Full course overview 1. Fourier analysis and analog filtering 1.1 Fourier Transform 1.2 Convolution and filtering 1.3 Applications of analog signal processing MAP 555 : Signal Processing 2. Digital signal processing Part 1 : Fourier analysis and analog filtering 2.1 Sampling and properties of discrete signals 2.2 z Transform and transfer function 2.3 Fast Fourier Transform R. Flamary 3. Random signals 3.1 Random signals, stochastic processes 3.2 Correlation and spectral representation 3.3 Filtering and linear prediction of stationary random signals December 7, 2020 4. Signal representation and dictionary learning 4.1 Non stationary signals and short time FT 4.2 Common signal representations (Fourier, wavelets) 4.3 Source separation and dictionary learning 5. Signal processing with machine learning 5.1 Learning the representation with deep learning 5.2 Generating realistic signals 1/108 2/108
Course overview Signal and function in Lp(R) space Fourier Analysis and analog filtering 4 Signals and definitions 4 Properties of signals Lp space Dirac distribution and convolution Lp(S) is the set of functions whose absolute value to the power of p has a finite Linear Time-Invariant systems integral or equivalently that Fourier transform 15
Fourier series p x p = x(t) dt < (1) Fourier Transform and properties k k | | ∞ Properties of the Fourier Transform ZS Frequency response and filtering 29 Convolution and Fourier Transform I L1(R) is the set of absolute integrable functions Filtering and frequency representation Representation of the FT and frequency response I L2(R) is the set of quadratically integrable functions (finite energy) First and second order systems I L (R) is the set of bounded functions Applications of analog signal processing 66 ∞ Analog filtering Modulation Signal and images Fourier optics In this course we will mostly study Digital signal processing 105 I 1D temporal signal with x(t) R, t R (or complex valued function). Random signals 105 ∈ ∀ ∈ I 2D images with x(v) R, v R2 Signal representation and dictionary learning 105 ∈ ∀ ∈ Signal processing with machine learning 105 3/108 4/108 Some properties of signals Classical signals (1)
Signal x(t) Causality Heaviside function Signal Γ(t) A signal x(t) is causal if 0.5 1.0 0.4 0 pour t < 0 0.8 x(t) = 0, x < 0 0.3 Γ(t) = 1/2 pour t = 0 (2) ∀ 0.6 0.2 Example: 1 pour t > 0 0.4 0.1 0 for t < 0 Also known as the step function. 0.2 0.0 2 x(t) = t 0.0 sin(t) exp for t 0 0.1 ( 2 4 2 0 2 4 t 0.2 − ≥ 4 2 0 2 4 t
Signal x(t) 1.1 Periodicity Rectangular function Signal PiT (t) 1.0
A signal x(t) is periodic if period T0 is 0.25 0.9
0.20 0.8 x(t kT0) = x(t), t R, k N 1/T pour t < T/2 − ∀ ∈ ∀ ∈ 0.7 | | 0.15 ΠT (t) = 1/2T pour t = T/2 (3) 0.6 | | 0.10 Example: 2 0.5 0 ailleurs (t kT0 1) 0.05 x(t) = exp − − for kT0 < t < 2 0.4 1 T T − I Π(t) =T (Γ(t 2 ) Γ(t + 2 )). 0.00 0.3 (k + 1)T0, k N 4 2 0 2 4 − − t 0.05 ∀ ∈ 4 2 0 2 4 I Finite energy signal (finite support). t
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Classical signals (2) Classical signals (3) Complex exponantial with z = τ + wi
τ w 2 0 2 Complex exponential \ − Signal exp(( 0.5 + 2j) t) Signal exp(( 0.5 +0j) t) Signal exp(( 0.5 +2j) t) − − ∗ − ∗ − ∗ let e (t) be the following function R C 12 z 5 5 → 10
8 0 0
ez(t) = exp(zt) (4) 6 0.5 − 5 4 5 where z is a complex number. When z = τ + wi the, 2 10 10 0
4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 t t t e (t) = (cos(w t) + i sin(w t)) exp(τt) z Signal exp((0.0 + 2j) t) Signal exp((0.0 +0j) t) Signal exp((0.0 +2j) t) − ∗ ∗ ∗ 1.0 1.0 ∗ ∗ ∗ 1.0
0.8 Special cases: 0.5 0.5 0.6
0.0 0.0 I z = τ real, then we recover the classical exponential. 0 0.4 0.5 0.5 0.2
0.0 1.0 1.0 ez(t) = exp(τt) 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 t t t
Signal exp((0.5 + 2j) t) Signal exp((0.5 +0j) t) Signal exp((0.5 +2j) t) − ∗ ∗ ∗ 12 z = wi imaginary then 5 I 5 10
8 0 0 e (t) = cos(w t) + i sin(w t) 6 z .5 ∗ ∗ ∗ 5 4 5
2
10 10 0
4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 t t t
7/108 8/108 Dirac delta Dirac delta (2)
Main properties of Dirac delta Signal δ(t) Dirac delta definition
1.0 I Model point mass at 0. I Let φ a function supported in [ 1, 1] of unit mass: ∞ φ(u)du = 1 0.8 − −∞ I Value outside 0 : δ(t) = 0, t = 0 1 t 0.6 I φT (t) = φ( ) has support on [ T,T ] and unit mass.R ∀ 6 T T − I δ is a tempered distribution. 0.4 I We can define the dirac delta δ as I Very useful tool in signal processing 0.2 δ(t) = lim φT (t) 0.0 T 0 I Can be seen as the derivative of the → 0.2 4 2 0 2 4 Heavyside function 1t 0(t) t ≥ I Integral Delta dirac in practice + + ∞ ∞ δ(t)dt = 1, x(t)δ(t)dt = x(0) (5) I Theoretical object in signal processing (impulse). Z−∞ Z−∞ I Used to model signal sampling for digital signal processing. I Dirac and function evaluation for signal x(t) and t0 R : ∈ I Used to model point source in Astronomy/image processing, point charge in δ(t t0)x(t) = δ(t t0)x(t0) Physics. − − + I Has a bounded discrete variant. ∞ x(t), δ(t t0) = x(t)δ(t t0)dt = x(t0) (6) h − i − Z−∞
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Convolution operator Linear Time-Invariant (LTI) systems
x(t) y(t) Definition System Let two signals x(t) and h(t). The convolution between the two signals is defined as
+ ∞ Definition x(t) ? h(t) = x(τ)h(t τ)dτ (7) − I A system describes a relation between an input x(t) and an output y(t). Z−∞ I Convolution is a bilinear mapping between x and h. I Properties of LTI systems:
I It models the relation between the input and the output of a Linear Time I Linearity x1(t) + ax2(t) y1(t) + ay2(t) Invariant system. → I Time invariance x(t τ) y(t τ) I If f L1(R) and h Lp(R), p 1 then − → − ∈ ∈ ≥ I A LTI system can most of the time be expressed as a convolution of the form: f ? h p f 1 h p k k ≤ k k k k y(t) = x(t) ? h(t) I The dirac delta δ is the neutral element for the convolution operator: where h(t) is called the impulse response (the response of the system to an input + ∞ x(t) = δ(t)) x(t) ? δ(t) = x(τ)δ(t τ)dτ = x(t) (8) − Z−∞ Examples x(t) ? δ(t t0) = x(t t0) (9) Passive electronic systems (resistor/capacitor/inductor) . − − I I Newtonian mechanics, Fluid mechanics, Fourier Optics.
