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
2 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
1. STATIONARY RANDOM PROCESS
1.1. Basic concepts (1) Continuous random process:
)}()}...(),(),({)( 321 txtxtxtxtx Kk , 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, … tN (N: number of sampling values of
sample function)
t
t
t t+
x1(t)
kth sample function
1st sample function
xk(t) Time shift
3 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
)}()}...(),(),()( 321 Nkkkkk txtxtxtxtx : 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 )(txk and
)(tyk
(1) Mean value (Expectation): First-order statistical moment
dttx
NLimtxEt kNkx )(1)]([)(
Random process or Stochastic field
Stationary process
Non-stationary processes
Ergodic process
Non-ergodic signals
Gaussian process
Non-Gaussian process
4 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
dtty
NLimtyEt kNky )(1)]([)(
(2) Variance and covariance: Second-order moment
)()]([]))()([()( 2222 ttxEttxEt kkxkx : Variance
)()]([]))()([()( 2222 ttyEttyEt ykyky : Variance
)(*)()](*)([))]()())(()([()(
tttytxEttyttxEtC
yxkk
ykxkxy
: Covariance
))]()())(()([()( ttxttxEC xkxkxx : Covariance
))]()())(()([()( ttyttyEC ykykyy : Covariance
Note: Zero mean value process xk(t): 0)( tx
]))([()( 22 txEt kx : Variance
]))([()( 22 tyEt ky : Variance
)()0( 2 tC xxx
)()0( 2 tC yyy
(3) Mean square and root mean square
)]([)( 2 txEtC kxx : Mean square
)]([)( 2 tyEtC kyy : Mean square
Note: Zero mean value process xk(t): 0)( tx
5 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Variance )(2 tx = Mean Square Cxx(t)
(4) Correlations: Second-order moment
)](*)([)( txtxER kkxx : Autocorrelation
)](*)([)( tytyER kkyy : Autocorrelation
)](*)([)( tytxER kkxy : Cross-correlation
: arbitrary time (time shift or time lag)
Note: xxxxx RC 2)()(
yyyyy RC 2)()(
yxxyxy RC )()(
Zero mean random process: 0)( tx , 0)( ty
)()( xxxx RC
)()( yyyy RC
)()( xyxy RC
(5) Correlation coefficients
)()(
)()()(
2
xx
xxx
xx
xxxx R
RRC
: Auto-correlation coefficient
)()(
)()(
)(2
yy
yyy
yy
yyyy R
RRC
: Auto-correlation coefficient
)()(
)()(
)(
xy
yxxy
xy
xyxy R
RRC
: Cross-correlation coefficient
6 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
x
xx
xx
xxxx
CRC
2)(
)()()(
y
yy
yy
yyyy
CRC
2
)()()(
)(
yx
xy
xy
xyxy
CRC
)(
)()(
)(
Note 1:
i) )()( xxxx RC
ii) )0()( xxxx CC
iii) )0()( xxxx RR
iv) 2)0( xxxC and
2)0( yyyC
v) )0(*)0(|)(| 2yyxxxy CCC
vi) 222|)(| yxxyC
vii) )0(*)0(|)(| 2yyxxxy RRR
viii) )()( xxxx RR and )()( yyyy RR
)()( yxxy RR
Note 2: 0
i) 2)0( xxxC : Variance
7 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
ii) 2)0( yyyC : Variance
iii) 22 )]([)0( xxx txER : Mean square
iv) 22 )]([)0( yyy tyER : 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:
deRfS fjxxxx
2*)()(
: Auto-spectral density function
deRfS fjyyyy
2*)()(
: Auto-spectral density function
deRfS fjxyxy
2*)()(
: Cross-spectral density
function
Inverse Fourier Transform:
dfefSR fjxxxx
2*)()(
: Auto-correlation
dfefSR fjyyyy
2*)()(
: Auto-correlation
dfefSR fjxyxy
2*)()(
: Cross-correlation
8 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Sxx(f), Syy(f), Sxy(f): Two-sided spectra, f[-,]
One-sided spectral densities
dfRdfRfS xxxxxx 2cos*)(22cos*)()(0
0
