the installed/operating costs?

I

Installation problems: Special installation problems, e.g., corrosive fluids, explosive mixtures, size and shape constraints, remote transmission questions, etc.

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Signal transmission I

Pneumatic: Pneumatic signals are normally 3-15 pounds per square inch (psi).

I

Electronic: Electronic signals are normally 4-20 milliamp (mA).

I

Optic: Optical signals are also used with fiber optic systems or when a direct line of sight exists.

I

Hydraulic

I

Radio

I I

Glossary: http: //lorien.ncl.ac.uk/ming/procmeas/glossary.htm http://www.sensorland.com/GlossaryPage001.html http://www.sensorland.com/

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2. Basic sensors designs and their mathematical models I

We can easily measures electrical quantities: I I I

Resistance: U = RI or u(t) = Ri(t) ∂u(t) 1 Capacitance: ∂u(t) ∂t = C i(t) or i(t) = C ∂t Inductance: u(t) = L ∂i(t) ∂t

I

Sensors and transducers are used to convert many physical quantities to changes in R, C or L.

I

Resistance: I I

Resistive Temperature Detectors (Thermistors) Strain Gauges (Pressure to resistance)

I

Capacitance: Capacitive Pressure Sensor

I

Inductance: Inductive Displacement Sensor

I

Thermoelectric Effects: Temperature Measurement

I

Hall Effect: Electric Power Meter

I

Photoelectric Effect: Optical Flux-meter

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Resistivity/Conductivity I

Resistance R : R = ρ l/s (Ohm) I I I I

I

ρ: Resistivity ohm/meter 1/ρ: conductivity Siemens/meter l: length meter s: section surface meter2

Dipole model: u(t) = R i(t)

I

Impedance U (ω) = R I(ω) −→ Z(ω) =

I

U (ω) =R I(ω)

Power dissipation P (t) = R i2 (t) = u2 (t)/R

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Capacity C I

Capacitance: C = I I I I

I

Q U

Φ = ε0 U (Farads)

Q Electric charge (coulombs) U Potential (volts) ε0 Electrical permittivity Φ Electric charge flux (weber)

Dipole model: 1 u(t) = C

Z

t

i(t0 ) dt0

0

∂u(t) 1 ∂u(t) = i(t) or i(t) = C ∂t C ∂t I(ω) = jωC U (ω) I

Impedance Z(ω) =

1 jωC

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Inductance L I

Inductance: L = I I

I

Φ I

(Henri)

Φ Magnetic flux (Weber) I Current (Amp)

Dipole model (Faraday) : u(t) = L

∂i(t) ∂t

U (ω) = jωL I(ω) I

Impedance U (ω) = jωL I(ω) −→ Z(ω) = jω L

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Measuring R, C and L I

Measuring R: I

Simple voltage divider

I

Bridge measurement systems I I I

I

Single-Point Bridge Two-Point Bridge (Wheatstone Bridge) Four-Point Bridge

Measuring C and L I

I

AC voltage dividers and Bridges (Maxwell Bridge) Resonant circuits (R L C circuits)

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Measuring R I

Wheatstone bridge:

At the point of balance: Rx R2 R2 = ⇒ Rx = · R3 R1 R3 R1   Rx R2 VG = − Vs R3 + Rx R1 + R2 I

See Demo here: http://www.magnet.fsu.edu/education/tutorials/java/wheatstonebridge/index.html

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Measuring R I

The Wien bridge: At some frequency, the reactance of the series R2 C2 arm will be an exact multiple of the shunt Rx Cx arm. If the two R3 and R4 arms are adjusted to the same ratio, then the bridge is balanced. ω2 =

Cx R4 R2 1 and = − . Rx R2 Cx C2 C2 R3 Rx

The equations simplify if one chooses R2 = Rx and C2 = Cx ; the result is R4 = 2R3 .