11/108 12/108 LTI systems and Ordinary Differential Equation Signal and frequencies
Ordinary Differential Equation (ODE) The system is defined by a linear equation of the form:
dy(t) dny(t) dx(t) dmx(t) a0y(t) + a1 + + an = b0x(t) + b1 + + bm (10) dt ··· dtn dt ··· dtm
I ODE based system with linear relations are an important class of LTI systems. I A signal is x(t) a function of time, an image x(v) a function of space. I Also called homogeneous linear differential equation. I Those functions are what we measure/observe but can be hard to interpret/process automatically. I n is the number of derivatives for y(t) and m for x(t). I Another representation for a signal is in the frequency domain (1/t). I max(m, n) is the order of the system. I Better representation for numerous applications. I The output of the system can be computed from the input by solving Eq. (10). I Linearity and time invariance are obvious from equation. Applications I Signal processing (biomedical, electrical). I Image processing (2D signals), filtering, reconstruction. I Colors are combination of waves of different frequencies.
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Fourier Series (1) Fourier series (2)
Decomposition as trigonometric series 2π One can express periodic x(t) of period T0 = integrable on the period as w0
a ∞ x(t) = 0 + [a cos(kw t) + b sin(kw t)] 2 k 0 k 0 kX=1 where ak and bk are the Fourier coefficients that can be computed as 2 2 ak = x(t) cos(kw0t)dt bk = x(t) sin(kw0t)dt T0 T0 ZT0 ZT0 I Representation of a periodic signal as an infinite number of coefficients corresponding to harmonic frequencies. I Can be interpreted as a change of basis from temporal to frequencies. History I Functions can be approximated with a finite number N of terms. I Trigonometric series used by Euler, d’Alembert, Bernoulli and Gauss. I Gibbs phenomenon appears for discontinuous functions I Introduced by Joseph Fourier in [Fourier, 1807]. [Hewitt and Hewitt, 1979]. I Fourier claimed that these series could approximate any function.
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1.0
0.8
0.6
0.4
0.2
0.0 x(t)
x0(t) 0.2 0 1 2 3 4 5
Example of Fourier series Complex Fourier series
Complex harmonic decomposition 2π One can express periodic x(t) of period T0 = integrable on the period as w0
∞ jkw0t 2π x(t) = cke avec w0 = T0 k= X−∞
where the coefficients ck are called the complex Fourier coefficients and can be computed with
T0 T0/2 1 jkw0t 1 jkw0t 1 jkw0t 1.0 ck = x(t)e− dt = x(t)e− dt = x(t)e− dt 0.8 T0 T0 T0 0 T0 T0/2 0.6 Z Z Z− 0.4 0.2 Relations between decompositions 0.0 0.2 Using the Euler formula we can show that ak and bk and the ck coefficients are related 0.4 0.6 by 5 a0 4 = c0 ak = ck + c k bk = j(ck c k) − − 0 3 2 − 1 2 2 3 Time t Frequency 4 1 5 Note that if x(t) is an even function then the bk = 0 , and if x(t) is odd then ak = 0. 6 0 Example : Square wave
I Square wave with T0 = 2 17/108 18/108 ∞ 1 I x(t) = i= 1[iT0,iT0+T0/2](t) 0.8 −∞ 0.6 I a0 = 1,P ak = 0 k > 0 0.4 ∀ Fourier transform 2 Interpretation of the Fourier transform 0.2 I bk = for k odd else bk = 0 0 πk 0 1 2 3 4 5
∞ i2πft Definition (Fourier Transform) x(t) = e X(f)df The Fourier Transform (FT) of a signal x(t) can be expressed as Z−∞ Harmonic representation ∞ i2πft [x(t)] = X(f) = e− x(t)dt (11) F I The FT represents the signal in the frequency domain. Z−∞ I X(f) is the magnitude of a sinusoidal signal for frequency f. When it exists the inverse Fourier transform is defined as | | I Arg(X(f) is the phase of the sinusoidal signal. 1 ∞ i2πft − [X(f)] = x(t) = e X(f)df (12) F I For a real signal x(t), X(f) = X( f)∗ and an informal interpretation would be Z−∞ − + + ∞ ∞ I Note that the ˆ operator is also often used for the Fourier transform xˆ of x. x(t) = X(f)ei2πftdf = X(f) ei2π(ft+Arg(X(f)))df (13) | | In signal processing the references often use j instead of i for the imaginary Z−∞ Z−∞ I + ∞ number (i is a measure of current). X(0) + 2 X(f) cos(2π(ft + Arg(X(f)))) (14) ≈ | | I The FT is a change of representation for the function x from the temporal Z0+ representation to the harmonic (frequency) representation. I The modulus and argument of the FT allow identification of the frequency content of the signal and its phase.
19/108 20/108 Examples of Fourier Transform (1) Examples of Fourier Transform (2)
Rectangular function Decreasing exponential Signal x(t)
Signal ΠT (t) 1.0 1/T if t < T/2 1 for t > 0 | | 0.5 at 0.8 ΠT (t) = 1/2T if t = T/2 (15) x(t) = e− Γ(t), Γ(t) = 1/2 for t = 0 0.4 | | 0.6 0 else 0 else 0.3 0.4 0.2 The Fourier transform is with a > 0 0.2 0.1 The Fourier transform is T/2 0.0 1 i2πft 0.0 [ΠT (t)] = e− dt 6 4 2 0 2 4 6 ∞ t F T 0.1 at at i2πft T/2 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − Z− t [e Γ(t)] = e e dt F X(f) i2πft T/2 0 0.6 e− Z Partie reelle [ΠT (t)] = − 3.5 F ∞ (a+i2πf)t Partie imaginaire Partie reelle = e− dt i2πfT T/2 3.0 Partie imaginaire 0.4 − 0 iπfT iπfT 2.5 Z (a+i2πf)t e e− 2.0 ∞ e− 0.2 = − = i2πfT 1.5 (a + i2πf) 0 sin(πfT ) 1.0 − 0.0 = = sinc(πfT ) 0.5 1 πfT = 0.0 a + i2πf 0.2
0.5 4 2 0 2 4 sin(t) f 1.0 with sinc(t) = and sinc(0) = 1 4 2 0 2 4 t f
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Properties of the Fourier Transform Properties of the Fourier Transform (2)
Linearity Frequency shift Let x(t) be a signal of FT X(f) then we have Ley x1(t) and x2(t) two signals of TF X1(f) and X2(f) respectively. For a R and b R, we have : i2πf0t ∈ ∈ e x(t) = X(f f0) F − [ax1(t) + bx2(t)] = aX1(f) + bX2(f) h i F Multiplication by a complex exponential of frequency f0, translates the TF by f0. Proof. Comes from the linearity of the integration. Proof. Regroup exponentials in the integral.