2cos*)(2)( dffSR xxxx
Changing the two-sided spectral density Sxx(f) with f[-,] to the one-sided spectral
density Gxx(f) with f[0,]
)(2)( fSfG xxxx
)(2)( fSfG yyyy
)(2)( fSfG xyxy
Thus,
dfRfG xxxx 2cos*)(4)(0
; f[0,]
dfffGR xxxx 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
9 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
)()(*)(2)( 2 fiQfCdeRfG xyxyfj
xyxy
)( fCxy : Co-spectrum
)( fQxy : Quadratic spectrum
Writing in standard form: )(|)(|)( fj
xyxyxyefGfG
Where:
)()(|)(| 22 fQfCfG xyxyxy
)()(
tan)( 1
fCfQ
fxy
xyxy
)(cos|)(|)( ffGfC xyxyxy
)(sin|)(|)( ffGfQ xyxyxy
dfffiQffCR xyxyxy ]2sin)(2cos)([)(0
dffCR xyxy
0
)()0(
(2) Spectra via Fourier transform
Fourier Transform (Kinchint-Weiner’s pair):
dtetxfX ftjkk
0
2)()(
dfefXfx ftjkk
0
2)()(
10 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Spectral density function:
]|)([|12)( 2fXET
LimfG kTxx
]|)([|12)( 2fYET
LimfG kTyy
)](*)([12)( fYfXET
LimfG kkTxy
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(|)(| 2yyxxxy RRR
)]([*)]([|)](*)([| 222 tyEtxEtytxE
yxyyxxxy CCC 222 *)0(*)0(|)(|
]))([(*]))([(|)])((*))([(| 222yxyx tyEtxEtytxE
Thus,
yx
xyxy
C
*
)()( : Correlation coefficient
1)(0 xy
:0)( xy x(t), y(t) Uncorrelated;
11 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
:1)( xy x(t), y(t) Correlated;
(2) Coherence:
)(*)(|)(| 2 fGfGfG yyxxxy
)(*)(|)(|
)(2
2
fGfGfG
fyyxx
xyxy
1)(0 2 fxy
(3) Role of coherence function )(2 fxy (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 fxy 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 fxy is similar to the correlation
coefficient function )(2 xy . In constant-parameter linear systems, the
coherence has some following possibilities:
1) 0)(2 fxy : x(t) and y(t) uncorrelated (unrelated)
2) 1)(2 fxy : x(t) and y(t) correlated (unrelated)
3) 1)(0 2 fxy : x(t) and y(t) some possible situations exist:
a. Extra noise
b. Non-linear system between input x(t) and output y(t)
12 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
c. SIMO or MISO
)(*)()()(
)(2
2
fGfGfiQfC
fyyxx
xyxyxy
)(*)()(
)(2
2
fGfGfC
fyyxx
xyxy : Coherence
)( fCxy : Co-spectrum (Real part of Cross-spectrum)
)(),( fGfG yyxx : Auto-spectra
13 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 and A/D conversion
Data Analysis and Post-data processing
Deterministic signals Periodic signals
Nonperiodic signals
Sinusoidal signals (2 cycle)
Complex periodic signals (T cycle)
Almost periodic signals
Transient signals
14 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Branches of random signals
Notes and comments:
i) Sinusoidal (harmonic) signals (2-periodic signals): )2()( ntxtx
ii) Complex periodic signals (T-periodic signals): )()( nTtxtx
The complex periodic signals may be expanded by a Fourier series into the
combination of harmonic components (sine and cosine functions) as follow:
)2cos()( 00 t
TXtx (Original signals)
1
0 ]12sin12cos[2
)(n
nn tT
nbtT
naatx (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 of nonstationary signals
15 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
(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
iix x
N 1
1 ;
N
iix x
Nrms
1
21
)()(1)( 001
txtxN
R i
N
iixx
Nnkj
N
kix ex
NfS /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.
