stationary random process and digital signal processing - theory

8/17/2019 Stationary Random Process and Digital Signal Processing - Theory

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1 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

NOTES ON

STATIONARY RANDOM PROCESS AND

DIGITAL SIGNAL PROCESSING

Prepared by Le Thai Hoa

2004


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1. STATIONARY RANDOM PROCESS

1.1. Basic concepts

(1) Continuous random process:

)}()}...(),(),({)( 321 t xt xt xt xt x K k , t

Where: { }: Ensemble of sample functions xk (t)

k: Index of sample function (k=1,2,3…K)

t: Time variable

Random process {xk (t)} = Ensemble of sample function xk (t)

(2)

The random process is called as the K-variate random process (multi-variate

random process)

Ensemble (sample records) of random signal

(3) For discrete sample function, discrete values of any sample random function are

measured at certain time points t1, t2, t3, … t N (N: number of sampling values of

sample function)

t

t

t t+

x1(t)

k th sample function

1st sample function

xk (t)

Time shift


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)}()}...(),(),()( 321 N k k k k k t xt xt xt xt x : Discrete sample function

1.2. Classification of random process

(1) Classification of random process can be widely expressed as follows

Classification of random processes

1.3. Representation of random process:

(1) Time-domain representation (as raw formats and sources)

(2) Frequency-domain representation (due to Fourier Transform)(3) Time-frequency representation (due to Wavelet Transform)

1.4. Characteristics of random process

Basic statistical characteristics of two arbitrary random processes )(t xk and

)(t yk

(1)

Mean value (Expectation): First-order statistical moment

dt t x N

Limt x E t k N k x )(1

)]([)(

Random processor Stochastic field

Stationary process

Non-stationary processes

Ergodic process

Non-ergodic signals

Gaussian process

Non-Gaussian process


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dt t y N

Limt y E t k N k y )(1

)]([)(

(2)

Variance and covariance: Second-order moment

)()]([]))()([()( 2222 t t x E t t x E t k k xk x : Variance

)()]([]))()([()( 2222 t t y E t t y E t yk yk y : Variance

)(*)()](*)([

))]()())(()([()(

t t t yt x E

t t yt t x E t C

y xk k

yk xk xy

: Covariance

))]()())(()([()( t t xt t x E C xk xk xx : Covariance

))]()())(()([()( t t yt t y E C yk yk yy : Covariance

Note: Zero mean value process xk (t): 0)( t x

]))([()( 22 t x E t k x : Variance

]))([()( 22 t y E t k y : Variance

)()0( 2 t C x xx

)()0( 2 t C y yy

(3) Mean square and root mean square

)]([)( 2 t x E t C k xx : Mean square

)]([)( 2 t y E t C k yy : Mean square

Note: Zero mean value process xk (t): 0)( t x


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Variance )(2 t x = Mean Square Cxx(t)

(4) Correlations: Second-order moment

)](*)([)( t xt x E R k k xx : Autocorrelation

)](*)([)( t yt y E R k k yy : Autocorrelation

)](*)([)( t yt x E R k k xy : Cross-correlation

: arbitrary time (time shift or time lag)

Note: x xx xx RC 2)()(

y yy yy RC 2)()(

y x xy xy RC )()(

Zero mean random process: 0)( t x , 0)( t y

)()( xx xx RC

)()( yy yy

RC

)()( xy xy RC

(5) Correlation coefficients

)(

)(

)(

)()(

2

xx

x xx

xx

xx xx

R

R

R

C : Auto-correlation coefficient

)()(

)()()(

2

yy

y yy

yy

yy

yy R

R

R

C : Auto-correlation coefficient

)(

)(

)(

)()(

xy

y x xy

xy

xy

xy R

R

R

C : Cross-correlation coefficient


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x

xx

xx

xx xx

C

R

C 2

)(

)(

)()(

y

yy

yy

yy

yy

C

R

C 2

)(

)(

)()(

y x

xy

xy

xy

xy

C

R

C

)(

)(

)()(

Note 1:

i) )()( xx xx RC

ii) )0()( xx xx C C

iii) )0()( xx xx R R

iv)2)0( x xxC and

2)0( y yyC

v) )0(*)0(|)(|2

yy xx xy C C C

vi)222|)(| y x xyC

vii) )0(*)0(|)(|2

yy xx xy R R R

viii) )()( xx xx R R and )()( yy yy R R

)()( yx xy R R

Note 2: 0

i)2)0( x xxC : Variance


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ii)2)0( y yyC : Variance

iii)22 )]([)0( x xx t x E R : Mean square

iv)22

)]([)0( y yy t y E R : Mean square

(6) Power spectral density (PSD) function in frequency-domain

1.5. Power spectral density (PSD)

PSD function can be computed by following methods: i) Via correlation function (by

definition), ii) Via Fourier transform and iii) Via filter-squaring-averaging computation

(1) Spectra via correlation (by Fourier Transform of correlation)

By definition of spectral density through the Fourier Transform:

d e R f S f j xx xx2*)()(

: Auto-spectral density function

d e R f S f j yy yy2*)()(

: Auto-spectral density function

d e R f S f j xy xy2*)()(

: Cross-spectral densityfunction

Inverse Fourier Transform:

df e f S R f j

xx xx

2

*)()(

: Auto-correlation

df e f S R f j yy yy 2*)()(

: Auto-correlation

df e f S R f j xy xy 2*)()(

: Cross-correlation


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Sxx(f), Syy(f), Sxy(f): Two-sided spectra, f [-,]

One-sided spectral densities

d f Rd f R f S xx xx xx 2cos*)(22cos*)()(0

0

2cos*)(2)( d f f S R xx xx

Changing the two-sided spectral density Sxx(f) with f [-,] to the one-sided spectral

density Gxx(f) with f [0,]

)(2)( f S f G xx xx

)(2)( f S f G yy yy

)(2)( f S f G xy xy

Thus,

d f R f G xx xx 2cos*)(4)(0

; f [0,]

df f f G R xx xx 2cos*)()(0

; f [0,]

Real part and imaginary part of one-sided cross-spectral density:

f Hz

Spectra

[-,0] [0,]

Gxx(f)=2Sxx(f): One-sided

Sxx(f): Two-sided


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)()(*)(2)( 2 f iQ f C d e R f G xy xy f j

xy xy

)( f C xy : Co-spectrum

)( f Q xy : Quadratic spectrum

Writing in standard form:

)(|)(|)(

f j

xy xy xye f G f G

Where:

)()(|)(| 22 f Q f C f G xy xy xy

)(

)(tan)( 1

f C

f Q f

xy

xy

xy

)(cos|)(|)( f f G f C xy xy xy

)(sin|)(|)( f f G f Q xy xy xy

df f f iQ f f C R xy xy xy ]2sin)(2cos)([)(0

df f C R xy xy

0

)()0(

(2)

Spectra via Fourier transform

Fourier Transform (Kinchint-Weiner’s pair):

dt et x f X ft j

k k

0

2)()(

df e f X f x ft j

k k

0

2)()(


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Spectral density function:

]|)([|1

2)( 2 f X E T

Lim f G k T xx

]|)([|12)( 2 f Y E T

Lim f G k T yy

)](*)([1

2)( f Y f X E T

Lim f G k k T xy

1.5. Coherence

Coherence plays the same role as the correlation coefficient. The correlation coefficient

is expressed in time domain, whereas the coherence in frequency domain.

(1) Correlation coefficient:

)0(*)0(|)(| 2 yy xx xy R R R

)]([*)]([|)](*)([|222

t y E t x E t yt x E

y x yy xx xy C C C 222 *)0(*)0(|)(|

]))([(*]))([(|)])((*))([(|222

y x y x t y E t x E t yt x E

Thus,

y x

xy

xyC

*)()( : Correlation coefficient

1)(0 xy

:0)( xy x(t), y(t) Uncorrelated;


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:1)( xy x(t), y(t) Correlated;

(2) Coherence:

)(*)(|)(| 2 f G f G f G yy xx xy

)(*)(

|)(|)(

22

f G f G

f G f

yy xx

xy

xy

1)(02

f xy

(3) Role of coherence function )(2

f xy (constant-parameter linear systems) can be

interpreted as fractional portion of the mean square value at the output y(t) that is

contributed by the input x(t) at frequency value f. In contrast, the quantity

)](1[2

f xy is the portion of mean square value of output y(t) not be contributed by

input x(t) at frequency f.

Note: The role of coherence function )(2

f xy is similar to the correlation

coefficient function )(2 xy . In constant-parameter linear systems, the

coherence has some following possibilities:

1) 0)(2

f xy : x(t) and y(t) uncorrelated (unrelated)

2) 1)(2 f xy : x(t) and y(t) correlated (unrelated)

3) 1)(02

f xy : x(t) and y(t) some possible situations exist:

a. Extra noise

b. Non-linear system between input x(t) and output y(t)


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c. SIMO or MISO

)(*)(

)()()(

2

2

f G f G

f iQ f C f

yy xx

xy xy xy

)(*)(

)()(

22

f G f G

f C f

yy xx

xy

xy : Coherence

)( f C xy : Co-spectrum (Real part of Cross-spectrum)

)(),( f G f G yy xx : Auto-spectra


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2. DIGITAL SIGNALS AND CLASSIFICATION

2.1. Digital signals

(1)

The signals and data measurements are the similar concepts for almost cases in the

physical measurements and experiments. The classification of signals is important

to the digital signal processing (DSP) or data processing, especially, this closely

relates to the digital filters and discrete signal analysis.

2.2. Classifications

(1) Signals can be commonly classified in the engineering application by some follows

categories: i) Continuous (analogue) and discrete signals (digital), ii) Deterministic

and random signals

Analogue and discrete signals

Branches of deterministic signals

Electric signals(by data acquisition)

Analogue Signals(Continuous)

Discrete Signals(Digital)

Sampling andA/D conversion

Data Analysis andPost-data processing

Deterministic signals Periodic signals

Nonperiodic signals

Sinusoidal signals(2 cycle)

Complex periodic signals(T cycle)

Almost periodic signals

Transient signals


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Branches of random signals

Notes and comments:

i) Sinusoidal (harmonic) signals (2-periodic signals): )2()( nt xt x

ii) Complex periodic signals (T-periodic signals): )()( nT t xt x

The complex periodic signals may be expanded by a Fourier series into the

combination of harmonic components (sine and cosine functions) as follow:

)2

cos()( 00

t T

X t x (Original signals)

1

0 ]12sin12cos[2)( nnn t

T nbt

T na

at x (Fourier series)

iii) Almost-periodic signals: can be expressed by the sum of sine functions that

their frequencies are not periodic.

iv) Transient signals: can be defined as totally non-periodic signals (In other

word, the transient signals can be considered as the deterministic signals but

out of any kinds of periodic and almost-periodic signals). Apart from

periodic and almost-periodic signals, however, the spectrum of transient

data only exists under form of continuous spectrum but the discrete spectrum

does not exist .

Random signals Stationary signals

Nonstationary signals

Ergodic signals

Non-ergodic signals

Specific classification ofnonstationary signals


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(2) The characteristics of the random signals can be clarified by some following

quantities:

i) Amplitude distribution: Mean value (expectation) or root mean square

(standard deviation) [first moment]ii) Correlation functions (auto- and cross-correlations) [joint or second moment]

iii) Power spectral density (PSD) [power contribution of each frequency

components]

(3)

For discrete signals, the above-mentioned quantities can be expressed by formulas:

N

i i x

x N 1

1

;

N

i

i

x

x N

rms1

21

)()(1

)( 001

t xt x N

Ri

N

i

i xx

N nk j N

k

i xe x

N f S

/2

1

1)(

; n=[1,N]

(4)

Some hints on types of random processes

i) Stationary signals are that their mean value and correlation of discrete

signals do not vary on time.

ii) Non-stationary signals, by contrast, their mean value and correlation vary on

time.

)(),(

)(

xx xx

x x

Rt R

t

iii) Weakly stationary signals (or stationary in the wide sense) are that mean

value and correlation (first and joint moments) are time invariant.


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)(),(

)(

xx xx

x x

Rt R

t

iv)

Strongly stationary signals (or stationary in the strict sense) are that all

moments and joint moments (not only first moment but also high-order

moments) are time invariant.

)(),(

)(

xx xx

x x

Rt R

t

(first moment and joint moment)

)(),(

)(

xx xxi

x xi

Rt R

t

th

th

(ith

moment and ith

joint moment)

v) Ergodic signal is stationary one (Mean value and correlation are time

invariant), moreover, its mean value and correlation are constant with

different samples of signal.

)()()(

)(

xxk

xx

x

k

x

R R

k: Index of k th sample of signal

vi) Non-ergodic signal is stationary one (Mean value and correlation are time

invariant), however, its mean value and correlation are differed with

different samples of signal.

)()()(

)(

xx

k

xx

x

k

x

R R

vii) Gaussian signal (Normal distributed signal) is ergodic stationary one with

zero-mean and standard deviation


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2)()(

)(

)()(

0

k

xx

k

xx

k

x

C R

viii) Non-gaussian signal is ergodic stationary one with non zero-mean and

standard deviation

2)(

0)(

xx

x x

C

t

Note 1:

Single random process: Signal of one physical quantity (phenomenon) at one

position

Multi-random processes: Signals of many physical quantities at different

positions

Sampling: Signal of random process at any time ( time interval)

Multi-dimensional process: are multi-variable function

Multi-variate process: Vector of many signals

Multi-variate and multi-dimensional process: Vector of many processes (signals)

in which each individual signal is multi-variable function

Ensemble: the collection of sample functions (any time interval) of one signal.