Time shift Time scaling Let x(t) be a signal of FT X(f). Let x(t) be a signal of FT X(f) and a a scaling a = 0 then we have For t0 R, let x(t t0) a time shift of x(t) then we have: 6 ∈ − i2πt0f 1 f [x(t t0)] = e− X(f) [x(at)] = X F − F a a | | Proof. Change of variable in the integral. Proof. Change of variable for separate cases a > 0 and a < 0.
23/108 24/108 Properties of the Fourier Transform (3) Properties of the Fourier Transform (4)
Derivation Let x(t) be a signal of FT X(f) then we have Even and odd signals dx(t) = j2πfX(f) F dt x(t) X(f) Even real Even real Odd real Odd imaginary Integration Even imaginary Even imaginary Let x(t) be a signal of FT X(f) such that ∞ x(t)dt = 0 then we have Odd imaginary Odd real −∞ t R 1 For a real signal x(t) : X(f) = X( f)∗ x(u)du = X(f) − F i2πf Z−∞ Conjugate signal If ∞ (x(t) c)dt = 0 where c is often called the constant term, we have Let x(t) be a signal of FT X(f) and x∗(t) its complex conjugate, then we have −∞ − R t 1 [x∗(t)] = X∗( f) x(u)du = X(f) + cδ(f) F − F i2πf Z−∞ where δ(f) is the Dirac delta. Those two properties can be used to solve Ordinary Differential Equations (ODE).
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Duality of the Fourier Transform Fourier Transform in Lp(R)
1 1 I For 1 p 2 the FT maps from Lp(R) to Lq(R) with + = 1. Let x(t) be a signal of FT X(f). When the inverse Fourier transform exists we can ≤ ≤ p q write I Consequence of the Riesz–Thorin theorem. + + ∞ j2πf( t) ∞ j2πft x( t) = X(f)e − df = X(f)e− df I The TF of an absolute integrable function is bounded (Example : rectangle). − Z−∞ Z−∞ Parseval-Plancherel identity in L2 I The last term is the TF of function X(f). The TF of an L2 function is L2. Note that L2 is a Hilbert space of inner product:
I This means that if [x(t)] = X(f) then ∞ F < x, y >= x(t)y∗(t)dt [X(t)] = x( f) Z−∞ F − 2 2 For two functions x, y L2(R) of respective TF X,Y L2(R) the I Applying twice the TF operator to x(t) returns x( t): [ [x(t)]] = x( t) ∈ ∈ − F F − Parseval-Plancherel identity states that Example ∞ ∞ < x, y >= x(t)y∗(t)dt = X(f)Y ∗(f)df (16) For the rectangular function ΠT (t) : Z−∞ Z−∞
∞ ∞ ΠT (t) sinc(πfT ) 2 2 → < x, x >= x(t) dt = X(f) df (17) sinc(πT t) ΠT ( f) = ΠT (f) | | | | → − Z−∞ Z−∞ which means that the energy of a signal is preserved by FT. More details in [Hunter, 2019, Chap. 5.A] and [Mallat et al., 2015, Chap. 1]
27/108 28/108 Convolution and Fourier Transform Dirac delta and Fourier Transform
Convolution and Fourier Transform Let two signals x(t) and h(t) of respective Fourier transform X(f) and H(f) then Fourier transform and Dirac delta
[x(t) ? h(t)] = X(f)H(f) (18) I Fourier Transform of δ(t) and δ(t t0): F − + I The TF of a convolution is a pointwise multiplication in frequency. ∞ i2πft 0 [δ(t)] = δ(t)e− dt = e = 1 I The complex exponential function is the eigenvector for the convolution operator. F Z−∞
I Easy interpretation of the effect of a linear filtering. i2πft0 [δ(t t0)] = e− F − Proof I By duality of FT we have: [1] = δ(t) F i2πf0t [e ] = δ(f f0) ∞ ∞ 2iπft F − [x(t) ? h(t)] = e− x(u)h(t u)dudt F − I Convolution Z−∞ Z−∞ [x(t) ? δ(t)] = 1X(f) = X(f) ∞ ∞ 2iπf(u+v) = e− x(u)h(v)dudv F ∞ Z−∞ Z−∞ [x(t)δ(t)] = X(f) 1 = X(f)df = x(0) F ∗ ∞ 2iπfu ∞ 2iπfv = e− x(u)du e− h(v)dv = X(f)H(f) Z−∞ Z−∞ Z−∞ with the change of variable v = t u. − 29/108 30/108
The dirac comb Fourier transform of periodic signals
Cosine I The dirac comb is expressed as x(t) = cos(2πf0t) with f0 > 0 ∞ XT (t) = δ(t kT ) (19) − k= X−∞ I Bounded signal with unbounded energy. where X is the Cyrilic Sha symbol. I Intuitively this signal contains only one frequency (f0) I The Fourier Transform of the dirac comb is I Its TF can be computed thanks to the dirac distribution.
∞ 2iπkT f 1 ∞ k 1 [XT (t)] = e = δ f = X 1 (f) (20) F T − T T T FT of trigonometric functions k= k= X−∞ X−∞ j2πf t j2πf t where the second equality comes from the Poisson summation formula. e 0 + e− 0 1 1 cos(2πf0t) = = δ(f f0) + δ(f + f0) F 2 2 − 2 I The dirac comb is used to perform a regular temporal sampling. j2πf t j2πf t e 0 e− 0 1 1 I Multiplying a signal by the dirac comb corresponds to a convolution by a dirac sin(2πf0t) = − = δ(f f0) δ(f + f0) F 2i 2i − − 2i comb in the Frequency domain (and vice versa). The FT of sine and cosine is equal to 0 everywhere except on the frequency f0 of the functions.