)(),()(
xxxx
xx
RtRt
iii) Weakly stationary signals (or stationary in the wide sense) are that mean
value and correlation (first and joint moments) are time invariant.
16 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
)(),()(
xxxx
xx
RtRt
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.
)(),()(
xxxx
xx
RtRt
(first moment and joint moment)
)(),(
)(
xxxxi
xxi
RtR
tth
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
xk
x
RR
k: Index of kth 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.
)()()(
)(
xxk
xx
xk
x
RR
vii) Gaussian signal (Normal distributed signal) is ergodic stationary one with
zero-mean and standard deviation
17 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
2)()(
)(
)()(
0
k
xxk
xx
kx
CR
viii) Non-gaussian signal is ergodic stationary one with non zero-mean and
standard deviation
2)(
0)(
xx
xx
Ct
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
18 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
19 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
(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 Records Analysis
Multiple Sample Records
Mean Values and Root Mean Square(RMS)
Auto-Correlation Function Analysis
Auto-spectral Density Analysis
Cross-Correlation Function Analysis
Cross-spectral Density Analysis
Coherence Function Analysis
Joint Probability Density Functions
Probability Density Functions
One sample of one process Pairs of two samples of one process or many processes
Frequency Response Function (FRF)
20 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
(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 noise
iii) 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.
21 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
SISO system
Independent MIMO system
Hxy(f)
Frequency Response Function (FRF)
Input signal x(t)
Output signal y(t)
Signal noise n(t)
Hxy,1(f) Input signal
x1(t) Output signal
y1(t)
Hxy,2 (f) Input signal
x2(t) Output signal
y2(t)
Hxy,k (f)
Input signal xk(t)
Output signal yk(t)
……………………..
Channel No.1
Channel No.2
Channel No.k
Signal noise n(t)
22 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 Sample Records
Multiple Sample Records
23 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
3.2.1. Data acquisition
Transducer: Device and sensor to transform from any physical phenomena
(force, pressure and motions: displacement, velocity and acceleration) to
electric 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 like
monocrystalline 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.
24 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 Records Analysis
Multiple Sample Records
Mean Values and Root Mean Square(RMS)
Auto-Correlation Function Analysis
Auto-spectral Density Analysis
Cross-Correlation Function Analysis
Cross-spectral Density Analysis
Coherence Function Analysis
25 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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):
nx n=1,2,3… N (N: Number of samples)
Equally spaced time interval (T: sampling time or sampling period) of samples in
discrete signal:
TnTTn *0 n=1,2,3… N
Continuous Signal A/D Conversion Data Sampling
Discrete Signal Discrete Data Record
Frequency Response Function
Signal Filter Input signal ui (t)
A/D Conversion Data Sampling
Output signal xn(t)
Continuous Data Discrete Data
26 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
T0: Initial time (T0 = 0) TnTn *
T: Sampling period
NT: Total time length of discrete signal; TN *
F: Sampling frequency (Hz); TF 1
F0: Fundamental sampling frequency; TNF
*1
0
Thus, )*()( 0 TnTxTxx ni n=1,2,3… N
Noting that the Limit sampling frequency (the Nyquist frequency) for eliminating the
Aliasing Errors
TFN *2
1
3.3.2. Data standardization
The purpose of data standardization is to transform the original data record nx to
new type of data record nx' (can be called the fluctuating data record) that has the
zero mean value.
1) Mean value and root mean square of sample record
N
nnx
Nx
1
1 n=1,2,3…N (Mean value)
27 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
N
nnx x
NSQRTs
1
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 nx to the new zero-mean data record nx' .
2) Establishment of the fluctuating data record (Zero-mean data record)
nx' : n=1,2,3…N
xnTxxxx nnn )(' n=1,2,3…N
Having: 0' x (Zero-mean value)
N
nnx x
N 1
2'
2 '1
1 (Mean Square value or Variance)
N
nnxx x
NSQRTs
1
2'' ]'
11[
(Root Mean Square or Standard Deviation)
3) Fourier Transform (Discrete Fourier Transform-DFT):
Fluctuating data record (Zero-mean data record) or standardized data record
nx' n=1,2,3…N
Discrete Fourier Transform
N
nkk TnFjTnxFX
1)]2exp(*)([1)(
28 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
X(Fk): DFT at computational frequency kF
k: Index of discrete frequencies
Fk: Computational frequency; )( fkFk
Note:
i) The frequency space )( f = The fundamental sampling frequency (F0)
NTFf 1
0
ii) Number of frequency space (K)
0FFFK sc
cF : Cut-off frequency
sF : Starting frequency
iii) Computational frequency Fk:
NTkFfkFF ssk )( ; k=0,2,3…K-1
NTkFk ; 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
29 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
N
nk Tn
NTkjTnx
NTkXFX
1)]2exp(*)([1)()(
;k=0,2,3…K-1
N
nk NnkjTnx
NTkXFX
1)]/2exp(*)([1)()(
;k=0,2,3…K-1
k=1:
N
ns NnjTnxFXFX
11 )]/2exp(*)([1)()(
k=2:
N
ns NnjTnx
NTFXFX
12 )]/22exp(*)([1)1()(
k=3:
N
ns NnjTnx
NTFXFX
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
kNnkj
NTkX
NnTx
1)]/2exp(*)([1)( ; n=0,1,2…N
DFT X(Fk)
f(Hz) F1=Fs k=1
X(F1) X(Fi)
Fi=Fs+i/NT k=i
F1=Fc k=K
30 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
nx' n=1,2,3…N
Auto-correlation function of data record nx' with delay s
N
nsnnxx ssxsx
NsR
1)()( )]()([1)( (Auto-correlation functions)
trs *
rN
nrnnxx ssxsx
rNtrR
1)]()([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 technique
Fluctuating data record (Zero-mean data record) or standardized data record
nx' n=1,2,3…N
31 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Definition of mean power:
NTN
nnxx dttx
NTx
NtxES
0
21
0
22 )(11)]([
1
0
2|)(|1)(N
nnxx fX
NTfS : Two-sided spectral density
)( fX : Fourier transform at frequency f
1
0]/2exp[*1)(
N
nn Nnkjx
NfX ; k=0,1,2…N-1
1
0
2|)(|2)(2)(N
nknkxxkxx fX
NTfSfG : One-sided spectral density
Computational procedures (Ensemble-averaging technique)
1) Data Sampling: N samples of process x(t) are picked out (Data record)
nx , n=1,2,3…N
2) Data Standardization: Compute the mean value of data record ( x ), then
reconstruct the Fluctuating Data Record nx' with zero-mean value
( 0' x )
t
x(t)
T 2T 3T 4T 5T 6T NT (N-1)T
32 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
nx' , 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 fk,
k=1,2,3…N by using FFT technique
)( kfX , k=1,2,3…N
5) Scale Factor Adjustment of )( kfX : Scale factor of )( kfX due to the loss
by tapering operation. (By Hanning tapering: Scale factor by 3/8 )
6) Spectral Density: Estimate the spectral density )( kxx fS from each data
blocks
33 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
nsnnNsnnxy ssysx
NLimyxEsR
E[] : Expected value or mean value
For the number of samples is taken large enough, we have following approximations:
N
nsnnxx ssxsx
NsR
1)()( )]()([1)( (Auto-correlation functions)
N
nsnnxy ssysx
NsR
1)()( )]()([1)( (Cross-correlation
functions)
s : Spatial interval or time shift
34 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
s: Spatial coordinates or time variable
N: Number of samples
Note: 1) Corresponding values of signals are sampled at same time.
2) Rxx(s)=Rxx(-s) and Rxy(s)=Rxy(-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
xyxx RR ,
is
)( ixx sR Rxx, Rxy vs. s
35 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
1
0
1
0)()()()( ])([1)]([1][][)(
N
n
N
nsnNnNsnnxy ssy
NLimsx
NLimyExEs
For the number of samples is taken large enough, we have following approximations:
N
n
N
nsnnxx ssx
Nsx
Ns
1 1)()( ])([1)]([1)( (Auto-covariance functions)
N
n
N
nsnnxy ssy
Nsx
Ns
1 1)()( )]([1)]([1)( (Cross-covariance
functions)
s : Spatial interval or time shift
s: Spatial coordinates or time variable
N: Number of samples
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
nsnnxx ssx
NSQRTsx
NSQRTs
1 1)(
2)(
2 ]})([1{)]}([1{)(
Cross-covariance function:
N
n
N
nsnnxy ssy
NSQRTsx
NSQRTs
1 1)(
2)(
2 ]})([1{)]}([1{)(
Computational procedure of covariance function:
Step 1: Setting parameters
+) Number of sample: N
36 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
+) Time shift or spatial interval: s
Step 2: Sampling
+) Sampling signals x, y
Step 3: Computing covariance function
Step 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:
)()(
)(ssR
sxx
xxxx
(Auto-correlation coefficient function)
)()(
)(ssR
sxy
xyxy
(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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
39 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 processing
and analysis on simple harmonic functions seem to be much easy than the original
signals. Mathematically, the Fourier series is known as the harmonic analysis.
N: Number of samples
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
Fo 1
Discrete Fourier series is expressed as the sum of harmonic functions as follow:
)]sin()cos([2 00
1
1
0 TmnbTnmaax m
M
mmn
0 : Fundamental frequency of series (rad/s), NTFo
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 NmnbNnmaax m
M
mmn
40 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Discrete Fourier Coefficients:
)/2cos(2 1
0Nnmx
Na
N
nnm
; 10 Mm
)/2sin(2 1
0Nnmx
Nb
N
nnm
; 11 Mm
Computational procedure:
Step 1: Setting parameters
- 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 10
41 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Discrete Fourier Transform: (to frequency domain)
N
n
Tnjk
kenTxX1
])([)(
N
n
Tnfjk
kenTxfX1
2 ])([21)(
(Note: In origin, n=0N-1)
k: Number of discrete frequencies in range
In Matlab: 0)( kkk [k=1-K]: equal spacing of frequency range
0 : Fundamental frequency (rad/s), NTFo
220
Note: n (pointer of samples) and k (pointer of frequency) have different
meaning
Computational procedure:
Step 1: Setting parameters
- T (cycle sampling), N (numbers of samples)
- Freq. range: fs (starting freq.), fc (cut-off freq.)
- Number of freq. interval: K kfsfc /)(
Step 2:
Loop 1: Frequency
For k=1 to K )( kk
Loop 2: Sampling
For n=1 to N
N
n
nTjkk enTxX
1
)()()(
Step 3: Plotting X(k) versus (k)
42 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Putting NTFo
220 into formulas:
N
n
nTjkenTxkX1
0 ])([)( 0
N
n
nTNT
jkenTxFkX
1
2
0 ])([)2(
N
n
NknjenTxkFX1
/20 ])([)(
Fourier Series (in time domain)
K
kkkkk tkbtkaath
1
0 )]sin()cos([2
)(
)}(Re{2 kHN
ak
)}(Im{2 kHN
bk
Note in processing using FFT:
1) )/(2 NT or )/(1 NTF
2) Maximum frequency: )2/(12/max TFF
3) Spectrum calculated at certain frequencies: 0, F, 2F, 3F… Fmax
Leakage Effect: Amplitudes will distribute on adjacent closed frequencies. This effects
occurs 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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 is
much 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
NknjenTxkFX
1
0
N
n
knNnk WxX ; k=[0,N-1]; Nj
N eW /2
Xk contains N2 components (N of xn and N of WN)
Note: sincos je j = x + jy (Euler’s Formula)