Thus random signal is the collection of these sample function of one signal

Note 2:

Process (Field): Display and illustration of one physical quantity at one position

(If measurement of the same physical quantity at one position differs from that

at another position due to its distribution and redistribution)

Sample: Display and illustration of a physical quantity at one position at any

time interval


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Record: Discrete display of a physical quantity at one position and one time

Signal: Electric display (continuous or discrete) of a physical quantity at one

position and one time

Difference between ergodic and non-ergodic processes

Ergodic stationary random process Non-ergodic stationary random process

- Different initial phase angles

- The same amplitude

- The same frequency

- Different initial phase angles

- Different amplitude

- The same frequency

Summary on classifications and definition of random signals

No. Items Definition Note

1 Stationary Mean value and correlation not

vary on time

Time-invariant

2 Stationary-Ergodic Mean value and correlation not

vary on time and sampling

Time-invariant

Sampling-independant

3 Stationary-

Nonergodic

Mean value and correlation not

vary on time, but sampling

Time-invariant

Sampling-dependant

4 Weak stationary Mean value and correlation

(first-order moments) not vary on

time

Time-invariant of only

first moments

5 Strong stationary All first-order and high-order

moments not vary on time

Time-invariant of first

and high-order

moments

6 Non-stationary Mean value and correlation vary

on time

Time-variant


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(5) The ensemble (collection of sampling records) of random signal can be divided by

two categories: Individual sample records (one process) and Multiple sample

records (many processes). The characteristics of individual and multiple sample

records of one and many processes can be expressed by figure hereinafter:

Characteristics of individual and multiple sample records of random signals

Note 3:

Coherence function: is the relation between the power auto-spectral density and

the power cross-spectral density.

Frequency response function (gain factors and phase factors): is also the linear

relation between the power auto-spectral density and the power cross-spectral

density.

Individual Sample RecordsAnalysis

Multiple Sample Records

Mean Values andRoot Mean Square(RMS)

Auto-Correlation FunctionAnalysis

Auto-spectral DensityAnalysis

Cross-Correlation FunctionAnalysis

Cross-spectral DensityAnalysis

Coherence FunctionAnalysis

Joint Probability DensityFunctions

Probability DensityFunctions

One sample of one process Pairs of two samples of one process or many processes

Frequency ResponseFunction (FRF)


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(6) Applications of statistical functions

From correlation functions

i) Similarity between 2 signals or 1 signal at different positions and time delays

ii)

Prediction of signals in noise, influence of noiseiii) Identification of propagation directions and velocities

iv) Measurements of time delays

From power spectral density (PSD) function

i)

Power contribution of each frequency components

ii) Identifications of system properties and input signals from output ones

iii)

Identification of noise and energy sources

From coherence function

i) Accuracy of linear input/output systems

ii)

Identification of propagation directions and velocities

From frequency response function (FRF)

i)

Identification of relationship between input and output signals

2.3. Relationships of input and output signals

(1) The input and output signal systems can be commonly classified by: i) Single input

and single output systems (SISO); ii) Single input and multi output systems

(SIMO); iii) Multi input and single output systems (MISO); iv.Multi input and

multi output systems (MIMO).

(2)

In the practical applications, many signal channels are simultaneously measured at

various positions or different time delays. For many cases of MIMO systems, many

input and output signals can be correlated or uncorrelated measured simultaneously.


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SISO system

Independent MIMO system

Hxy(f)

Frequency ResponseFunction (FRF)

Input signalx(t)

Output signaly(t)

Signal noisen(t)

Hxy,1(f)

Input signalx1(t)

Output signaly1(t)

Hxy,2 (f)

Input signalx2(t)

Output signaly2(t)

Hxy,k (f)

Input signalxk(t)

Output signalyk(t)

……………………..

Channel No.1

Channel No.2

Channel No.k

Signal noisen(t)


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3. DATA ACQUISITION, PROCESSING AND ANALYSIS

3.1. Introduction

(1)

In order to understand and clarify the physical measurements, signal processing and

analysis for the buffeting experiments and other specified measurements as well in

the wind tunnel, it is important to understand the digital signal processing (DSP)

and measurement procedures and instruments in wind tunnels.

(2) This study hinges on some following points:

1) Instrumental systems their by function: data acquisition, A/D conversion, data

qualification and analysis

2)

A/D conversion and sampling theorem for eliminating the aliasing errors

3) Data analysis techniques

3.2. Data processing procedure

(1) The digital signal processing procedure can be expressed by the following diagram:

Signal processing procedures for measurement systems

(3) Three following main steps of the digital signal processing procedure will be

overviewed as follows:

Data Acquisition Data Conversion Data Qualification Data Analysis

Transducer

Signal Conditioning

Signal Calibration

A/D Conversion

Aliasing Errors

Quantization Errors

Classification

Validation

Editing

Individual SampleRecords

Multiple SampleRecords


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3.2.1. Data acquisition

Transducer: Device and sensor to transform from any physical phenomena

(force, pressure and motions: displacement, velocity and acceleration) toelectric signals. Transducers commonly are employed two kinds of materials:

piezoelectric and strain-sensitive materials.

+) Piezoelectric materials: frm physical quantities to electric charge, such

as naturally polarized crystals like quartz and artificially polarized

ferroelectric ceramics like barium titanate.

+) Strain-sensitive materials: from physical quantities to resistance, such

as metallic like copper-nickel alloy and semiconductor likemonocrystalline silicon.

Signal conditioning: Change from the electric signals (charge and resistance)

to voltage.

3.2.2. Data conversion

Analogue to digital converter (ADC): Transforms from continuous analogue

signal to digital signal

Aliasing errors: Eliminated by sampling theorem: sampling frequency (F) or

sampling time interval (t)

Note: ADC can be stored under two types of codes: binary and ASCII codes

Binary code: By numbers 0 and 1 (8 bytes)

ASCII code: By numbers from 0 to 9 (1byte). However, almost data of

discrete signals (after ADC) has been stored under this ASCII code in

application, because it is easily red by any applied programs such as Matlab

and Fortran for data post-processing.


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3.2.3. Data qualification

Many imperfections of received signals with noise can be reduced by the

digital or analogue filters.

Data qualification consists of: classification, validation and editing.

3.2.4. Data analysis

Individual sample records (Uni-variate process): Sample collection of

measurement data of one physical phenomenon at one point in various time

intervals.

Multiple sample records (Multi-variate processes): Sample collection of many

individual records of one physical phenomenon at numerous points or of some physical phenomena at one point in various time intervals.

Fig. 2. Individual and multiple sample record analysis

Individual sample record and uni-variate process

Mean and root mean square (RMS) value computation:

Auto-correlation function computation

Auto-spectral density function computation

Individual Sample RecordsAnalysis

Multiple Sample Records

Mean Values andRoot Mean Square(RMS)

Auto-Correlation FunctionAnalysis

Auto-spectral DensityAnalysis

Cross-Correlation FunctionAnalysis

Cross-spectral DensityAnalysis

Coherence FunctionAnalysis


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Multiple sample record and multi-variate processes

Cross-correlation function computation

Cross-spectral density function computation

Coherence function computation

3.3. Data analysis procedures

3.3.1.

Data sampling and data preparation

Multi input and multi output system (MIMO) = Combination of independent single

input and single output systems (SISO)

n: Index of ensemble (sample record) or signal or random process

Multi sample record (of multi signals) = Collection of number of individual sample

records (of individual signal). Thus analysis of the multi sample record can be carried

out by analysis of the individual sample record and analysis of the pairs of correlated

two individual sample records.

