INTRODUCTION TO SIGNAL PROCESSING

Transcription

1 INTRODUCTION TO SIGNAL PROCESSING Iasonas Kokkinos Ecole Centrale Paris Lecture 7 Introduction to Random Signals

2 Sources of randomness Inherent in the signal generation Noise due to imaging Prostate MRI Thermal noise: Denoised Version Movement of electrons inside resistor in equilibrium

3 Sources of randomness Doppler Radar Send out wave of frequency ω Listen to its reflection Reflected frequency will depend on targets speed Noise: Birds, insects ( angels ) Sea motion Wind on trees How can we decide if a target is present?

4 Sources of randomness Speech generation: deterministic system Complex mechanical models Turbulence for fricatives For engineering: simple model + noise Texture modeling in images Same feel as natural images Sufficient for computer vision, maybe not for graphics

5 Random Signals Speech signal: Scatter plots of successive samples

6 Random Signals What can we learn from the one signal about the other? How can we understand if signals are regular?

7 Random Signals How can we identify when the signals behavior changes?

8 Random Signals How can we remove daily fluctuations? How can we predict the signal?

9 Random Signal Analysis Extract information for understanding & classification Spectral Estimation Parametric Modelling Applications: Signal Compression/Transmission Pattern recognition (e.g. speech recognition) Signal Detection (e.g. presence of sinusoid/target) Signal Estimation (e.g. frequency of sinusoid/speed)

10 Random Signal Processing Denoising Signal corrupted by noise Goal: recover signal Signal: possibly random Tracking Dynamical system Noisy observations of state Goal: Recover actual state

11 7 th Lecture Layout Randomness in Signals Introduction to Stochastic Processes First and Second order statistics Power Spectrum LTI Systems & Stochastic Processes

12 Stochastic Process examples Motion of a particle in a liquid ζ: the particle we chose to observe Voltage of AC generator with unknown phase ζ: the signal we get by plugging in Also known as: random process, random signal

13 Stochastic Process Family of signals Second order distribution: Density: N-th order distribution: joint distribution of

14 Discrete-time Stochastic Process ω: Realization of the random process For any ω: x is a discrete-time signal At any time n: x[n] is a random variable

15 Description of Stochastic Processes In general: joint distribution for any order and any time. In practice: only a few statistics are used Mean at time t: expected value of x(t): Autocorrelation:

16 7 th Lecture Layout Randomness in Signals Introduction to Stochastic Processes First- and Second- order statistics Power Spectrum LTI Systems & Stochastic Processes Spectral Factorization Normal & Predictable Processes

17 Discrete-time stochastic process x[n] Expected Value Variance Autocorrelation Autocovariance

18 Stationary Processes Strict-sense-Stationary process: Joint distribution of any k observations does not change with time Wide-sense-stationary (WSS) process: Constant mean Autocorrelation is a function of Bounded variance

19 Autocorrelation of WSS processes Properties Hermitian Symmetry: Maximum: Average Power: Positive Semidefinite:

20 Autocorrelation Matrix Consider Autocorrelation matrix: Properties: Hermitian: Toeplitz: Positive Semidefinite: For any vector, Autocovariance Matrix: where

21 WSS White Noise Process v(n) WSS White Noise: Zero mean: Auto-correlation: Autocorrelation matrix: Diagonal Observations are uncorrelated White Gaussian Noise (WGN) Each sample is independent of other samples Each sample follows zero-mean Gaussian distribution

22 White Gaussian Noise White noise: sequence of uncorrelated random variables WGN: Gaussian variables WGN Figure Credit: Manolakis, Ingle & Kogon

23 Autocorrelation matrix of sinusoid + WGN Sinusoid signal: Autocorrelation of noise: Autocorrelation of x[n]: Autocorrelation matrix will have the form: where

24 DT-processes x[n], y[n] x[n] = rainfall on day n y[n] = #umbrellas sold on day n Cross-Correlation Cross-Covariance Relation between Covariance & Correlation: Consider: y[n] temperature on day n.

25 7 th Lecture Layout Randomness in Signals Introduction to Stochastic Processes First and Second order statistics Power Spectrum LTI Systems & Stochastic Processes

26 Power Spectrum Definition of power spectrum of a WSS stochastic process x[n]: Wiener-Khintchine theorem: The power spectrum of a WSS stochastic process equals the DTFT of its autocorrelation

27 Power Spectrum of white noise Autocorrelation function: Power Spectrum: Equal power on all frequencies Just like white Light Red, pink, violet noises..

28 7 th Lecture Layout Randomness in Signals Introduction to Stochastic Processes First and Second order statistics Power Spectrum LTI Systems & Stochastic Processes

29 Stochastic Processes & LTI Systems

30 Stationary Processes and LTI systems WSS process x(n) drives an LTI system: Mean: Input-output crosscorrelation:

31 Stationary Processes and LTI systems Output autocorrelation: Power Spectrum:

32 Systems & Random Signals System: Deterministic Input: Stochastic Process Output: Stochastic Process System s Effect: