bg physiological signal processing 07 random2020psp01mk 9815

Random Processes a Statistical Specification of Random Processes b Stationary Random Processes c Response of LTI Systems to Random Processes d Power Spectral Densities e Some Useful Rand

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Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

Random Signals

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I Introduction

II Introduction to Electrophysiology

III Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

VI The z-Transform

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Random Signals

1 Probability Models & Random Variables

a) Randomness & Statistical Regularity

b) Random Variables

c) Probability Distributions

d) Statistical Averages

e) Two Useful Random Variables

2 Jointly Distributed Random Variables

3 Linear Estimation

4 Random Processes

5 Estimation of Mean, Variance, & Covariance

6 Sources of Noise & Interference

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Randomness & Statistical

Regularity (1)

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cialist_republic_1000_dong_2003_vantaa brass/1029811/Default.aspx

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Randomness & Statistical

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Random Signals

1 Probability Models & Random Variables

b) Random Variables

3 Linear Estimation

4 Random Processes

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Random Variables (1)

• Sample space: S = {1, 2, 3, 4, 5, 6}.

• Event: a particular subset of sample space.

• For instance: the event (X < 4) is the subset

{1, 2, 3} of S.

( )

Number of occurrences of event

Total number of trials

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A random variable is a function from a sample

space S to the real numbers.

S = {EEE, EEP, EPP, EPE, PPP, PPE, PEE, PEP}

The total number of emblem X = {0, 1, 2, 3}

The total number of pagoda Y = {0, 1, 2, 3}

The total number of emblem minus the total

number of pagoda Z = {–3, –1, 1, 3 }

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• A random variable is a function from a sample

space S to the real numbers.

• A continuous random variable can take as a

value any real number in a specified range.

• A discrete random variable can take values

from a finite or countably infinite set of

numbers.

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Random Signals

b) Random Variables

c) Probability Distributions

3 Linear Estimation

4 Random Processes

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Probability Distributions (1)

To construct the histogram

of a set of observations:

1 Divide the horizontal axis into

intervals of appropriate size

Δ x,

2 Determine the number n i of

observations in the i th interval,

3 Draw over each interval a

rectangle with area

proportional to the relative

Histogram

1 2 3 4 5 6 7

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Random Signals

b) Random Variables

d) Statistical Averages

3 Linear Estimation

4 Random Processes

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i i

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Random Signals

b) Random Variables

e) Two Useful Random Variables

3 Linear Estimation

4 Random Processes

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Two Useful Random Variables (1),

2

b a

A random variable X is said to be uniformly

distributed over the interval ( a, b), if f X ( x) is given by:

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Two Useful Random Variables (2),

σ π

− −

and variance σ 2 , denoted X ∼ N(m, σ 2 ) or N(x; m, σ 2 ), if

f X

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Random Signals

2 Jointly Distributed Random Variables

a) Probability Functions

b) Covariance & Correlation

c) Linear Combinations of Random Variables

3 Linear Estimation

4 Random Processes

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Probability Functions (2)

The random variables X and Y are statistically independent

if the likelihood of the values of one does not depend upon

the likelihood of the other.

, ( , ) ( ) ( )

f x y = f x f y

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Random Signals

a) Probability Functions

b) Covariance & Correlation

c) Linear Combinations of Random Variables

3 Linear Estimation

4 Random Processes

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Random Signals

a) Probability Functions

b) Covariance & Correlation

c) Linear Combinations of Random Variables

3 Linear Estimation

4 Random Processes

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Linear Combinations

of Random Variables (2)

T p

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T p

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Random Signals

3 Linear Estimation

4 Random Processes

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Random Signals

3 Linear Estimation

4 Random Processes

a) Statistical Specification of Random Processes

b) Stationary Random Processes

c) Response of LTI Systems to Random Processes d) Power Spectral Densities

e) Some Useful Random Process Models

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Statistical Specification

of Random Processes (1)

• Recall: a random variable X is a rule for assigning

to every outcome ζ of a random experiment a real

number X(ζ).

• The random variable may take different values if

we repeat the experiment, and we do not know in advance which value will occur.

• Similarly, a random vector assigns a vector to

each outcome ζ of the sample space.

• Random process: to each ζ we assign a time

function X(t, ζ ) (continuous-time stochastic

process) or a sequence X[n, ζ ] (discrete-time

stochastic process).

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Statistical Specification

of Random Processes (2)

• Random process: to each ζ we assign a time

function X(t, ζ ) (continuous-time stochastic

process) or a sequence X[n, ζ ] (discrete-time

stochastic process).

• A random process is not one function (or

sequence), just as a random variable is not one

number.

• Each time we perform the random experiment, we observe only one realization or sample function

(sequence) of the process.

• x[n] is used to denote the discrete-time stochastic

process X[n, ζ ].