Discrete valued signal (or random process):

n x n=1,2,3… N (N: Number of samples)

Equally spaced time interval (T: sampling time or sampling period) of samples in

discrete signal:

T nT T n *0 n=1,2,3… N

Continuous Signal A/D ConversionData Sampling

Discrete Signal Discrete Data Record

FrequencyResponse Function

Signal FilterInput signal ui (t)

A/D ConversionData Sampling

Output signal xn(t)

Continuous Data Discrete Data


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T0: Initial time (T0 = 0) T nT n *

T: Sampling period

NT: Total time length of discrete signal; T N *

F: Sampling frequency (Hz);T

F 1

F0: Fundamental sampling frequency; T N F

*

10

Thus, )*()( 0 T nT xT x x ni n=1,2,3… N

Noting that the Limit sampling frequency (the Nyquist frequency) for eliminating the

Aliasing Errors

T F N

*2

1

3.3.2. Data standardization

The purpose of data standardization is to transform the original data record n x to

new type of data record n x' (can be called the fluctuating data record ) that has the

zero mean value.

1)

Mean value and root mean square of sample record

N

n

n x

N x

1

1 n=1,2,3…N (Mean value)


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N

n

n x x N

SQRT s1

2 ]1

1[ n=1,2,3…N (Root Mean Square

value)

Note: For the stationary ergodic data record (time-invariant and sampling-

independent), it is very convenient and common to transform the initial data

record n x to the new zero-mean data record n x' .

2) Establishment of the fluctuating data record (Zero-mean data record)

n x' : n=1,2,3…N xnT x x x x nnn )(' n=1,2,3…N

Having: 0' x (Zero-mean value)

N

n

n x x N 1

2'

2 '1

1 (Mean Square value or Variance)

N

n

n x x x

N SQRT s

1

2'' ]'

1

1[

(Root Mean Square or Standard Deviation)

3) Fourier Transform (Discrete Fourier Transform-DFT):

Fluctuating data record (Zero-mean data record) or standardized data record

n x' n=1,2,3…N

Discrete Fourier Transform

N

n

k k T nF jT n xF X

1

)]2exp(*)([1

)(


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X(Fk ): DFT at computational frequency k F

k: Index of discrete frequencies

Fk : Computational frequency; )( f k F k

Note:

i) The frequency space )( f = The fundamental sampling frequency (F0)

NT F f

10

ii) Number of frequency space (K)

0F

F F K sc

cF : Cut-off frequency

sF : Starting frequency

iii) Computational frequency Fk:

NT

k F f k F F

ssk )( ; k=0,2,3…K-1

NT

k F

k ; k=0,2,3…K-1

Note: Index of computational frequency starts from 0

Frequency value starts from Fs (???)

iv) DFT of x(t) at the computational frequency


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N

n

k T n NT

k jT n x

NT

k X F X

1

)]2exp(*)([1

)()(

;k=0,2,3…K-1

N

nk

N nk jT n x NT

k X F X

1 )]/2exp(*)([

1

)()( ;k=0,2,3…K-1

k=1:

N

n

s N n jT n xF X F X 1

1 )]/2exp(*)([1

)()(

k=2:

N

ns

N n jT n x NT

F X F X 12 )]/22exp(*)([

1

)

1

()(

k=3:

N

n

s N n jT n x

NT F X F X

13 )]/32exp(*)([

1)

2()(

Note: +) In the DFT formula, factor (1/) appears (DFT standardization)

+) In Matlab command, X=FFT(x) (without standardization)

Inverse Discrete Fourier Transform:

N

k

N nk j NT

k X

N nT x

1

)]/2exp(*)([1

)( ; n=0,1,2…N

DFTX(Fk )

f(Hz)

F1=Fs k=1

X(F1)X(Fi)

Fi=Fs+i/NTk=i

F1=Fc k=K


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4) Auto-correlation function

Auto-correlation function can be computed by i) Direct computation by definition and

ii) Indirect computation via FFT. Note that the second method is more efficiency

because of application of FFT algorithm, however, the first one is easier to compute but more time consuming.

Method 1: Direct computation of auto-correlation function

Fluctuating data record (Zero-mean data record) or standardized data record

n x' n=1,2,3…N

Auto-correlation function of data record n x' with delay s

N

n

snn xxss xs x

N s R

1)()( )]()([

1)( (Auto-correlation functions)

t r s *

r N

n

r nn xx ss xs xr N

t r R1

)]()([1

)*(

5) Auto-spectral density function

Auto-spectral density function can be computed by i) Ensemble Averaging and ii)

Frequency Averaging

Method 1: Ensemble-averaging techniqueFluctuating data record (Zero-mean data record) or standardized data record

n x' n=1,2,3…N


31/62


Definition of mean power:

NT N

n

n xx dt t x NT

x N

t x E S 0

21

0

22 )(11

)]([

1

0

2

|)(|1

)(

N

n

n xx f X NT f S : Two-sided spectral density

)( f X : Fourier transform at frequency f

1

0

]/2exp[*1

)( N

n

n N nk j x

N f X ; k=0,1,2…N-1

1

0

2|)(|2

)(2)( N

n

k nk xxk xx f X NT

f S f G : One-sided spectral density

Computational procedures (Ensemble-averaging technique)

1) Data Sampling: N samples of process x(t) are picked out (Data record)

n x , n=1,2,3…N

2) Data Standardization: Compute the mean value of data record ( x ), then

reconstruct the Fluctuating Data Record n x' with zero-mean value

( 0' x )

t

x(t)

T 2T 3T 4T 5T 6T NT(N-1)T


32/62


n x' , n=1,2,3…N

3) Data Blocking: Taper data record by each data blocks using Window

functions

4)

Fourier Transform: Compute the Fourier Transform at frequency f k ,

k=1,2,3…N by using FFT technique

)( k f X , k=1,2,3…N

5) Scale Factor Adjustment of )( k f X : Scale factor of )( k f X due to the loss

by tapering operation. (By Hanning tapering: Scale factor by 3/8 )

6) Spectral Density: Estimate the spectral density )( k xx f S from each data

blocks


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4. CORRELATION FUNCTION

4.1. Introduction

(1) The correction functions evaluate the statistical independence between the signals

(time-dependant) or stochastic processes (space-dependant). The cross correlation

function is a measure to tell us how much two processes or two signals are like each

other, whereas the auto correlation function tell us how much a process or a signal at

time t is like itself at time t+ (: time shift or time lag) or how much a process or a

signal at location (x,y,z) is like itself at another location (x+x,y+y,z+z). In addition,

to evaluate how much two processes or two signals are like each other, the correlation

coefficient function also is given.

4.2. The discrete correlation function

The discrete correlation function of two processes or two signals x, y can be expressed

as follow:

1

0

)()()()( )]()([1

][)( N

n

snn N snn xy ss ys x

N

Lim y x E s R

E[] : Expected value or mean value

For the number of samples is taken large enough, we have following approximations:

N

n

snn xxss xs x

N s R

1)()( )]()([

1)( (Auto-correlation functions)

N

n

snn xy ss ys x N

s R1

)()( )]()([1)( (Cross-correlation

functions)

s : Spatial interval or time shift


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s: Spatial coordinates or time variable

N: Number of samples

Note: 1) Corresponding values of signals are sampled at same time.