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Measuring C I

Maxwell Bridge:

I

R1 and R4 are known fixed entities. R2 and C2 are adjusted until the bridge is balanced. R3 =

R1 · R4 −→ L3 = R1 · R4 · C2 R2

To avoid the difficulties associated with determining the precise value of a variable capacitance, sometimes a fixed-value capacitor will be installed and more than one resistor will be made variable. A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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Forward modeling and simulation of circuits

I

Input-Output model

I

R-C and R-L circuits

I

L-C and R-L-C circuits

I

Transfert function and impulse response

I

General linear systems

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Forward modeling and simulation of circuits Input-Output model: Examples of R − C circuits:

− − − − − R − −− − − − − − | g(t) f (t) C | − − − − − − − −− − − − − − ∂g(t) 1 (f (t) − g(t)) = i(t), i(t) = ∂t C R ∂g(t) 1 ∂g(t) = (f (t) − g(t)) −→ g(t) + RC = f (t) ∂t RC ∂t G(ω) 1 G(ω) + RCjωG(ω) = F (ω) −→ H(ω) = = F (ω) 1 + jRCω

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Forward modeling and simulation of circuits Input-Output model: Examples of R − L circuits

− − − − − R − −− − − − − − | g(t) f (t) L | − − − − − − − −− − − − − − ∂i(t) (f (t) − g(t)) = g(t), i(t) = ∂t R L ∂(f (t) − g(t)) L ∂g(t) L = g(t) −→ g(t) + = − f (t) R ∂t R ∂t R L L G(ω) 1 + jωL/R G(ω) + jωG(ω) = F (ω) −→ H(ω) = = R R F (ω) L/R L

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Forward modeling and simulation of circuits L − C circuits

− − − − − L − −−− − − − − − | f (t) C g(t) | − − − − − − − −−− − − − − − ∂g(t) 1 = i(t), ∂t C

L

∂i(t) = (f (t) − g(t)) ∂t

∂ 2 g(t) ∂ 2 g(t) = (f (t) − g(t)) −→ g(t) + LC = f (t) ∂t2 ∂t2 G(ω) 1 = G(ω) − LCω 2 G(ω) = F (ω) −→ H(ω) = F (ω) 1 − LCω 2 LC

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Forward modeling and simulation of circuits R − L − C circuits

− − − −R − −L − −− − − − − − | f (t) g(t) C | − − − − − − − −− − − − − − − 1 ∂g(t) = i(t), ∂t C RC

Ri(t) + L

∂i(t) = (f (t) − g(t)) ∂t

∂g(t) ∂ 2 g(t) + LC = (f (t) − g(t)) ∂t ∂t2

g(t) + RC

∂g(t) ∂ 2 g(t) + LC = f (t) ∂t ∂t2

G(ω)+RCjωG(ω)−LCω 2 G(ω) = F (ω) −→ H(ω) =

G(ω) 1 = F (ω) 1 + jRCω − LCω 2

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Resonant circuits

I

The resonant pulsation is: r ω0 =

1 LC

which gives: f0 =

ω0 1 = √ 2π 2π LC

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Forward modeling and simulation of circuits General linear systems:

f (t) −→ H(ω) −→ g(t)

I

H(ω) is called Transfert function of the system.

I

h(t) = IFT{H(ω)} is the impulse response of the system.

I

Given f (t) and h(t) or H(ω) we can compute g(t). G(ω) = H(ω)F (ω) −→ g(t) = h(t) ∗ f (t)

I I I

f (t) = δ(t) −→ g(t) = h(t)  Rt 0 t)2

i

Entropy H(X) = −

X

pi ln pi

i

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Discrete variables probability distributions I

Bernouilli distribution: A variable with two outcomes only X = {0, 1}, P (X = 1) = p, P (X = 0) = q = 1 − p p q 6 6

0 I

-

X

Bernoulli trial B(n, p): n independent trials of an experiment with two outcomes only 0010001100000010 I I

I

1

p probability of success q = 1 − p probability of failure

Binomial distribution Bin(.|n, p) : The probability of k successes in n trials:   n P (X = k) = pk (1 − p)n−k k

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Binomial distribution Bin(.|n, p) The probability of k successes in n trials:   n P (X = k) = pk (1 − p)n−k , k = 0, 1, · · · , n k E {X} = n p,

Var {X} = n p q = n p (1 − p)

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Poisson distribution

I

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed X ∼ Bin(n, p)

lim

n7→∞,np7→λ

X ∼ P(λ)

λk exp [−λ] k! Var {X} = λ

P (X = k|λ) = E {X} = λ, I

If Xn ∼ Bin(n, λ/n) and Y ∼ P(λ) then for each fixed k, limn→∞ P (Xn = k) = P (Y = k).