31/108 32/108 Fourier transform of periodic signals (2) Fourier Transform in Rd
The Fourier Transform can be naturally extended to functions in Rd. Fourier transform of periodic signal Fourier Tansform in Rd d Let x(t) be a periodic signal of period T , it can be expressed as the following complex Let x(v): R C, the Fourier Transform of x can be expressed as 0 → Fourier series: k i2π T t 2iπ
k 1 2iπ
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Examples of Fourier Tranform in 2D Examples of Fourier Tranform in 2D (Modulus and Phase)
35/108 36/108 Fourier transform and angular frequency How to compute a Fourier Transform ?
I The FT in this course is a function of frequency f (in Hz). I Another common way to represent frequency is the angular frequency w (in rad/s) such that w Usual steps w = 2πf, f = 2π 1. Use known FT pairs if possible. I When using angular frequency the FT is non-unitary meaning that : 2. Express the function as a composition of operations with known properties:
∞ iwt I Linearity, time shift ˜[x(t)] = X˜(w) = e− x(t)dt F I Convolution Z−∞ I Duality 1 1 ∞ iwt ˜− [X(f)] = x(t) = e X˜(w)dw 3. Use the properties of FT on the composition. F 2π Z−∞ 4. Check properties (FT of even/odd function) to detect easy mistakes. I There exists a unitary angular frequency FT that scales both FT and IFT by 1 . √2π As a rule : try to avoid computing the integral but sometime you have to do it. I In the following we will sometime use the FT as a function of the angular frequency: w X˜(w) = X 2π
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Frequency response of LTI systems Frequency response and static gain
x(t) y(t) Response to a mono-frequency signal h(t) I For a system of impulse response h(t) with an input x(t) = e2jπf0t
+ ∞ 2jπf0h(t τ) Impulse response and frequency response y(t) = h(τ)e − dτ Z−∞ + I Most LTI systems can be expressed as a convolution of the form: 2jπf t ∞ 2jπf hτ = e 0 h(τ)e− 0 dτ y(t) = x(t) ? h(t) Z−∞ 2jπf0t = e H(f0) = x(t)H(f0) where h(t) is called the impulse response (the response of the system to an input x(t) = δ(t)) I An input signal with unique frequency f0 is multiplied by H(f0).
I The Fourier transform of the LTI system relation is I Its amplitude is multiplied by H(f0) and a phase Arg(H(f0)) is added. | | I The complex exponential is an eigenvector of the convolution operator. Y (f) = H(f)X(f) (23)
I The frequency response H(f) (also called transfer function) of the LTI system is Static gain the Fourier transform of h(t): The complex static gain is the constant K such that
+ Y (f) ∞ H(f) = (24) K = H(0) = h(t)dt X(f) Z−∞
39/108 40/108 LTI systems and Ordinary Differential Equation Representation of the frequency response
Frequency interpretation of the frequency response Ordinary Differential Equation (ODE) The system is defined by an equation of the form: I The frequency response of a system gives information on the transformations due to the system. dy(t) dny(t) dx(t) dmx(t) a0y(t) + a1 + + an = b0x(t) + b1 + + bm (25) I Quantities that can be plotted : dt ··· dtn dt ··· dtm H˜ (w) = Re(H˜ (w)) + jIm(H˜ (w)) ˜ Frequency response of an ODE = H˜ (w) ejArg(H(w)) | | I We recall the properties of the FT for th n-th derivative of a function: I H˜ (w) modulus of the frequency response. (n) | | 1 Im(H˜ (w) d x(t) I Arg(H˜ (w)) = H˜ (w) = tan− phase in radian. = (2iπf)nX(f) = (iw)nX(w) ∠ Re(H˜ (w) F dtn I The Frequency response of the ODE can be expressed as Graphical representation of systems m I Bode plot (Modulus+Argument). Y (w) b0 + b1jw + + bm(jw) H(w) = = ··· n (26) X(w) a0 + a1jw + + an(jw) I Nichols/Black plot (Modulus VS Argument). ··· I Nyquist plot (Real VS Imaginary)
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Bode plot Properties of the Bode plot
The logarithm and the argument allows for simple diagrams for combination of systems Multiplication
If two LTIs H˜1(w) and H˜2(w) are in series the the equivalent system is H˜ (w) = H˜1(w)H˜2(w)
I G˜(w) = G˜1(w) + G˜2(w)
I Φ(˜ w) = Φ˜ 1(w) + Φ˜ 2(w) Definition The Bode plot of a system is composed of two plots that are function of w: Division H˜ (w) I Magnitude (or gain) in decibels (dB) If and LTI can be expressed as H˜ (w) = 1 then H˜2(w) G˜(w) = 20 log ( H˜ (w) ) 10 I G˜(w) = G˜1(w) G˜2(w) | | − I Φ(˜ w) = Φ˜ 1(w) Φ˜ 2(w) I Phase in degrees or radians − This is particularly useful for rational frequency responses such as ODE. Φ(˜ w) = Arg(H(w) ) = H˜ (w) | ∠| | The scale of the radial frequency w is logarithmic, which means that for a rational frequency response H one will be mostly piecewise linear. 43/108 44/108 Diagramme de Black Nyquist plot
D´efinition Definition The Nichols plot (Diagramme de Black in France) is a parametric plot of H˜ (w) with The Nyquist plot is a parametric plot of H˜ (w) with Real(H˜ (w)) on x-axis and 20 log H˜ (w) on y-axis and phase Φ(˜ w) on x-axis. Imag(Φ(˜ w)) on y-axis. 10 | | I Show theM odulus/Phase trajectory as a function of w. I Show the trajectory of H˜ in the complex plane. I Can be ploted following the Bode plot w. I Used in system control to study the stability of systems.