2) R xx(s)=R xx(-s) and R xy(s)=R xy(-s)

3) Above approximate formulas are accuracy only if N

Computational procedure of correlation function:

Step 1: Setting parameters

+) Number of sample: N

+) Time shift or spatial interval: s

Step 2: Sampling

+) Sampling signals x, y

Step 3: Computing correlation function

Step 4: Plotting Rxx, Rxy versus n [1,N]

4.3. The discrete covariance function

The discrete covariance function of two processes or two signals x, y can be expressed

as follow:

Span-wise

s

s ss

s

xy xx R R ,

is

)( i xx s R R xx, R xy vs. s


35/62


1

0

1

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

1)]([

1][][)(

N

n

N

n

sn N n N snn xy ss y N

Lims x N

Lim y E x E s

For the number of samples is taken large enough, we have following approximations:

N

n

N

n

snn xxss x

N s x

N s

1 1)()( ])([

1)]([

1)( (Auto-covariance functions)

N

n

N

n

snn xyss y

N s x

N s

1 1)()( )]([

1)]([

1)(

(Cross-covariance

functions)

s

: Spatial interval or time shifts: Spatial coordinates or time variable


If x, y are the zero-mean processes or signals, these mean that E[x(s)]=0 and E[y(s)]=0,

then the Root mean square (R.M.S) value must be replaced to the mean value (or

Expected value):

Auto-covariance function:

N

n

N

n

snn xxss x

N SQRT s x

N SQRT s

1 1

)(2

)(2 ]})([

1{)]}([

1{)(

Cross-covariance function:

N

n

N

n

snn xyss y

N SQRT s x

N SQRT s

1 1

)(2

)(2 ]})([

1{)]}([

1{)(

Computational procedure of covariance function:


+) Number of sample: N


36/62


+) Time shift or spatial interval: s

Step 2: Sampling

+) Sampling signals x, y

Step 3: Computing covariance functionStep 4: Plotting xx, xy versus n [1,N]

4.4. The discrete correlation coefficient function

The discrete correlation coefficient function of two processes or two signals x, y can be

expressed as follow:

)(

)(

)( s

s R

s xx

xx

xx

(Auto-correlation coefficient function)

)(

)()(

s

s Rs

xy

xy

xy

(Cross-correlation coefficient function)

Note: ]1,1[)( s : 1 Full-correlated; 0 No correlated

4.5. Examples

Example 1: Correlation function

0 50 100 150-1

-0.5

0

0.5

1

Number of sample

Initial signals x,y

0 50 100 150-0.2

-0.1

0

0.1

0.2

Number of sample

Cross-correlation Rxy by definition

0 100 200 300-0.2

0

0.2

0.4

0.6

Number of sample

Auto-correlation Rxx,Ryy by Xcorr(x)

0 100 200 300-0.2

-0.1

0

0.1

0.2

Number of sample

Cross-correlation Rxy by Xcorr(x,y)


37/62


Example 2: Correlation of signal with noise

0 1 2 3

x 10-3

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Initial sinal+noise

0 0.2 0.4 0.6 0.8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Correlation functions of signal with noise


38/62


5. FOURIER TRANSFORM

5.1. Introduction

(1) It is well known that the Fourier Series Transform and Fourier Spectral Transform

have been widely applied for almost kinds of natural and physical phenomena.

Applications and contributions of the Fourier Series and Fourier Transform concentrate

on the problem of the Digital Signal Processing (D.S.P) in the data processing and

analysis of measurements and the buffeting response prediction in which the spectral

representation cant be required.

(2) In summary, the Fourier Series and the Fourier Transform will be studied for such

purposes as follows:

1) Data processing and analysis of measurement processes in the Digital Signal

Processing (D.S.P)

2) Spectral representation in the buffeting response prediction

(3) Data measurements can be expressed under the continuous or discrete processes.

However, almost measured signals have been collected under the discrete signal for the

data post-processing and analysis. The discretization of measured signals is well

known as the sampling processes.

(4) Some following DSP techniques will be studied hereinafter:

1) The discrete Fourier series (DFS)

2) The discrete Fourier transform (DFT)

3) The fast Fourier transform (FFT)

4) Amplitude spectrum and phase spectrum

5) The discrete inverse Fourier transform (IFT)

6) The sampling technique


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5.2. The Discrete Fourier Series

The Fourier series purposes to decompose any periodic or non-periodic signals into

combination of simple harmonic signals (sine and cosine functions). Post processingand analysis on simple harmonic functions seem to be much easy than the original

signals. Mathematically, the Fourier series is known as the harmonic analysis.


T: Period (Time step of sampling) of a sample (s)

F: Frequency of a sample (Hz),

T

F 1

NT: Fundamental period of series (s)

Fo: Fundamental frequency of series (Hz) NT

Fo1

Discrete Fourier series is expressed as the sum of harmonic functions as follow:

)]sin()cos([2

00

1

1

0 T mnbT nmaa

xm

M

m

mn

0 : Fundamental frequency of series (rad/s), NT F o

220

m: Times of fundamental frequency (m=1: fundamental harmonic term with 0 ,

m=2: 2nd harmonic term with 2 0 , mth harmonic term with m 0 )

M: Cutting-off number of frequency

ao, am, bm: Fourier coefficients of series

n: Pointer of samples

)]/2sin()/2cos([2

1

1

0 N mnb N nmaa

xm

M

m

mn


40/62


Discrete Fourier Coefficients:

)/2cos(2 1

0

N nm x

N

a N

n

nm

; 10 M m

)/2sin(2 1

0

N nm x N

b N

n

nm

; 11 M m

Computational procedure:


-

Initial signal

- Number of samples N

- Number of series M

Step 2: Sampling

Step 3: Calculating Fourier Coefficients

-

am, bm

Step 4: Simulating Fourier series and Plotting Simulated signal vs. number of

samples (n)

5.3. The Discrete Fourier Transform

x(t): time-dependant signal or stochastic process

T: Sampling cycle (time interval for a sample)

F: Sampling frequency (F=1/T: number samples per a second)

N: Total samples

NT: Total time for sampling or fundamental period of transform

Fo: Fundamental frequency of transform NT

F 1

0


41/62


Discrete Fourier Transform: (to frequency domain)

N

n

T n j

k k enT x X

1

])([)(

N

n

T n f j

k k enT x f X

1

2 ])([2

1)(

(Note: In origin, n=0 N-1)

k: Number of discrete frequencies in range

In Matlab: 0)( k k k [k=1-K]: equal spacing of frequency range

0 : Fundamental frequency (rad/s), NT F o

220

Note: n (pointer of samples) and k (pointer of frequency) have different

meaning

Computational procedure:


- T (cycle sampling), N (numbers of samples)

- Freq. range: fs (starting freq.), fc (cut-off freq.)