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Poisson distribution

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Continuous case I I

Cumulative Distribution Function (cdf): Measure theory

F (x) = P (X < x)

P (a ≤ X < b) = F (b) − F (a) P (x ≤ X < x + dx) = F (x + dx) − F (x) = dF (x) I

If F (x) is a continuous function p(x) =

I

∂F (x) ∂x

p(x) probability density function (pdf) Z b P (a < X ≤ b) = p(x) dx a

I

Cumulative distribution function (cdf) Z x F (x) = p(x) dx −∞

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Continuous case I

Expected value Z E {X} =

I

Variance Z Var {X} =

I

x p(x) dx =< X >

(x − E {X})2 p(x) dx = (x − E {X})2

Entropy Z H(X) =

I I

− ln p(x) p(x) dx = h− ln p(X)i

Mode: Mode(X) = arg maxx {p(x)} Median Med(X): Z Med(X) Z +∞ p(x) dx = p(x) dx −∞ Med(X)

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Uniform and Beta distributions I

Uniform: X ∼ U(.|a, b) −→ p(x) = E {X} =

I

a+b , 2

Var {X} =

x ∈ [a, b] (b − a)2 12

Beta: X ∼ Beta(.|α, β) −→ p(x) = E {X} =

I

1 , b−a

α , α+β

1 xα−1 (1−x)β−1 , x ∈ [0, 1] B(α, β)

Var {X} =

αβ (α +

β)2 (α

+ β + 1)

Beta(.|1, 1) = U(.|0, 1)

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Uniform and Beta distributions

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Gaussian distributions Different notations: I

classical one with mean and variance: 2

X ∼ N (.|µ, σ ) −→ p(x) = √ E {X} = µ, I



1 exp − 2 (x − µ)2 2 2σ 2πσ 1



Var {X} = σ 2

mean and precision parameters:   λ λ 2 X ∼ N (.|µ, λ) −→ p(x) = √ exp − (x − µ) 2 2π E {X} = µ,

Var {X} = σ 2 =

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1 λ

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Generalized Gaussian distributions I

Gaussian: 1

"

1 X ∼ N (.|µ, σ 2 ) −→ p(x) = √ exp − 2 2 2πσ I



(x − µ) σ

2 #

Generalized Gaussian: "   # β |x − µ| β X ∼ GG(.|α, β) −→ p(x) = exp − 2αΓ(1/β) α E {X} = µ,

I

Var {X} =

α2 Γ(3/β) γ(1/β)

β > 0, β = 1 Laplace, β = 2: Gaussian, β 7→ ∞: Uniform

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Gaussian and Generalized Gaussian distributions

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Gamma distributions I

Forme 1: β α α−1 −βx x e for x ≥ 0 Γ(α)

p(x|α, β) = E {X} = I

α , β

Var {X} =

α , β2

Mod(X) =

α−1 α+β−2

Forme 2: θ = 1/β p(x|α, θ) =

I

α = 1:

I

0 1, for ν > 2,

Interesting relation between Student-t, Normal and Gamma distributions: Z S(x|µ, 1, ν) = N (x|µ, 1/λ) G(λ|ν/2, ν/2) dλ Z S(x|0, 1, ν) =

N (x|0, 1/λ) G(λ|ν/2, ν/2) dλ

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Student and Cauchy − ν+1  2 x2 p(x|ν) ∝ 1 + ν

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Vector variables

I

Vector variables: X = [X1 , X2 , · · · , Xn ]0 p(x) probability density function (pdf) Expected value Z E {X} = x p(x) dx =< X >

I

Covariance

I I

Z

(X − E {X})(X − E {X})0 p(x) dx

= (X − E {X})(X − E {X})0

cov[X] =

I

Entropy Z E(X) =

I

Mode:

− ln p(x) p(x) dx = hln p(X)i

Mode(p(x)) = arg maxx {p(x)}

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Vector variables X = [X1 , X2 ]0

I

Case of a vector with 2 variables:

I

p(x) = p(x1 , x2 ) joint probability density function (pdf)

I

Marginals Z p(x1 ) =

p(x1 , x2 ) dx2 Z

p(x2 ) = I

p(x1 , x2 ) dx1

Conditionals p(x1 |x2 ) = p(x2 |x1 ) =

p(x1 , x2 ) p(x2 ) p(x1 , x2 ) p(x1 )

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Multivariate Gaussian Different notations: I

mean and covariance matrix (classical): X ∼ N (.|µ, σ)   1 0 −1 −n/2 −1/2 p(x) = (2π) |Σ| exp − (x − µ) Σ (x − µ) 2 E {X} = µ,