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Frequency response of electronic systems First order system (1)
Principle I System Ohm’s law can be extended to capacitors and inductors using what is called complex x(t) = Ri(t) + y(t) called electrical impedance. 1 t The linear system i(t) u(t) is expressed as y(t) = i(v)dv → C Z−∞ U˜(w) = H˜ (w)I˜(w) = Z˜(w)I˜(w) x(t) = RCy0(t) + y(t) I Frequency response Y (f) 1 Resistor Capacitor Inductor H˜ (f) = = X(f) 1 + RC2jπf 1 t di(t) I u(t) = Ri(t) I u(t) = i(u)du I u(t) = L dt C I Using complex impedance ˜ ˜ −∞ I U(w) = RI(w) ˜ 1 ˜ I U˜(w) = jLwI˜(w) I U(w) = jCwR I(w) Y˜ (w) = ZcI˜(w) et X(w) = (ZR + ZC )I˜(w) I ZR = R 1 I ZL = jLw I ZC = jCw ˜ ˜ Y (w) Zc 1 1 H(w) = = = Z = The frequency response of passive electronic systems can be computed with simple ˜X(w) ZC + ZR 1 + R 1 + RCjw computation of complex numbers. ZC
47/108 48/108 First order system (2) First order system (3)
Normalized system We reformulate the frequency response as as: 1 H(w) = (27) Diagramme de Bode 1 + j w w0 Argument
1 1 1 where w0 = τ = RC . 1. H˜ (w) = w 1+j w Diagramme de Bode 0 1 2. Φ(˜ w) = arg(H(w)) = arg(1 + jw) = tan− (w) Modulus − − 3. limw 0 Φ(˜ w) = 0 ˜ 1 → 1. H(w) = 1+j w w0 4. limw Φ(˜ w) = π/2 →∞ − ˜ 1 ˜ 1 2. H(w) = 2 5. When w = w0, Φ(w) = tan− (1) = π/4 ( 45◦) | | 1+ w w2 − − − r 0 when w = 10w0, Φ(˜ w) = 84◦ w2 − ˜ when w = .1w0 Φ(˜ w) = 6◦ 3. G(w) = 20 log10( H(w) ) = 10 log10(1 + w2 ) | | − 0 −
4. limw 0 G˜(w) = 0 → ˜ w2 5. limw G(w) = 10 log10( w2 ) = 20 log10(w) + 20 log10(w0) →∞ − 0 −
6. When w = w0, G˜(w) = 10 log (2) = 3dB − 10 − 49/108 50/108
First order system (4) First order system (5)
Bode plot Nichols plot
Bode Diagram Nichols Chart 0
−10 10 3 dB 6 dB −3 dB −20 0 −6 dB
Magnitude (dB) −30 −10 −12 dB
−40 −20 dB 0 −20
Open−Loop Gain (dB) −30
−45 −40 −40 dB Phase (deg)
−50 −90 −2 −1 0 1 2 −360 −315 −270 −225 −180 −135 −90 −45 0 10 10 10 10 10 Frequency (rad/sec) Open−Loop Phase (deg)
51/108 52/108 2 zwn + wn z 1 − − p
First order system (6) Second order system (1)
I Complex Impedance L Y˜ (w) = ZcI˜(w) Nyquist plot X˜(w) = (ZL + ZR + ZC )I˜(w) Nyquist Diagram
0.5 10 dB6 dB4 dB2 dB0 dB−2 dB−4 dB −10 dB −6 dB I Frequency response
1 Y˜ (w) Z 20 dB −20 dB H˜ (w) = = C = jCw ˜ 1 0 X(w) ZL + ZR + ZC jCw + R + jLw Imaginary Axis I Normalized frequency response
−0.5 1 k −1 −0.5 0 0.5 1 ˜ H(w) = 2 = jw jw Real Axis 1 + RCjw + LC(jw) 1 + 2z + ( )2 wn wn
I k Static gain : k = 1 R C I z damping ratio of the system : z = 2 L 1 wn natural frequency of the system : wqn = I √LC 53/108 54/108
Second order system (2) Second order system (3)
Linear differential equation Response of the system for z > 1 The second order differential equation corresponding to the system is I c1 and c2 are real coefficients. d2y(t) dy(t) + 2zw + w2 y(t) = kw2 x(t) (28) I The FT can be expressed as dt2 n dt n n M M H˜ (w) = (32) jw c1 − jw c2 Factorization − − w The second order system can be factorized as with M = n , 2√z2 1 − kw2 I The impulse respons of the system is H˜ (w) = n (29) (jw c1)(jw c2) − − h(t) = M(ec1t ec2t)Γ(t) − with I The step response of the system is c1 = (30) ec1t ec2t 2 c2 = zwn wn z 1 (31) e(t) = 1 + M Γ(t) c1 − c2 − − − p c1 and c2 are called the poles of the transfer function.
55/108 56/108 Second order system (4) Second order system (5)
Response of the system for z < 1 Response of the system for z = 1 The FT becomes: I In this case the damping is weak and oscillations appear. 2 kwn H˜ (w) = (33) I This comes from teh fact that when z < 1 coefficients c1 and c2 are complex. (jw + w )2 n The impulse response is that is the square of one first order system. The impulse response fo the system can be expressed as h(t) = M(ec1t ec2t)Γ(t) − 2 wnt h(t) = wnte− Γ(t) I The step response is
The step response can be expressed as zwnt wne− 2 h(t) = sin wnt 1 z Γ(t) 2 wnt wnt √1 z − e(t) = (1 e− wnte− )Γ(t) − − − p that is a sine with an exponentially decreasing magnitude.
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Second order system (6) Second order system (7)
Impulse and step responses Bode plot We can plot the Bode plot using the normalized frequency response: Réponse impulsionnelle d’un systeme du second ordre 0.6 k z=0.5 H(w) = (34) z=1 2 0.4 jw + 2z jw + 1 z=1.5 wn wn 0.2 Modulus ˜ k 0 1. H(w) = 2 . jw +2z jw +1 wn wn −0.2 −5 0 5 10 15 20 2. H˜ (w) = k . 2 2 2 | | w 2 w 1 w +4z w Réponse indiciele d’un systeme du second ordre s − n n 1.5 3. G˜(w) = 20 log ( H˜ (w) ) = z=0.5 10 | | z=1 2 2 2 1 z=1.5 w 2 w 10 log10 1 w + 4z w + 20log(k) − − n n ! 0.5 4. limw 0 G˜(w) = 20log(k) → ˜ w4 5. limw G(w) = 10 log10( 4 ) = 40 log10(w) + 40 log10(wn) 0 →∞ − wn − −5 0 5 10 15 20 6. En w = w0, G˜(w) = 20 log (2z) + 20 log(k). − 10 59/108 60/108 Second order system (8) Second order system (9)
Properties of the modulus Bode plot The modulus of the frequency response for z < (2)/2 has a maximum at the I Argument following frequency p k 2 ˜ wmax = wn 1 2z 1. H(w) = jw 2 jw . +2z +1 − wn wn I The value of the modulus at this frequencyp is 2 2z w ˜ jw jw 1 wn 2. Φ(w) = arg(H(w)) = arg( + 2z + 1) = tan− 2 . k wn wn 1 w − − − w2 H˜ (wmax) = n ! | | 2z√1 z2 − 3. limw 0 Φ(˜ w) = 0 → I The cutoff frequency at -3dB is equal to 4. limw Φ(˜ w) = π( 180◦) →∞ − − 1 5. En w = w0, Φ(˜ w) = tan− (1) = 90◦, 2 2 4 w 3 = wn 1 + 2z + 2 4z + 4z − − − − q p
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Second order system (10) Second order system (11)
Bode plot Nichols plot
Bode Diagram Nichols Chart
50 40 0 dB 0.25 dB z=0.1 0.5 dB z=1 20 0 1 dB −1 z=10dB 3 dB 6 dB −3 dB −50 0 −6 dB −12 dB
Magnitude (dB) −100 −20 −20 dB
−150 −40 −40 dB 0 z=0.1 −60 −60 dB −45 z=1 Open−Loop Gain (dB) z=10 −80 −80 dB −90
Phase (deg) −135 −100 −100 dB
−120 dB −180 −120 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 −180 −135 −90 −45 0 Frequency (rad/sec) Open−Loop Phase (deg)
63/108 64/108 Second order system (12) Applications of analog signal processing
Nyquist plot R2
Nyquist Diagram - R + 6 0 dB z=0.1 z=1 CR z=10 1 4
2 dB −2 dB 2 Applications of analog signal processing 4 dB −4 dB 6 dB −6 dB 10 dB−10 dB I Analog signal filtering. 0 I Electronic passive and active filters. Imaginary Axis I Modeling and filtering with physical systems. −2 I Telecommunications. I Amplitude modulation. −4 I Multiplexing. I Fourier opticsL −6 −6 −4 −2 0 2 4 6 I Light propagation in perfect lens/mirror systems. Real Axis I Point spread functions of telescope and cameras.