)/2sin()/2cos(/2 NknjNkne Nknj
01
12
21
0
2
Nj
NNj
N
n
Nnj
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
0 8/02je
8/22je
8/52je 8/72je
Re
Im
44 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Based on above-mentioned notes, we can easily obtain:
),( Nmrem
Nm
N WW ; m[0,N-1]; rem(m,N) is the remainder after dividing m
by N mNjm
N eW )/2( ; m[0,N-1]
))(/2(),( miNNjNmremN eW
; i=1,2,..N
knMax )(
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
knNn
knNnk WxWxX ; k=[0:N-1]
Due to: k
Nk
N WW 22/2/ , thus
12/
0
12/
02/122/2
N
n
N
n
knNn
knNnk WxWxX ;
168
88
08 WWW
178
98
18 WWW 19
811
83
8 WWW
208
128
48 WWW
218
138
58 WWW
228
148
68 WWW
238
158
78 WWW
188
108
28 WWW
45 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Number of components reduces to 2)2
( N < 2N of original problem. Depending on N,
efficiency of FFT algorithm is different. For example, if N=2K then complex
components in FFT is NNKN2log
22 (in comparison of N2 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
NknjenTxkFX01
/20 ])([)(
; k=[0:N-1]:
1
0
/20 ])([1)(
N
k
NknjekFXN
nTx ; 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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 from
samples ’
signalsampling FF 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 60 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
number of samples
Num
ber o
f ser
ies
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
Num
ber o
f ser
ies
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
Num
ber o
f ser
ies
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
Num
ber o
f ser
ies
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
Num
ber o
f ser
ies
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
Num
ber o
f ser
ies
M=5
0
Initial Signal Impulse Function
Signal from D.F.S
47 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Example 2: FFT vs. SPECTRUM
Fourier Transform by FFT and Spectrum
Note:
Fourier Transform Spectrum
dtetxjXxCFT tj )()()(
1
0)()(
N
n
nTjnexjXxDFT
dtetxjXxCSPEC tj )()()( 2
1
0
2)()(N
n
nTjnexjXxDSPEC
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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
NknjenTxkFX1
/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
NknjenTxkFX1
/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
Sig
nal
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
DFT
Mag
nitu
de
DFT by definition
0 20 40 600
2
4
6
8
Frequency Hz
Spectrum command
49 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
Sig
nal
0 20 40 60-0.4
-0.2
0
0.2
0.4
0.6
Frequency(Hz)
Am
plitu
de
Spectrum
0 20 40 60-4
-2
0
2
4
Frequency(Hz)
Pha
se(ra
d)
Phase spectrum
0 20 40 600
5
10
15
20
Frequency(Hz)
Pha
se(ra
d)
Phase spectrum
fftshift(H)
unwrap() Angle(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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
- Mean value (Expectation):
1
0
1][N
iix x
NxE
- Mean square :
1
0
222 1][N
iix x
NxE
- Root mean square : )1(][1
0
22
N
iix x
NsqrtxE
- Variance:
21
0
1
0
22222 ]1[]1[][])[(
N
ii
N
iixxx x
Nx
NxExE
- Standard deviation: )][(]))[(( 222xxx xEsqrtxEsqrt
Thus zero-mean random signals ( 0x ) 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: 222xx
xy
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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
The instantaneous power of signal x(t) at any time t can be defined as: 2|)(| txPower
For the complex signals, we must use the mean power or expected power:
2/
2/
22/
2/
22 )(1)(1)]([T
T
T
TT dttx
Tdttx
TLimtxEpowerMean
(when T is long enough)
NTN
nnxx dttx
NTx
NtxESpowermeanDiscrete
0
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
NknjekFXN
nTx
Thus, the discrete mean power can be written as follows:
1
0
21
0
/20