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Random Signals

3 Linear Estimation

a) Statistical Specification of Random Processes

b) Stationary Random Processes

c) Response of LTI Systems to Random Processes

d) Power Spectral Densities

e) Some Useful Random Process Models

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Stationary Random Processes (1)

• A stochastic process is said to be strictly stationary if the

sets of random variables x[n 1 ], , x[n p ] and x[n 1 + k], , x[n p + k] have the same joint probability distribution for any

• For p = 1, this implies that f(x[n]) = f(x[n + k]), that is, the

marginal probability distribution does not depend on time

that is:

• For p = 2, stationarity implies that all bivariate distributions

f(x[n], x[m]), depend only upon the lag (time difference) ℓ =

n − m, n ≥ m Thus, the covariance of x[n] and x[m] depends

on the lag ℓ, that is:

c xx [ n, m] = cov(x[n], x[m]) = c xx [ℓ] for all m, n

2

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1 The ACRS and ACVS have even symmetry:

2 The cross-correlation and cross-covariance

are NOT even sequences:

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3 The cross-correlation and cross-covariance

sequences are bounded by:

2

[ ] [0] E( [ ]) [ ] [0]

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4 The correlation matrix R x of the random

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5 The correlation matrix of a random vector

consisting of p consecutive samples of a

stationary process is symmetric and Toeplitz:

⋯

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Random Signals

3 Linear Estimation

b) Stationary Random Processes

c) Response of LTI Systems to Random Processes

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Response of LTI Systems to

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Random Signals

3 Linear Estimation

d) Power Spectral Densities

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Power Spectral Density (1)

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k k

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Random Signals

3 Linear Estimation

i White Noise Processes

ii Linear Processes

iii Autoregressive Moving Average (ARMA) Processes

iv Harmonic Process Models

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White Noise Processes

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Random Signals

3 Linear Estimation

i White Noise Processes

ii Linear Processes

iii Autoregressive Moving Average (ARMA) Processes

iv Harmonic Process Models

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Random Signals

3 Linear Estimation

ii Linear Processes

iii Autoregressive Moving Average (ARMA) Processes

iv Harmonic Process Models

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ARMA Processes (1)

• An ARMA(p, q) process defined by:

• The PSD of an ARMA(p, q):

• If q = 0: an all-pole system which generates an

autoregressive process of order p, AR(p).

• If p = 0: an all-zero system which generates a

moving average process of order q, MA(q).

• Given the ACRS r [ℓ], how to find { a , b }?

b e

a e

ω ω

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p

k k

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r r r

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Random Signals

3 Linear Estimation

ii Linear Processes

iii Autoregressive Moving Average (ARMA) Processes

iv Harmonic Process Models

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Harmonic Process Models

1 2

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Random Signals

3 Linear Estimation

4 Random Processes

5 Estimation of Mean, Variance, & Covariance

a) Basic Concepts & Terminology

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Basic Concepts & Terminology

(1)

• The random variable X is called an estimator for the mean μ of a random variable X from N observations x 1 ,

…, x N

• The numerical value x calculated from a particular set

of observations, is called an estimate.

• The estimator is the formula, while the value which it produces for a particular set of observations is the

estimator.

• The distribution of an estimator is called sampling

distribution to be distinguished from the distribution of

the random variables used for the estimation.

• A “good” estimator should have a distribution that is

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(2)

• The bias of an estimator θ of a parameter θ is:

• If B = 0 then the estimator is said to be

unbiased.

• Otherwise: biased.

• An unbiased estimator gives the correct value

on the average when used a large number of times.

B ( ) θ ˆ = E( ) θ θ ˆ −

θ ˆ

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(3)

• The variance of an estimator is:

• It shows the spread of its values about its

expected value; hence, in general, the variance should be small.

• A “good” estimator should have zero bias and the smallest possible variance.

2

θ ˆ = θ ˆ − θ ˆ var( ) E{[ E( )] }

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Random Signals

3 Linear Estimation

4 Random Processes

a) Basic Concepts & Terminology

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Sample Mean

1

k k

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Random Signals

3 Linear Estimation

4 Random Processes

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x m N

4 4

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Random Signals

3 Linear Estimation

4 Random Processes

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ˆ ˆ

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Random Signals

3 Linear Estimation

4 Random Processes

6 Sources of Noise & Interference

a) Noise in Biopotential Signal Measurements

b) Interference from External Electrical Field

c) Interference from External Magnetic Field

d) Conductive Interference

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Noise in Biopotential Signal

Measurements (1)

• (A.Y.K Chan Biomedical Device Technology – Principles and

Design Charles C Thomas, 2016)

• Biopotential signals are produced as a result of action potentials at the cellular level.

• Noise is simply defined as any signal other than the desired signal.

• Two different sources of noise and interference:

1 Artificial sources from the surrounding environment such as

electromagnetic interference (EMI) or mechanical motion For example, artifacts on an EEG recording caused by fluorescent lighting or unshielded power supply voltages are considered artificial interference.

2 Natural biological signal sources from the patient For example, in

ECG measurement, any signals other than ECG that arise from other biopotentials of the body are considered as natural noise These

include muscle artifact from the patient or electrical activity of the brain Brain activity is noise when measuring ECG; an ECG signal is considered noise when brain waves (EEG) are measured.

Tiêu đề	Random Signals
Tác giả	Nguyễn Công Phương
Trường học	[Unknown School/University]
Chuyên ngành	Physiological Signal Processing
Thể loại	Thesis
Năm xuất bản	2020

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Số trang	104
Dung lượng	737,62 KB