- Number of freq. interval: K k fs fc /)(

Step 2:

Loop 1: Frequency

For k=1 to K )( k k

Loop 2: Sampling

For n=1 to N

N

n

nT jk

k enT x X 1

)()()(

Step 3: Plotting X(k ) versus (k )


42/62


Putting NT F o

220 into formulas:

N

n

nT jk enT xk X

1

0 ])([)(0

N

n

nT NT

jk

enT xF k X 1

2

0 ])([)2(

N

n

N kn jenT xkF X 1

/20 ])([)(

Fourier Series (in time domain)

K

k

k k k k t k bt k a

at h

1

0 )]sin()cos([2

)(

)}(Re{2

k H N

ak

)}(Im{2

k H N

bk

Note in processing using FFT:

1) )/(2 NT or )/(1 NT F

2) Maximum frequency: )2/(12/max T F F

3) Spectrum calculated at certain frequencies: 0, F, 2F, 3F… Fmax

Leakage Effect: Amplitudes will distribute on adjacent closed frequencies. This effectsoccurs in cases that total sampling time NT does not coincide the integer multiple of

the sampling cycle T. To prevent the ‘leakage effect’ by Averaging Method or Window

Functions


43/62


5.4. The Fast Fourier Transform

It is very well-known that the Fast Fourier Transform (FFT) algorithm has been

powerfully used for solving the Discrete Fourier Transform. FFT is not a new

transform itself due to using the same DFT formula, however, its FFT algorithm ismuch faster than the conventional DFT. In principle, the FFT algorithm eliminates the

component repetition to make the faster computation.

DFT formula:

1

0

/20 ])([)(

N

n

N kn jenT xkF X

1

0

N

n

kn

N nk W x X ; k=[0,N-1];

N j

N eW /2

X k contains N 2 components (N of xn and N of W N )

Note: sincos je j = x + jy (Euler’s Formula)

)/2sin()/2cos(/2 N kn j N kne N kn j

0

1

12

2

1

0

2

N j

N

N j

N

n

N

n j

e

ee

Sum can be expressed in the complex plane of 2=3600 (N=8 for example)

Complex plane (N=8) Complex plane

00 k=0

450 k=1

900 k=2

1350 k=3

1800 k=4

2250 k=5

2700 k=6

3150 k=7

je

08/02 je

8/22 je

8/52 je 8/72 je

Re

Im


44/62


Based on above-mentioned notes, we can easily obtain:

),( N mrem N

m

N W W ; m[0,N-1]; rem(m,N) is the remainder after dividing m

by N

m N jm

N eW )/2( ; m[0,N-1]

))(/2(),( miN N j N mrem N eW

; i=1,2,.. N

kn Max )(

For example, N=8, n=[0:7], k=[0:7], max(kn)=49, components W are expressed by

the complex plane

Complex plane (N=8)

Thus, among 49 components, 8 of these are unique.

Then, suppose N is a multiple of 2, we decompose the samples into two vectors

containing even- and odd-numbered samples as follows:

12/

0

12/

0

212

22

N

n

N

n

kn

N n

kn

N nk W xW x X ; k=[0:N-1]

Due to:k

N

k

N W W 2

2/2/ , thus

12/

0

12/

02/122/2

N

n

N

n

kn

N n

kn

N nk W xW x X ;

168

88

08 W W W

178

98

18 W W W

198

118

38 W W W

208

128

48 W W W

218

138

58 W W W

228

148

68 W W W

238

158

78 W W W

188

108

28 W W W


45/62


Number of components reduces to 2)2

( N

< 2 N of original problem. Depending on N,

efficiency of FFT algorithm is different. For example, if N=2K then complex

components in FFT is N

N

K

N

2log22 (in comparison of N

2

components in DFT)

5.5. The Discrete Inverse Fourier Transform

The discrete inverse Fourier transform of X(f) can be expressed as follows

N

n

N kn jenT xkF X 01

/20 ])([)( ; k=[0:N-1]:

1

0

/20 ])([

1)(

N

k

N kn jekF X N

nT x ; n=[0:N-1]: Inverse Transform

5.6. Amplitude spectrum and phase spectrum

The amplitude spectrum is defined as the vector of DFT component amplitudes,

whereas the phase spectrum is vector of DFT component phase angles in radian unit.

5.7. The Discrete Sampling Theory

The problem in the sampling technique is to require 2 necessary points:

1)

Sampling values are required enough for the data processing and analysis.

2) Sampling the pick-up values from either continuous signals or discrete signals

represents the original one. This means that from the sampling values can

reconstruct the similar original signals.


46/62


The sampling parameter: Sampling rate or sampling frequency Fs(Hz).

The sampling theorem may be stated as follows: ‘ If continuous signals is sampled at a

sampling rate or sampling frequency that is twice higher than their highest frequency

component, then it is possible to recover and reconstruct the original signals fromsamples ’

signalsampling F F max,2

5.8. Examples

Example 1: D.F.S

Initial signal (impulse function) and signal due to Discrete Fourier Series at various

number of series (M=1,3,5,10,10,50)

0 10 20 30 40 50 60 70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

number of samples

N u m b e r o f s e r i e s M

= 1

0 10 20 30 40 50 60 70

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

number of samples

N u m

b e r o f s e r i e s M

= 3

0 10 20 30 40 50 60 70

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

number of samples

N u m

b e r o f s e r i e s

M

= 5

0 10 20 30 40 50 60 70-0.2

0

0.2

0.4

0.6

0.8

1

1.2

number of samples

N u m


M

= 1 0

0 10 20 30 40 50 60 70-0.2

0

0.2

0.4

0.6

0.8

1

1.2

number of samples

N u m


M

= 2 0

0 10 20 30 40 50 60 70-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

number of samples

N u m


M

= 5 0

Initial SignalImpulse Function

Signal fromD.F.S


47/62


Example 2: FFT vs. SPECTRUM

Fourier Transform by FFT and Spectrum

Note:

Fourier Transform Spectrum

dt et x j X xCFT t j )()()(

1

0

)()( N

n

nT j

ne x j X x DFT

dt et x j X xCSPEC t j )()()( 2

1

0

2)()( N

n

nT jne x j X x DSPEC

Amplitude

Noise

f1=8Hz

f1=33Hz

No noise

0 0.2 0.4 0.6 0.8-10

0

10

20

time [t]

signal

0 20 40 600

2

4

6

8

frequency [Hz]

spectra by using FFT

0 0.2 0.4 0.6 0.8-10

0

10

20

time [t]

signal

0 20 40 600

5

10

15

20

25

frequency [Hz]

spectrum by using spectrum


48/62


Example 3: DFT

Results: +) Fast Fourier Transform (FFT) is not a new form of Fourier transform, the

applied formula is exactly such an expression

N

n

N kn jenT xkF X

1

/20 ])([)(

, however,

the modified algorithm is used for much faster computation of DFT

+) Result by DFT’s defined formula is exactly same to that by FFT’s Matlab

command. Moreover, Magnitudes of DFT and FFT are absolute values (abs(H))

+) Results by DFT and FFT are different from that by Spectrum

Notes: +) DFT definition

N

n

N kn jenT xkF X

1

/20 ])([)(

:

H0=x*exp(-j*2*pi*k'*n/N);

n: pointer of sample, k: pointer of frequency, k must be transposed (k’)

