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 2 : Digital Signal Processing 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 1/80 2/80 Course overview Digital Signal Processing Fourier Analysis and analog filtering 4 Digital signal processing 4 Sampling Analog/Digital conversion 5 Sampling Reconstruction of analog signals Aliasing Digital filtering 13 Discrete Convolution Discrete Time Fourier Transform (DTFT) Z-transform and transfer function Finite signals 31 Circular convolution Digital Signal Processing Discrete Fourier Transform (DFT) I Microprocessors widely available and cheap since the 70s. Fast Fourier Transform and fast convolution Applications of DSP 54 I Analog-to-digital converter (ADC) and digital-to-analog converter (DAC). Digital filter design I Standard signal processing : ADC DSP DAC. Sinc interpolation → → Digital Image Processing I DSP more robust/stationary. Random signals 77 I Analog SP is faster but sensitive to physics (temperature). Signal representation and dictionary learning 77 I Digital Signal Processing can be done on dedicated hardware or processors. Signal processing with machine learning 77 3/80 4/80 Sampling Sampling in the Fourier domain Principle I Sampling is the reduction of a continuous-time signal to a discrete-time signal. I A discrete signal sampled for period T can is expressed as ∞ xT (t) = x(nT )δ(t nT ) (1) − n= Fourier transform of sampled signal X−∞ 1 T is the sampling period (or interval), f = is the sampling frequency. I If x(nT ) is bounded for n Z, xT (t) is a tempered distribution. I s T ∈ Due to the properties of the dirac δ the sampled signal is equal to I The FT of xT (t) can be expressed as a function of X(f) = [x(t)]: I F ∞ 1 1 ∞ 1 xT (t) = x(t) δ(t nT ) = x(t)XT (t) (2) [xT (t)] = [x(t)XT (t)] = X(f) ? X 1 (t) = X f (3) − F F T T T − T n= n= X−∞ X−∞ where XT (t) is the dirac comb of period T . I The regular sampling leads to a periodization in the Fourier domain. 5/80 6/80 Nyquist/Shannon sampling Theorem Nyquist/Shannon sampling Theorem Proof Let xT (t) = x(t) n∞= δ(t nT ) be the sampled signal. Its Fourier Transform is −∞ − P 1 ∞ 1 XT (t) = X f T − T n= X−∞ 1 1 1 1 Since we know that X is of support [ 2T , 2T ] it means that f [ 2T , 2T ] we have 1 − ∀ ∈ − XT (f) = T X(f). Now if we want to reconstruct the signal we can multiply in the Fourier domain by the Theorem [Shannon, 1949][Nyquist, 1928] ideal filter: T if f < 1 Let x(t) be a signal of Fourier transform X(f) that has a support in [ 1 , 1 ]. Then H(f) = | | 2T − 2T 2T 0 else the signal x(t) can be reconstructed from its sampling with Using the bounded support of X we have now f ∀ ∞ x(t) = hT (t nT )x(nT ) = xT (t) ? hT (t) (4) X (f)H(f) = X(f) − T n= X−∞ Which in the temporal domain means where πt πt sin( T ) ∞ ∞ hT (t) = sinc = πt (5) T x(t) = xT (t) ? h(t) = h(t) ? x(nT )δ(t nT ) = x(nT )h(t nT ) T − − n= n= For a signal of frequency support [ B, B], B is often called the Nyquist frequency X−∞ X−∞ − (half the sampling rate necessary for reconstruction). πt 7/80 where h(t) = sinc T is the inverse TF of H. 8/80 Aliasing Example of aliasing Aliasing in the time domain Aliasing in the time domain 1.0 1.0 0.5 x(t) x(t) 0.0 xT(t) * hT(t) 0.5 xT(t) * hT(t) xT(t) xT(t) 0.5 0.0 1.0 2 1 0 1 2 6 4 2 0 2 4 6 I Let x(t) = cos(2πf0t) be a signal that we want to sample. We suppose that fs < f < f = f . Aliasing I 2 0 s s I We have I The FT of xT (t) is a weighted sum of X(f) = [x(t)]: F 1 1 XT (f) = (δ(f f0 kfs) + δ(f + f0 kfs)) ∞ T 2 − − − 1 1 n k [xT (t)] = [x(t)XT (t)] = X(f) ? X 1 (t) = X f (6) X F F T T T − T n= f f X−∞ I The only components of the spectrum in [ s , s ] are: − 2 2 1 1 I When the support of X(f) is not in [ 2T , 2T ] the repeated shapes will overlap 1 − XT (f)H(f) = (δ(f f0 + fs) + δ(f + f0 fs) in frequency. 