I

cov[X] = Σ

mean and precision matrix: X ∼ N (.|µ, Λ)   1 −n/2 1/2 0 p(x) = (2π) |Λ| exp − (x − µ) Λ(x − µ) 2 E {X} = µ,

cov[X] = Λ−1

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Multivariate normal distributions

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Multivariate Student-t −1/2

p(x|µ, Σ, ν) ∝ |Σ|

I

p=1 f (t) =

I

 (ν+p)/2 1 0 −1 1 + (x − µ) Σ (x − µ) ν

−(ν+1) Γ((ν + 1)/2) √ (1 + t2 /ν) 2 Γ(ν/2) νπ

p = 2, Σ−1 = A Γ((ν + p)/2) √ f (t1 , t2 ) = Γ(ν/2) ν p π p

I

|A|1/2 2π

 1 +

p X p X

 −(ν+2) 2

Aij ti tj /ν 

i=1 j=1

p = 2, Σ = A = I f (t1 , t2 ) =

−(ν+2) 1 (1 + (t21 + t21 )/ν) 2 2π

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Multivariate Student-t distributions

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Multivariate normal distributions

Normal

Student-t

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Dealing with noise, errors and uncertainties I

Sample averaging: mean and standard deviation N

x ¯=

1X xn n n=1

v u u S=t

N

1 X (xn − x ¯ )2 n−1 n=1

I

Recursive computation: moving average 1 x ¯k = n

k X

xi ,

i=k−n+1

x ¯k = x ¯k−1 +

x ¯k−1

k−1 1 X = xi n i=k−n

1 (xk − xk−n ) n

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Dealing with noise I

Exponential moving average 1 x ¯k = n

k X i=k−n+1

xi ,

x ¯k+1

1 = n+1

k+1 X

xi

i=k−n+1

n 1 x ¯k + xk+1 n+1 n+1 n 1 x ¯k = x ¯k−1 + xk = α¯ xk−1 + (1 − α)xk n+1 n+1 The Exponentially Weighted Moving Average filter places more importance to more recent data by discounting older data in an exponential manner x ¯k+1 =

I

x ¯k = α¯ xk−1 + (1 − α)xk = α[α¯ xk−2 + (1 − α)xk−1 ](1 − α)xk x ¯k = α¯ xk−1 + (1 − α)xk = α2 x ¯k−2 + α(1 − α)xk−1 (1 − α)xk x ¯ k = α3 x ¯k−3 + α2 (1 − α)xk−2 + α(1 − α)xk−1 + (1 − α)xk A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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Dealing with noise I

Exponential moving average x ¯k = α¯ xk−1 + (1 − α)xk x ¯ k = α2 x ¯k−2 + α(1 − α)xk−1 (1 − α)xk x ¯ k = α3 x ¯k−3 + α2 (1 − α)xk−2 + α(1 − α)xk−1 + (1 − α)xk

I

The Exponentially Weighted Moving Average filter is identical to the discrete first-order low-pass filter:

I

Consider the Laplace transform function of a first-order low-pass filter, with time constant τ : x ¯(s) 1 ∂x ¯(t) = −→ τ +x ¯(t) = x(t) x(s) 1 + τs ∂t     ∂x ¯(t) x ¯k − x ¯k−1 τ Ts = −→ x ¯k = x ¯k−1 + xk ∂t Ts τ + Ts τ + Ts

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Other Filters I

First order filter: 1 x ¯(s) = H(s) = x(s) (1 + τ s)

I

I

I

Second order filter: H(s) =

1 (1 + τ s)2

H(s) =

1 (1 + τ s)3

Third order filter:

Bode diagram of the filter transfer function as a function of τ and as a function of the order of the filter.

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Exercise 1 I

Consider the following signals: 1. 2. 3. 4. 5. 6. 7. 8.

f (t) = a sin(ωt) f (t) = a cos(ωt) PK f (t) = k=1 [ak sin(ωk t) + bk sin(ωk t)] PK f (t) = k=1 ak exp  [−j(ωk t)] f (t) = a exp −t2   PK f (t) = k=1 ak exp − 21 (t − mk )2 /vk f (t) = a sin(ωt)/(ωt) f (t) = 1, if |t| < a, 0 elsewhere

I

For each of these signals, first compute their Fourier Transform F (ω), then write a Matlab program to plot these signals and their corresponding |F (ω)|.

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Consider the following signals: 1. f (t) = a exp [−t/τ ] , t > 0 2. f (t) = 0, t ≤ 0, 1

I

For each of these signals, compute their Laplace Transform g(s).