H0=H0/N; H0 must be standardized by dividing by N (number of samples)

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Number of samples or time interval

S i g n a l

0 20 40 60-0.4

-0.2

0

0.2

0.4

0.6

Frequency Hz

FFT command

0 20 40 60-0.4

-0.2

0

0.2

0.4

0.6

Frequency Hz

D F T M a g n i t u d e

DFT by definition

0 20 40 600

2

4

6

8

Frequency Hz

Spectrum command


49/62


Example 4: Amplitude and phase spectrum

Example 5: Amplitude and phase spectrum

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Number of samples

S i g n a l

0 20 40 60-0.4

-0.2

0

0.2

0.4

0.6

Frequency(Hz)

A m p l i t u d e

Spectrum

0 20 40 60-4

-2

0

2

4

Frequency(Hz)

P h a s e ( r a d )

Phase spectrum

0 20 40 600

5

10

15

20

Frequency(Hz)

P h a s e ( r a d )

Phase spectrum

fftshift H

unwraAn le H

0 50 100-0.5

0

0.5

1

1.5signal

0 50 1000

1

2

3

4FFT

0 50 100-4

-2

0

2

4phase spectrum

0 50 1000

50

100

150unwrapped p.s

0 50 100-0.5

0

0.5

1

1.5signal

0 50 1000

1

2

3

4swapped FFT

0 50 100-4

-2

0

2

4p.s of swapped FFT

0 50 1000

50

100

150unwrapped p.s


50/62


6. POWER SPECTRAL DENSITY FUNCTION

6.1. Introduction

(1) The random signals can not be determined exactly. It means that the random

signals always differ from each other at the different observations, moreover, the

random signals contain the random parameters that can not be described by the means

of analytical quantities and determinant-parameter methods, but they are only able to

be described by the terms of statistical parameters that can differ from one random

signal to another.

(2) Thus the questions are that what terms of statistical parameters can be able to

describe the characteristics of the random signals. It is well known that the two most-

commonly-used means are:

i) amplitude distribution functions of random signal

ii) power spectral density of random signal (or equivalent as the correlation

function)

(3) In the term of amplitude distribution, the commonly-used statistical quantities are:

i) mean value (expectation); ii) mean square (variance when zero-mean signals) or root

mean square (standard deviation when zero-mean signals).

(4) It is broadly said that the square of DFT magnitudes of any function x(t) is

considered as power contribution of any frequency components in x(t) over the

frequency domain.

6.2. Characteristics of amplitude distribution

The most-commonly-used statistical terms of the amplitude distribution characteristics

are defined underneath:


51/62


- Mean value (Expectation):

1

0

1][

N

i

i x x N

x E

- Mean square :

1

0

222 1

][

N

i

i x

x N

x E

- Root mean square : )1

(][1

0

22

N

i

i x x

N sqrt x E

- Variance:

21

0

1

0

22222 ]1

[]1

[][])[(

N

i

i

N

i

i x x x x

N x

N x E x E

- Standard deviation: )][(]))[(( 222 x x x x E sqrt x E sqrt

Thus zero-mean random signals ( 0 x

) have the following deductions:

Mean square = Variance

Root mean square = Standard deviation

Linear relationship of y and x as y=ax+b, we easily obtain: 222 x x

x y

a

ba

Some distribution probability functions of measurements are widely used as

i) uniform probability distribution:

ii)

normal (Gaussian) probability distribution:

6.3. The Power Spectral Density (PSD)

As above-mentioned, the power of random signals is expressed by the square of DFT

magnitudes of x(t) at any time t, whereas the power spectral density is expressed in the

frequency domain . Some concepts of power have been used:


52/62


The instantaneous power of signal x(t) at any time t can be defined as:

2|)(| t xPower

For the complex signals, we must use the mean power or expected power:

2/

2/

22/

2/

22 )(1

)(1

)]([T

T

T

T

T dt t xT

dt t xT

Limt x E power Mean

(when T is long enough)

NT N

n

n xx dt t x NT

x N

t x E S power mean Discrete0

21

0

22 )(11

)]([

(Definition:This is only approximate estimation, that is why it is called as the

spectral estimation)

We have the discrete inverse Fourier transform:

1

0

/20 ])([

1)(

N

k

N kn jekF X N

nT x

Thus, the discrete mean power can be written as follows:

1

0

21

0

/20

1

0

2 ])(1

[11 N

n

N

k

N kn j N

n

n xxekF X

N N x

N S

1

0

1

0

1

0

/)(2

3

1

0

21

0

/203

1])([

1 N

k

N

m

N

n

N nmk j

mk

N

n

N

k

N kn j

xx e X X N

ekF X N

S

1

0

2

2

1 N

k

k xx X

N S


53/62


We have:

1

0

21 N

k

k xx x N

S

As a result, the relationship between the mean power of signal in the time domain and

power spectrum, or the mean power of signal can be presented in the term of the power

spectrum. This expression has been well known as the Parseval’s Theorem.

1

0

21

0

2 1 N

k

k

N

k

k X

N x (Parseval’s Theorem)

In DSP, it is defined the periodogram that has N components given by:

21)( k xx X

N k P

; k=[0,N-1]

)()( k N Pk P xx xx

The power spectrum can be obtained:

1

0

1

0

2 )(11 N

k

xx

N

k

k xxk P

N x

N S

Note: In the same way that 2k

x represents a measure of signal power at a point in

the time domain, Pxx(k) represents the measure of signal power at the point in

the frequency domain.

Therefore, the vector Pxx=[Pxx(k)], k=[0:N-1] is considered as the measure of

estimation of the power spectral density (PSD) of signal x(t).

Some expressions in the Power Spectral Density (PSD) S of signal x(t)


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Discrete PSD: 1)

1

0

2

2

1 N

k

k xx X N

S (Frequency domain)

Xk is DFT amplitude of x(t) at frequency k

2)

1

0

21 N

k

k xx x

N S (Time domain)

xk is discrete value of signal x(t) at time t

3)

1

0

1

0

2 )(11 N

k

xx

N

k

k xxk P

N x

N S (Periodogram)

21

)( k xx X

N k P

; m=[0,N-1]

Parseval’s Theorem:

1

0

21

0

2 1 N

k

k

N

k

k X

N x (Parseval’s Equality)

Continuous PSD: 1) NT

xx dt t x NT

S 0

2 )(1

(Frequency domain)

2) T

T

xx xx df f PS

2/1

2/1

)( or T

T

xx xx d PS

/

/

)(

The relationship of the power spectrum to the auto-correlation: The discrete auto-

correlation of vector x from sampled values of signal x(t) with extended period has

been defined as follows:

1

0

1)(

N

n

snn xx x x N s R ; s=[0:N-1]

By definition of the mean power, we have:


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1

0

21)0( N

n

xxn xx power MeanS x N

R

Discrete Fourier transform of the auto-correlation can be calculated by definition:

1

0

1

0

/21)}({ N

s

N

n

N sk j

snn xx Re x x

N s R DFT S

; k=[0:N-1]

1

0

/2 N

n

N kn j

nk e x X ; k=[0:N-1] (Definition of DFT)

)(1)}({ 2 k P X N

s R DFT S xxk xx R ; k=[0:N-1]

In the conclusion, the periodogram of vector x with periodic extension is the DFT of

the auto-correlation function of x, that means:

)(1

)}({2

k P X

N

s R DFT S xxk xx R

Role of the auto-correlation function gives us information on how much

dependence of given sample xn on nearby samples xn+1 , xn+2 , and so on.

