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 5/108 6/108 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−∞ 9/108 10/108 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. 13/108 14/108 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.