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Analog filtering Applications of analog filtering
x(t) y(t) Filter
Definition Signal processing system that aim at selecting part of the signal and attenuating another part (noise). Analog filtering as opposed to digital filtering (next course) I High end audio, amplifiers, (equalizer, echo). I Car suspension. Objectives I Seismic protection. I Find a system that transform a signal x(t) to extract pertinent information. I Band-pass before Analog-to-Discrete conversion. I Attenuate noise in a signal. I Fourier optics, telescope modeling. I Separate several components of a signal (when different frequency bands).
67/108 68/108 Signal to Noise Ratio (SNR) Filtering and bandwidth Gain and Attenuation I In order to characterize a filter one uses its Gain/Phase (Bode plot). Aditive noise The recording of a signal often contains additive noise: G˜DB (w) = 20 log ( H˜ (w) ) et Φ(˜ w) = Arg(H˜ (w)) 10 | |
y(t) = x(t) + n(t) I Attenuation is also often used A˜(w) = G˜DB (w) − y(t) is the recorded signal, x(t) is the signal of interest and n(t) is the noise. Bandwith and passband Signal to Noise Ratio The band with of a filter is the set of frequency for which the Gain is over a reference (usually -3dB). Bandwith at 3dB: P − SNR = x ou SNR(dB) = 10 log (R ) (35) P 10 S/B H˜ (w) n BW = w 20 log | | 3 | max( H˜ (w) ) ≥ − I Px is the power of the signal and Pn is the power of the noise. | | 2 Ax I When signals are cosine the SNR is SNR = 2 where Ax and An are the An Types of filters amplitudes. I Low-pass, BW = [O, fc] with fc cutoff frequency I The objective of filtering is often to maximize the SNR. I High-pass, BW = [fc, ] ∞ I Band-pass, BW = [fc1 , fc2 ]
I Band-stop, BW = [0, fc ] [fc , ] 1 ∪ 2 ∞ 69/108 70/108
Filter distortion Filter distortion (2)
Undistorted transmission A signal is considered undistorted when the output of the system is With
y(t) = Cx(t t0) I C a constant gain. − I t0 > 0 is a delay. Phase distortion Let a system of frequency response A system with no distortion has the following FT and impulse response H˜ (w) = H(w) ejφ(w) ˜ | | X(w) jwt0 H˜ (w) = = Ce− et h(t) = Cδ(t t0) Y˜ (w) − We can deduce that for
With x(t) = cos(ωt) I H˜ (w) = C or else amplitude distortion. y(t) = H˜ (ω) cos(ωt + φ(ω)) = H˜ (ω) cos(ω(t + φ(ω)/ω)) | | | | | | I Arg(H˜ (w)) = wt0 or else phase distortion. − The delay φ(ω)/ω is also called propagation time of frequency delay. For it to be Note that the argument of the frequency response varies linearly with the frequency. independent from frequency it is necessary that φ(ω) = cte = τ φ(ω) = ωτ ω → 71/108 72/108 Ideal low pass filter Filter design
Definition I The ideal low-pass filter is often a theoretical object in signal processing. Real filter I Perfect to use when the noise and signal have non-overlapping spectra. I Ideal filters are non causal and cannot be implemented in practice . I The frequency response of the ideal filter is I We search for an approximation of the ideal filter. I the approximation has to respect constraints (Gabarit in french). 1 if f < fc H(f) = | | 0 else Constraints of a filter where fc is the cutoff frequency. Parameters: I The impulse response of the filter is I Bandwidth BP and rejected band sin(2πfct) h(t) = 2fc = 2fcsinc(2πfct) I Oscillations : 2πfct I ε in passing bandwidth Realizable filter I δ in attenuated bandwidth I A realizable temporal filter is causal and stable (absolute integrable). I Ideal filter is neither of those and cannot be used for 1D (time) filtering. The constraints define the area that are acceptable for a given application. I For images (2D) causality is not necessary.
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Simple example of filter design Simple example of filter design
I Application to Brain computer interface. I Interesting signal for event related potentials below 12Hz (ws = 2π 12). ≈ ∗ I SNR= G˜(ws) G˜(wedf ) I Electrical noise (EDF) at 50Hz − I SNR is a decreasing function of w0 . (wedf = 2π 50). ∗ I What is the best value for w0? I Two low power signals As An. ≈ I Maximum attenuation of signal at -3dB. I Filtering with first order filter. Choix que w0
I With the maximum attenuation of 3dB constraint. ws w0 . I Frequency response I Before filtering: → ≤ ≤ ∞ A SNR= 20 log s = 0 I For w0 = wedf RS/B = 2.76dB ˜ 1 10 An → H(w) = w 1 + j After filtering : I For w0 = (wedf + ws)/2 = 37 2 π RS/B = 4.07dB w0 I ∗ ∗ → SNR= G(ws) G(wedf ) I Pour w0 = ws RS/B = 9.63dB I Gain in Db − → Choice of w ? w0 = ws respects the constraint and maximizes the SNR. I 0 → w2 G˜(w) = 10 log 1 + − 10 w2 0
75/108 76/108 Approximating a low pass filter (1) Approximating a low-pass filter(2) I Need for an approximation function that respects the constraints constrained optimization. I Criterion is optimized (for instance maximization of SNR). I Two approaches are usually used:
Maximally flat frequency response I Minimal distortion is achieved when the passband is flat. I Let H˜ (w) be the modulus of the frequency response of an order k filter. Constraints for a low-pass filter | | I H˜ (w) is maximally flat in w = 0 if all the Kthderivatives are null | | I Passband: 1 ε H˜ (w) 1 pour w < wp − ≤ | | ≤ dK H˜ (w) | | = 0 I wp: passing frequency. dwK I ε: passband margin parameter (ε = 1/2 3dB). → − I Stopband : H˜ (w) δ pour w > wa Equiripple filter | | ≤ I wa: attenuation frequency. I A better rollof (sharper decrease) can be achieved at teh cost of oscillations. I δ: stopband margin parameter. I Oscilations can occur in the passband (leading to distortion) of cutband (limited I wa wc is the transition band. − attenuation). I An equiripple filter has constant magnitude for its oscillations in the bandpass.