2 − − I In this case the signal cannot be reconstructed and some information is lost. I Reconstructed signal: x(t) = cos(2π(fs f0)t) 9/80 − 10/80 Aliasing in real life Analog to Digital Conversion ADC circuits Aliasing I Sampling frequency has to be twice the maximum frequency in the signal. I Low pass filtering before sampling (analog). I When sampling high frequency real life signals. I Several sources of noise : jitter (non perfect clock), non-linearity, I Always needs a low-pass filter (analog) before sampling. I For images CCD or CMOS (smartphones) sensors count photons. I Can be solved by oversampling (followed by filtering then subsampling). I Anti-aliasing filters in graphic cards (and digital cameras). Quantization I Computers are discrete, digital signal are discrete both in time and value. I Quantization is the conversion from continuous value to a finite bit format. I Number of bits has an important impact on SNR after reconstruction. 11/80 12/80 Discrete signal (1) Discrete signal (2) Dirac discrete [n] 1.00 Notations 0.75 I x(t) with t R is the analog signal. 0.50 ∈ 0.25 I xT (t) with t R is the sampled signal of period (T) but still continuous time: 0.00 ∈ 15 10 5 0 5 10 15 ∞ Discrete dirac xT (t) = x(nT )δ(t nT ) We note the discrete dirac δ[n] defined as − n= X−∞ 1 for n = 0 δ[n] = (7) I x[n] with n Z is the discrete signal sampled with period T such that: (0 else ∈ x[n] = x(nT ) Discrete signal Any discrete signal x[n] can be decomposed as a sum of translated discrete diracs: I Obviously one can recover xT (t) from x[n] with ∞ x[n] = x[k]δ[n k] (8) − ∞ k= xT (t) = x[n]δ(t nT ) X−∞ − n= X−∞ The discrete diracs are an orthogonal basis of L2(Z) of scalar product and corresponding norm I In order to simplify notations we will suppose T = 1 in the following. ∞ 2 ∞ 2 In this course we suppose that x[n] is bounded. < x[n], h[n] >= x[k]h∗[k], x[n] =< x[n], x[n] >= x[k] . I k k | | | | k= k= X−∞ X−∞ 13/80 14/80 Discrete Convolution Transfer function Filter h[n] in the time domain Transfer function magnitude H(e2i f) 0.20 Convolution between discrete signals h[n] Real(H(e2i f)) 1.5 Imag(H(e2i f)) Let x[n] and h[n] two discrete signals. The convolution between them is expressed as: 0.15 1.0 0.10 ∞ x[n] ? h[n] = x[k]h[n k] (9) 0.5 0.05 − 0.0 k= −∞ X 0.00 0.5 8 6 4 2 0 2 4 6 8 1.0 0.5 0.0 0.5 1.0 Digital filter properties Transfer function of a discrete filter Let the discrete system/operator/filter L described by its impulse response h[n]. I Let L be a digital filter of impulse response h[n]. i2πfk I Causality L is causal if h[n] = 0, n 0. L is causal if I For an input ef [k] = e the output of the filter is ∀ ≤ 1 for n 0 ∞ i2πf(n k) i2πfn ∞ i2πfk h[n] = h[n]Γ[n], where Γ[n] = ≥ (10) Lef [n] = e − h[k] = e e− h[k] (12) (0 else k= k= X−∞ X−∞ i2πfk I Stability A system is stable if the output of a bonded input is bounded. A I ef [k] = e are then the eigenvectors of the discrete convolution operator. necessary and sufficient condition is that I The Transfer function of the filter is defined as the following Fourier series: ∞ h[n] < (11) i2πf ∞ i2πfk | | ∞ H(e ) = e− h[k] (13) n= X−∞ k= X−∞ 15/80 16/80 This actually corresponds to the Fourier transform of the signal. Discrete Time Fourier Transform (DTFT) Fourier Transform and discrete convolution Discrete time domain Fourier domain 0.20 1.0 Fourier transform x[n] h[n] The Discrete Time Fourier Transform of the discrete signal x[n] is defined as 0.8 0.15 y[n] 0.6 0.10 i2πf ∞ i2πfk X(e ) = e− x[k] (14) 0.4 0.05 k= 0.2 X−∞ 0.00 0.0 8 6 4 2 0 2 4 6 8 1.0 0.5 0.0 0.5 1.0 I It is periodic and equivalent to the Fourier transform of xT (t).