A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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Exercise 2 Consider the following system: − − − − − R − −− − − − − − | f (t) C g(t) | − − − − − − − −− − − − − − with RC = 1. 1. Write the expression of the transfer function H(ω) =

G(ω) F (ω)

2. Write the expression of the impulse response h(t) 3. Write the expression of the relation linking the output g(t) to the input f (t) and the impulse response h(t) 4. Write the expression of the relation linking the Fourier transforms G(ω), F (ω) and H(ω) 5. Write the expression of the relation linking the Laplace transforms G(s), F (s) and H(s) A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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Exercise 2 (continued)

1. Give the expression of the output when the input is f (t) = δ(t) 2. Give the expression of the  output when the input is a step 0 ∀t < 0, function f (t) = u(t) = 1 ∀t ≥ 0 3. Give the expression of the output when the input is f (t) = a sin(ω0 t) 4. Give thePexpression of the output when the input is f (t) = k fk sin(ωk t) 5. Give thePexpression of the output when the input ist f (t) = j fj δ(t − tj )

A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

Master MNE 2014,

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Exercise 3

x ¯ = N1 Let note N x ¯N −1 =

PN

n=1 x(n), 1 PN −1 n=1 x(n), N −1

vN = N1 vN −1 =

PN

(x(n) − x ¯ N )2 n=1 P N −1 1 ¯N )2 n=1 (x(n) − x N −1

Show that I Updating mean and variance: x ¯N = vN = I

N −1 ¯N −1 + N1 x(n) = x ¯N −1 + N1 (x(n) N x N −1 N −1 ¯N )2 N vN −1 + N 2 (x(n) − x

−x ¯N −1 )

Updating inverse of the variance: −1 −1 −2 N vN = NN−1 vN ¯ N )2 vN −1 + (N −1)(N +ρN ) (x(n) − x −1 −1 with ρN = (x(n) − x ¯N )2 vN −1

I

Vectorial data xn

¯ N −1 + N1 x(n) = x ¯ N −1 + N1 (x(n) − x ¯ N −1 ) ¯ N = NN−1 x x N −1 N −1 ¯ N )(x(n) − x ¯ N )0 VN = N VN −1 + N 2 (x(n) − x −1 −1 −1 N N VN = N −1 VN −1 + (N −1)(N +ρN ) VN −1 (x(n) − x ¯N )(x(n) − x ¯N )0 VN−1 −1 0 ¯ N )0 VN−1 ¯ with ρN = (x(n) − x (x(n) − x ) N −1 A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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Exercise 4 Consider the following system: f (t) −→ H(ω) −→ g(t) with H(ω) =

1 1+jω .

1. Find h(t). 2. For a given input f (t) give the general expression of the output g(t). 3. f (t) give the general expression of the output g(t). 4. Give the expression of the  output when the input is a step 0 ∀t < 0, function f (t) = u(t) = 1 ∀t ≥ 0 5. Give the expression of the output when the input is f (t) = a sin(ω0 t) A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

Master MNE 2014,

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Exercise 5 Consider the following general system: f (t) −→ h(t) −→ g(t) 1. For a given input f (t) give the general expression of the output g(t). 2. Give thePexpression of the output when the input is f (t) = N n=0 f n δ(t − jn∆) with ∆ = 1. K X 3. Suppose that h(t) = hk δ(t − k∆) with ∆ = 1, Compute k=0

the output g(t) for t = 0, · · · , M ∆ with ∆ = 1 and M > N . 4. Show that if g(t) is sampled at the same sampling period P δ = 1, we have g(t) = M m=0 g m δ(t − m∆). Then show that gm =

K X

hk f nk

k=0 A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

Master MNE 2014,

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Exercise 5 (continued) 1. Show that the relation between f = [f 0 , · · · , f N ]0 , h = [h0 , · · · , hK ]0 and g = [g 0 , · · · , g M ]0 can be written as g = Hf or as g = F h. give the expressions and the structures of the matrices H and F . 2. What do you remark on the structure of these two matrices? 3. Write a Matlab programs which compute g when f and h are given. 4. Let name this program g=direct(h,f,method) where method will indicate different methods to use to do the computation. Test it with creating different inputs and different impulse responses and compute the outputs.

A. Mohammad-Djafari, Sensors, Measurement systems, Signal processing and Inverse problems,

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