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Butterworth filter (1) Butterworth filter (2)
I Butterworth filters are maximally flat [Butterworth et al., 1930]. I The Butterworth filter is monotonically decreasing with the frequency. I The amplitude of the frequency response can be expressed as I The amplitude of the frequency response can be expressed as Butterworth 2n 4n 6n 1 ˜ 1 w 3 w 5 w ˜ 1 H(w) = 1 + + ... H(w) = (36) | | − 2 wc 8 wc − 16 wc | | 2n w 0.8 1 + w c The derivative in w = 0 is then null up to order k = 2n 1. r 0.6 I with − 0.4 I The frequency response of a (normalized) Butterworth filter can be expressed as I n: order of the filter. ˜ 1 H(w) = B (w) where Bn(w) is a Butterworth polynomial : 0.2 n I wc: cutoff frequency. n 2 2 2k+n 1 0 (jw)) 2jw cos − π + 1 if n = even k=1 − 2n 0 0.2 0.4 0.6 0.8 1 ,Bn(w) = n 1 h −2 2 2k+n i1 (Qjw + 1) (jw)) 2jw cos − π + 1 if n = odd I The passing wp and attenuation wa frequencies are: k=1 − 2n Q h i For H˜ (w) = 1 ε For H˜ (w) = δ Order Polynomial | | − | | 1 1 + jw 1/2n 1/2n ε 1 δ 2 (jw)2 + √2jw + 1 wp = wc wa = wc − 1 ε δ 3 (jw + 1)((jw)2 + jw + 1) −
79/108 80/108 Chebyshev filter Filter implementation
Chebyshev type 1 Chebyshev type 2 Implementation of the filter consist in finding the physical components that recovers the selected frequency response H˜ (w). 1 1
0.8 0.8 Passive filter
0.6 0.6 I Only passive components (R, C, L).
0.4 0.4 I No energy source, no amplification (conservation of energy).
0.2 0.2 I The input and output impedance has an effect on the frequency response (impedance matching). 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Active filter I Better rolloff than Butterworth of same order but leads to oscillations in the I Use an energy source and Operational Amplifiers (OA). bandpass (type 1) or in the stopband (type 2). I OA has near infinite impedance but limited bandwidth (typically 100KHz). I Equiripple filter. I Saturation can occur (non-linearity). I Amplitude of the frequency response: 1 I Stability can be a problem (due to feedback) H˜ (w) = | | Tn( ): Chebyshev polynomial of 2 2 w I 1 + ε Tn · Rarely use inductors in practice (price, resistance, space, mutual inductance) ! wc order n. r
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Passive filters (1) Passive filters (2)
Example filter Butterworth Filter I Brain-Computer Interface application. I Corresponding frequency response with the Cauer topology. I w0 = ws = 2π 12 ∗ I For an order n filter with cutoff frequency wc = 1 the following structure: 1 1 I w0 = RC RC = 2 π 12 0.01326 → ∗ ∗ ≈ L2 L4 Ln-1 I What to choose for R and C ? I Price and space constraints.
C1 C3 Cn Filter transformation I low-pass high-pass → With the values : 1/jCw jLw et jLw 1/jCw → → 2k 1 I Ck = 2 sin( 2−n π) for k odd. 2k 1 I low pass band-pass I Lk = 2 sin( − π) for k even. → 2n Assuming the input and output have a 1 Ohm resistance. 1/jCw B/C(jw + 1/jw) et jLw L/B/(jw + 1/jw) I → →
83/108 84/108 Active filters (1) Active filters (2) First order active filter (with amplification)
R2 Second order active filter (Structure from [Sallen and Key, 1955]) r2 C2
- R R - R + K +
r1 CR1 C1
I Frequency response I Frequency response K A H˜ (w) = H˜ (w) = 2zjw (jw)2 jw 1 + + 2 1 + wn wn w0 where where 1 C1 3 K r1 + r2 R1 + R2 1 wn = et z = − et K = A = et w0 = R√C1C2 C2 2 r1 R1 RC r
I Parameters: R,C1,C2, r1, r2. I Parameters: R,C,R1,R2 I Permute R and C for a high-pass filter.
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Analog filtering : mechanical filter Modulation
Modulation is an encoding method that allows to transport a band-limited signal. Virgo Gravitational waves detector [Acernese et al., 2014] Demodulation is the reverse operation. I Interferometer to detect gravitational waves. Motivations I Attenuate vibrations from the earth [Braccini et al., 1996] I Raw signal transmission often not efficient (electromagnetic waves). 9 I Objective : attenuations of 10− for high frequencies. I The change in frequencies allow transmitting several band-limited signals in I Use a mirror in a chamber with mechanical filters. parallel. I Use a series of mechanical filters for the attenuation. I Use only of an authorized bandwidth. I Active correction for remaining low frequencies. Definitions I Modulating signal x(t) is a band limited signal we want to transmit (X(f) = 0 pour f > fx). | | I Carrier is the periodic base signal p(t) used for transportation often :
p(t) = cos(2πfpt)
I Modulated signal y(t) is a band-limited signal that can be transported in the physical medium (cable, air, optical fiber)
87/108 88/108 Amplitude Modulation (1) Amplitude Modulation (2)
h <1 y(t) Modulation index a(t) I Envelope of the modulated signal.
x(t) y(t) a(t) = Ac 1 + ksx(t) | | h =1 I Maximum amplitude of modulating signal: Definition: Amplitude Modulation (AM) Mx = max x(t) The carrier is multiplied by the modulating signal x(t) t | | I The index of modulation is defined as y(t) = Ac(1 + ksx(t)) cos(2πfpt + φm) h >1
h = ksMx I ks: modulation factor
I fp: carrier frequency I h < 1: under-modulation. I h > 1: over-modulation. I φm: phase (usually added during transmission).
