DSP_ch8

DSP_ch8 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Copyright (c) Andreas Spanias Ch 8 - 1 DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter 8 [email protected] http://www.eas.asu.edu/~spanias Copyright (c) Andreas Spanias Ch 8 - 2 Introduction to Random Signal Processing Deterministic Signals: are characterized explicitly by a mathematical equation and their value at any time can be predicted exactly. Random Signals: are not modeled explicitly by a mathematical equation, however, they are usually characterized by their statistics. Most “real-life” signals, such as those encountered in speech, communications,, and radar are random Our discussion concentrates on signals whose D.C. content and power spectra do not change with time are constant. Also we will be concerned with signals. whose statistics can be measured by time averaging. We call such signals wide-sense stationary and ergodic .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Copyright (c) Andreas Spanias Ch 8 - 3 Random Signal Processing - Some Definitions µ x N nN N Exn N xn == + →∞ =− [() ] () lim 1 21 The mean is defined as: Statistical expectation; for ergodic signals it is computed as a time average The variance σµ xx x x 22 2 = [ (() ) ) ] [ () ] The variance is a measure of dispersion from the mean Copyright (c) Andreas Spanias Ch 8 - 4 The mean, , is the DC component. The mean squared is the DC power. The mean square is the total average power. The variance is the AC power. The standard deviation is the rms value. x x 2 Ex n [( ) ] 2 σ x 2 x Examples, Interpretations, and Analogies to electrical signals 0 10 20 30 40 50 60 70 80 90 100 -4 -3 -2 -1 0 1 2 3 σ 2 1 > σ 2 2 σ 2 1 σ 2 2 µ 1 = µ 2 =0
Background image of page 2
3 Copyright (c) Andreas Spanias Ch 8 - 5 The Autocorrelation rm E xnm xn N xn mxn xx N nN N ( ) [ ( ) ( )] lim ( ) ( ) =+ = + + →∞ =− 1 21 Basic Properties: ) ( ) 0 ( ) ( ) ( m r r m r m r xx xx xx xx = uncorrelated i/p (e.g., white noise) Digital Filter correlated o/p (e.g., colored noise) The filter acts as a correlator The autocorrelation is a measure of predictability of the signal, i.e., a correlated signal would be one whose future values can be predicted from past values Copyright (c) Andreas Spanias Ch 8 - 6 The Cross-correlation and the Cross-covariance () [( )() ] () xy xy yx rm E x nm y n r m = The cross-correlation is a measure of similarity of two signals. Correlated signals x y
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 Copyright (c) Andreas Spanias Ch 8 - 7 The Power Spectral Density (PSD) The PSD is the Fourier transform of the autocorrelation, i.e., −∞ = = m m j xx j xx e m r e R ) ( ) ( The PSD is real and positive and shows how power is distributed across frequency Digital Filter 0 20 40 60 80 100 120 140 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 20 40 60 80 10 120 140 0 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 measured Input PSD measured Output PSD freq. response Copyright (c) Andreas Spanias Ch 8 - 8 White Noise White noise is uncorrelated and its spectrum (PSD) is constant over the entire frequency range. rm m ww w () = σδ 2 2 ) ( w j ww e R σ = Ideal white noise sequences can not be generated. In practice, we can generate pseudo-white noise.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/02/2009 for the course EEE DSP taught by Professor Ap during the Fall '09 term at ASU.

Page1 / 18

DSP_ch8 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online