1
0
2 ])(1[11 N
n
N
k
NknjN
nnxx ekFX
NNx
NS
1
0
1
0
1
0
/)(23
1
0
21
0
/203
1])([1 N
k
N
m
N
n
Nnmkjmk
N
n
N
k
Nknjxx eXX
NekFX
NS
1
0
22
1 N
kkxx X
NS
53 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
We have:
1
0
21 N
kkxx x
NS
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
kk
N
kk X
Nx (Parseval’s Theorem)
In DSP, it is defined the periodogram that has N components given by:
21)( kxx XN
kP ; k=[0,N-1]
)()( kNPkP xxxx
The power spectrum can be obtained:
1
0
1
0
2 )(11 N
kxx
N
kkxx kP
Nx
NS
Note: In the same way that 2kx 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)
54 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Discrete PSD: 1)
1
0
22
1 N
kkxx X
NS (Frequency domain)
Xk is DFT amplitude of x(t) at frequency k
2)
1
0
21 N
kkxx x
NS (Time domain)
xk is discrete value of signal x(t) at time t
3)
1
0
1
0
2 )(11 N
kxx
N
kkxx kP
Nx
NS (Periodogram)
21)( kxx X
NkP ; m=[0,N-1]
Parseval’s Theorem:
1
0
21
0
2 1 N
kk
N
kk X
Nx (Parseval’s Equality)
Continuous PSD: 1) NT
xx dttxNT
S0
2 )(1 (Frequency domain)
2)
T
Txxxx dffPS
2/1
2/1
)( or
T
Txxxx dPS
/
/
)(
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
nsnnxx xx
NsR ; s=[0:N-1]
By definition of the mean power, we have:
55 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
1
0
21)0(N
nxxnxx powerMeanSx
NR
Discrete Fourier transform of the auto-correlation can be calculated by definition:
1
0
1
0
/21)}({N
s
N
n
NskjsnnxxR exx
NsRDFTS
; k=[0:N-1]
1
0
/2N
n
Nknjnk exX
; k=[0:N-1] (Definition of DFT)
)(1)}({ 2 kPXN
sRDFTS xxkxxR ; 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 kPXN
sRDFTS xxkxxR
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: kkxy YXN
kP '1)( ; k=[0:N-1]
56 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
- Mean cross-power:
1
0
1 N
nnnxy yx
NS
1
0
1
02 )(1'1)0(
N
k
N
kxynnxyxy kP
NYX
NRS
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 )(kPxy , is
the measure of the coherence of signals x(t) and y(t) at different frequencies.
- Coherence function: )()(
)()(
2
2
fPfPfP
fCohyyxx
xyxy
6.4. Examples Example 1: FFT and SPECTRUM
0 0.5 10
0.2
0.4
0.6
0.8
1Impulse 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
Ampl
itude
FFT
0 0.5 10
0.2
0.4
0.6
0.8
1
Time (s)0 50
0
2
4
6
8x 10
-5
Frequency (Hz)
X2
0 500
2
4
6
8
Frequency (Hz)
Spectrum
Ampl
itudeThe same
Difference in shape? Different in value?
57 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
Note: 1) Using FFT command
H=FFT(x);
H=[H(1) 2*H(2:N/2)]/N; %Vector length limit and
standardization
2) 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 definition
S=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 intervals Frequency intervals
Sampling process (Step 2)
FFT (Step 3)
SPECTRUM (Step 4)
Digital Signal or Random Process
58 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
20normal signal
0 500
2
4
6
8
10FFT
0 500
2
4
6
8
10FFT with boxcar
0 500
1
2
3
4
5FFT with hamming
0 500
1
2
3
4FFT with bartlett
0 500
1
2
3
4FFT with hann
59 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
(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 two
signals
(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)}
61 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
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
)()(
)()(
fSfSfS
fCohyyxx
xyxy (Complex Coherence Function)
)()(|)(|
|)(|2
2
fSfSfS
fCohyyxx
xyxy (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
2212
21
uu
uuuu SS
SCoh
)()()(
)(21
2121
uu
uuuu SS
SCoh
62 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing
)()(
)()(
,0,
,,
yuu
yuyu SS
SCoh
: 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 40 600
2
4
6
8
Frequency [Hz]
Spectrum with FFT
0 20 40 60-0.5
0
0.5
1
Time[s]
Input Signal