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Amplitude Modulation (3) Amplitude Modulation (4)
Synchronous demodulation Interpretation in the Fourier domain Done with multiplying the signal with the carrier:
I Multiplication Convolution. X(f) w(t) = y(t) cos(2πf t + φ ) → p d Y (f) = X(f) ?P (f) = As(1 + ksx(t)) cos(2πfpt + φm) cos(2πfpt + φd)
-fx fx As As = (1 + ksx(t)) cos(φm φd) + (1 + ksx(t)) cos(4πfpt + φm + φd) I The spectrum of the modulating P(f) 2 − 2 signal is moved around the After low pass filtering (and removing of the constant) one can recover frequency fp. -fp fp Simple way to transmit a band As I Y(f) xˆ(t) = ksx(t) cos(φm φd) limited signal in a given bandwidth. 2 − I Modulated signal spectrum is -fp fp contained in fp fx. I cos(φm φd) = 1 if φm = φd. ± − I Very important to have a good synchronization (requires active components).
91/108 92/108 Amplitude Modulation (5) Applications of amplitude modulation (1)
y(t) a(t) ˆx(t)
C R
Low frequency radio broadcasting Asynchrone demodulation I First AM transmission R. Fessenden on 23 December 1900 at Cobb Island, Maryland (1.6Km). I Synchronous demodulation can require complex active components. I 1907 Lee de Forest invents the triod vacuum tube allowing for a better I A coarse approximation of the envelope of the signal can be done with a simple amplification [De Forest, 1908]. diode/RC system. I Weather bulletin emitted from the Eiffel Tower in february 1922. I Requires under-modulation because if h < 1 then I France Inter grandes ondes
a(t) = Ac 1 + ksx(t) = Ac + Acksx(t) | | I Emitted between 1 January 1947 and 31 December 2016. I Allouis longwave transmitter (2000KW), now used for TDF time signal. I Can require a lot of power for transmission.
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Applications of amplitude modulation (2) Frequency modulation (1)
X1 (f) X2 (f) X3 (f)
-fx fx -fx fx -fx fx
Definition Frequency modulation (FM) consists in modifying the frequency of the carrier using x(t). The modulated signal has the following form:
-2fp -3fp -fp 0 fp 2fp 3fp t y(t) = cos 2π f(τ)dτ Frequency-division multiplexing Z0 f(t) = f + f x(t) is the instantaneous frequency of the signal. I Multiplexing: transmission of several signals in parallel. I p ∆ I If x(t) = 0 we recover the carrier. I Use of a different fp for each signal. I When x(t) = 0 the instantaneous frequency is modified by x(t) I Every signal is band limited : if ∆fp > 2fx then no loss of information. 6 I f∆ is the frequency deviation (equivalent to ks in AM). I Frequency Hoping: Experimented by G. Marconi, Patent by N. Tesla [Tesla, 1903], proposed for secret communication by [Kiesler and George, 1942].
95/108 96/108 Frequency modulation (2) Fourier Optics
Properties of Frequency Modulation Principle I More robust than AM (noise, att´enuation)but propagation distance limited. Introduction to Fourier Optics: [Perrin and Montgomery, 2018, Goodman, 2005] I More complex to implement (requires a Voltage Controled Oscillator VCO). I Huygens–Fresnel principle for wave propagation. I Intuitively the spectrum of the modulated signal should be = 0 only in the band I 6 fp f∆Mx, BUT I When the source is at infinity, one can use the Fraunhofer diffraction (far field). ± I Continuous variation of the frequencies imply a spectrum on all frequencies. I Several optical elements corresponds to linear operations and can be defined as LTI and modeled/interpreted through Fourier Transform. I The Carson bandwidth rule states that most of the signal power (98%) is in the band I Difference between coherent VS incoherent sources. b = 2(f∆ + fx) Applications of Fourier transform in optics Application of Frequency Modulation I Analog image processing techniques. I FM radio broadcasting. I MRI : sampling of an image in the Fourier domain. I Frequency modulation synthesis (chiptunes). I Astronomy : modeling of telescopes, source detection, coronarography. I Magnetic tape storage.
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2D Fourier Transform using a lens The 4F correlator
I Place a transmissible object at the focal length of a lens. I The FT of the object is formed on the focal plane behind the lens. I FT computed at the speed of light, depends on the precision of the optics.
I Let i(v) be the 2D image in the focal plane before the lens and I(u) its FT. I Here f is the focal length of the lens (in optics ν or υ are often used to denote frequency). One can use the FT provided by Optical lenses to perform analog image filtering. I λ is the wavelength of the light source. I p I The Filtering is done with a mask in the Fourier plane of the image. I The the image formed in the right focal plane will be I( λf ) where p is the position in the focal plane. I Equivalent to a convolution (correlation). I The output image is mirrored due to the two FT instead of an inverse FT. I Active research domain in Optical Neural Networks [Zuo et al., 2019] 99/108 100/108 Telecope and Point Spread Function (PSF) Airy disk and angular resolution
Circular Aperture I The PSF for a circular aperture often called the Airy disk comes from the FT of the circle 2D[circ(r)] = J1(2pir0)/r0 . F I The diameter of the circle defines the maximum resolution.
I The source point in the images are considered incoherent to the observed image Angular resolution (intensity) is the sum of the responses of each source. I Minimal angle that allows discriminating two point sources. I A telescope can be considered as a LTI system (at least close to the axis). I Given by the Rayleigh criterion I The relation between the true image and the image observed in the focal plane is λ always a convolution by what is called the Point Spread Function: θ = 1.22 D y(v) = x(v) ? h(v) λ is the wavelength and D is the diameter of the telescope.
I The PSF h can be obtained as I It corresponds to the first zero of the Bessel J1 function.
1 2 h(v) = − [A(u)] F where A(u) is the aperture shape of the telescope. 101/108 102/108
Fourier Optics in astronomy Bibliography
I Signals and Systems [Haykin and Van Veen, 2007]. I Signals and Systems [Oppenheim et al., 1997]. I Signal Analysis [Papoulis, 1977]. I Fourier Analysis and its applications [Vretblad, 2003].
Real life telescopes I Polycopi´esfrom St´ephaneMallat and Eric´ Moulines [Mallat et al., 2015]. I New telescopes have several small mirrors : more complex PSF. I Th´eorie du signal [Jutten, 2018]. I Fourier Optics model only for perfect optics. I Distributions et Transformation de Fourier [Roddier, 1985] I Lenses/mirrors have optical aberrations and a surface roughness introducing scattering. I Ground telescope have to compensate for atmospheric turbulence (deformable mirrors with adaptive optics).
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