The Cross Spectrum: involves two time-dependant signals (two time series). Suppose x

and y are vectors of length N sampled from two signals x(t) and y(t).

- The cross-periodogram: k k xyY X

N k P '

1)(

; k=[0:N-1]


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- Mean cross-power:

1

0

1 N

n

nn xy y x N

S

1

0

1

02

)(1'1)0( N

k

N

k

xynn xy xy k P N

Y X N

RS

Role of the cross spectrum gives us information on how the cross

correlation function is distributed over the frequency scale.

Coherence function: The magnitude of the cross periodogram Pxy(k), that is )(k P xy , is

the measure of the coherence of signals x(t) and y(t) at different frequencies.

- Coherence function: )()(

)()(

2

2

f P f P

f P f Coh

yy xx

xy xy

6.4. Examples

Example 1: FFT and SPECTRUM

0 0.5 10

0.2

0.4

0.6

0.8

1

Impulse signal

0 500

0.1

0.2

0.3

0.4

0.5DFT

0 500

0.1

0.2

0.3

0.4

0.5

A m p l i t u d e

FFT

0 0.5 10

0.2

0.4

0.6

0.8

1

Time (s)

0 500

2

4

6

8x 10

-5

Frequency (Hz)

X2

0 500

2

4

6

8

Frequency (Hz)

Spectrum

A m p l i t u d e

The same

Difference in shape?Different in value?


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Note: 1) Using FFT command

H=FFT(x);

H=[H(1) 2*H(2:N/2)]/N; %Vector length limit and

standardization2) Using DFT’s definition

H=x*EXP(-j*2*pi*k'*n/N); %DFT by definition

H=[H(1) 2*H(2:N/2)]/N;

3) Using SPECTRUM command

S=SPECTRUM(x);

S=[S(1) 2*S(2:N/2)]; %Vector length limit

4) Using spectrum’s definitionS=ABS(H).^2; %Square of absolute amplitude

S=[S(1) 2*S(2:N/2)]/ (N^2); % Vector length limit and

standardize

Example 2:FFT and SPECTRUM of some types of functions

Time intervalsFrequency intervals

Sampling process(Step 2)

FFT(Step 3)

SPECTRUM(Step 4)

Digital Signal orRandom Process


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Example 3: Some built-in filter function (window functions) in FFT

0 5-10

0

10normal signal

0 50

5

10uniform signal

0 5-2

0

2sine signal

0 50

0.5

1impulse signal

0 500

0.2

0.4FFT

0 500

2

4

0 500

0.5

1

1.5

0 500

0.5

1

0 500

20

40spectrum

0 500

20

40

0 500

100

200

0 500

20

40

0 5-20

-10

0

10

20

normal signal

0 500

2

4

6

8

10

FFT

0 500

2

4

6

8

10

FFT with boxcar

0 500

1

2

3

4

5

FFT with hamming

0 500

1

2

3

4

FFT with bartlett

0 500

1

2

3

4

FFT with hann


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7. COMPUTATION OF COHERENCE FUNCTION

7.1. Introduction

(1)

Coherence function: Defined as the measure to evaluate the statistical

independence of two stochastic processes or two digital signals (but in the

frequency-domain)

(2) Coherence function: Computed by using the cross-spectrum (two signals) and

auto-spectrum (one signal) of two stochastic processes

(3)

Correlation function: Similar to the coherence function, the correlation function

also is the measure to evaluate the statistical independence of two stochastic

processes or two digital signals (but in the time-domain)

7.2. Cross spectrum and coherence function

(1)

The computation of cross spectra has been used in the field of Digital Signal

Processing (D.S.P) and Discrete Data Processing (D.D.P)

(2) The Auto-spectrum represents the Fourier transform of the Auto correlation

Stochastic process A

or digital signal A

Stochastic process B

or digital signal B

Correlation function (time domain)Coherence function (frequency domain )

Inter-relation or inter-influence


60/62


(3) The Cross-spectrum represents the Fourier transform of the Cross correlation

(4) The Cross-spectrum is the complex valued function whose magnitude and phase

are used in the signal processing to indicate the degree of correlation between twosignals

(5) The magnitude of the cross spectrum indicates whether frequency components of

one signal are associated with large or small amplitudes at the same frequency in

the second signal.

(6) The phase of the cross spectrum indicates whether the phase lag (tre) or lead (dan

truoc) of one signal with respect to the second signal for a given frequency

component.

(7) The cross-spectrum is used to determine the coherence function between two

signals

(8)

Cross spectrum and coherence function can be determined by the MATLAB as

powerful engineering program for Digital Signal Processing

7.3. Computation of coherence function

(1) Let x(t) and y(t) be two discrete data measurements of two signals, the problem is

that to determine i) Cross spectrum and auto spectrum and ii) Coherence function

(2) The Fourier transforms of discrete signals

X(f)=F{x(t)}


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Y(f)=F{y(t)}

(3) Auto spectrum of x(t) or y(t)

Sxx(f)=X(f) X*(f)Syy(f)=Y(f) Y*(f)

(4)

Cross spectrum of x(t) and y(t)

Sxy(f)=X(f) Y*(f)

Syx(f)=Y(f) X*(f)

(5)

Coherence function

)()(

)()(

f S f S

f S f Coh

yy xx

xy

xy (Complex Coherence Function)

)()(

|)(||)(|

2

2

f S f S

f S f Coh

yy xx

xy

xy (Magnitude Squared Coherence Function)

Note: Here two discrete signals are taken into account. In cases, many pairs of two

discrete signals are counted, the summation must be used, called the average cross

spectrum and the average coherence function. Complex cross spectrum and

coherence function consists of two parts: i) magnitude and ii) phase

(6) Some simplified formulas

)()()()(

21

2

21221

uu

uuuu

S S S Coh

)()(

)()(

21

2121

uu

uu

uuS S

S Coh


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)()(

)()(

,0,

,

,

yuu

yu

yuS S

S Coh

: The span-wise coherence

Where: )(, yuCo is co-spectrum (real part) Fourier-transformed from

correlation.

7.4. Examples

Example 1: FFT vs. SPECTRUM

0 0.2 0.4 0.6 0.8-10

0

10

20

time [t]

signal

0 20 40 600

2

4

6

8

frequency [Hz]

spectra by using FFT

0 0.2 0.4 0.6 0.8-10

0

10

20

time [t]

signal

0 20 40 600

5

10

15

20

25

frequency [Hz]

spectrum by using spectrum

0 0.2 0.4 0.6 0.8

-10

0

10

20

Time[s]

Input Signal

0 20 40 60

0

2

4

6

8

Frequency [Hz]

Spectrum with FFT

1

Input Signal

stationary random process and digital signal processing - theory

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