Fundamentals of
Measurement Technology
(3)
Prof,Wang Boxiong
1,Unit impulse function
Assuming a rectangular pulse pΔ(t) of
a width Δ and an amplitude 1/Δ,its area is
equal to 1,As Δ→0,the limit of pΔ(t) is
called the unit impulse function or delta
function denoted by δ(t),
2.2.6.4 Fourier transforms of power signals
Fig,2.36 Rectangular pulse function and delta function δ(t)
δ(t) is a pulse with unbounded amplitude and
zero time duration,This impulse function must
be treated as a so-called generalized function.
Properties:
1)
2)
The two properties for the impulse function can
be conveniently summarized into one defining
equation for δ(t),
provided x(t) is continuous at t=t0,
2.2.6.4 Fourier transforms of power signals
0,0
0,)(
t
tt? (2.96)
1)()(
tdt?
(2.97)
)()()( 00 txdttttx? (2.99)
The Fourier transform of the impulse
function δ(t):
Fourier transform pair:
2.2.6.4 Fourier transforms of power signals
1)()()( dtettFX tj (2.100)
1)(?t?
(2.101)
Fig,2.37 δ(t) and its Fourier transform
2.2.6.4 Fourier transforms of power signals
0)( 0 tjett
(2.102)
Fig,2.38 δ(t-t0) and its Fourier transform
)(2 00tje
Using the symmetry property,we can derive
the transform pairs:
(2.103)
2.2.6.4 Fourier transforms of power signals
)(21
(2.104)
Fig,2.39 The unity and its Fourier transform
Furthermore,we have the following
relation,
2.2.6.4 Fourier transforms of power signals
)()()()()( txtxtttx (2.105)
)(
)()(
)()()()(
tx
dtx
txtttx
)()()()()( 000 ttxtxtttttx (2.106)
Fig,2.40 Convolution of an arbitrary function with a unit impulse
2,Sinusoidal functions
Using the transform pair
we see
Similarly,
2.2.6.4 Fourier transforms of power signals
2c o s
00
0
tjtj ee
t
(2.109)
)(2 00tje
)()(c o s 000t (2.110)
)()(s in 000 jt (2.111)
2.2.6.4 Fourier transforms of power signals
Fig,2.42 Sinusoidal functions and their spectra
3,The Signum Function
The signum function,denoted by sgn(t),
is defined as
If
then
Suppose we differentiate the signum
function,Its derivative is 2δ(t):
2.2.6.4 Fourier transforms of power signals
0,1
0,0
0,1
)s g n (
t
t
t
t
(2.112)
)()(?Xtx?
)()( Xjdt tdx?
)(2)s g n ( ttdtd
The transform of
where
Thus the transform of sgn(t) is
The transform pair:
2.2.6.4 Fourier transforms of power signals
)sgn( tdtd
)()s g n ( Xjtdtd?
2)(2)( tFXj
jX
2)(?
jt
2)s g n (? (2.113)
2.2.6.4 Fourier transforms of power signals
Fig,2.43 sgn(t) and its spectra
5,Periodic functions
Periodic functions can be represented as
a sum of complex exponentials; because
we can transform complex exponentials by
means of Eq.(2.103),we should be able to
represent a periodic function using the
Fourier integral,Assuming x(t) is periodic of
period T,then
2.2.6.4 Fourier transforms of power signals
n
tjn
n eCtx
0)(?
The Fourier transform of x(t),
where
2.2.6.4 Fourier transforms of power signals
n
n
n
tjn
n
n
tjn
n
nC
eFC
eCF
txFX
)(2
][
][
)]([)(
0
0
0
(2.117)
dtetxTC tjn
T
Tn
0)(1 2
2
Example 12:
A special kind of periodic function is the unit
impulse train shown in Figure 2.47,This
function is useful in applications involving
sampling of time waveforms,
2.2.6.4 Fourier transforms of power signals
Fig,2.47 Periodic impulse sequence
Solution,The function has the form:
Expand it in a Fourier series as
where
the Fourier series representation of the unit
impulse train is
2.2.6.4 Fourier transforms of power signals
n
kTttx )()(?
(2.118)
tjnn eCtx 0)(?
TdtetTdtetxTC tjn
T
T
tjn
n
1)(1)(1 00 2
2
n
tjne
Ttx 0
1)(? (2.119)
Take the Fourier transform on both sides of
(2.119),
That is,
2.2.6.4 Fourier transforms of power signals
]1[)( 0
n
tjne
TFX
n T
nTX 2),(2)( 00
n n
nkTt )()( 00
(2.120)
Periodic impulse sequence and its frequency spectrum
A signal is said to be random if it depends on
probabilistic laws,Such signals:
– have unpredictable instantaneous values;
– cannot be desired by analytical time models;
– can be characterized by their statistical and
spectral properties,
An observed random signal must be seen as
a particular experimental realization of a set
(referred to as the ensemble) of similar
signals that can all be produced by the same
phenomenon or stochastic process,
2.2.7 Random signal description
2.2.7.1 Introduction
Sample function,the record of every
long-time observation for the time process
of a random signal,denoted by x(t)(Fig,
2.48),
2.2.7.1 Introduction
Fig,2.48 Random process and sample function
Sample record,a sample function in a
finite time interval,
Random process,the set (or ensemble)
of all sample functions under the same
experimental conditions,that is,
2.2.7.1 Introduction
),(,),(),()( 21 txtxtxtx i? (2.121)
The commonly used statistical characteristic
parameters:
– mean value,mean square value,variance,
probability density function,probability
distribution function,and power spectral density
function,etc,
– They are deduced in terms of set average,
They are calculated by averaging the simultaneous
observations for all sample functions at a certain time
instant ti(set average or ensemble average).
E.g.,mean value μx(t1) and mean square value ψ2x
(t1),
2.2.7.1 Introduction
N
i
iNx txNt
1
11 )(
1lim)(? (2.122)
N
i
iNx txNt
1
1
2
1
2 )(1lim)(?
(2.123)
A random process is said to be stationary if
all its statistical properties are time-invariant.
Strictly speaking,if the n-dimensional
distribution function of a random process,
{x(t)},for the n values t1,t2,…,t n of time and
any real number,satisfies the following
relationship,
{x(t)} is called a stationary random
process or stationary process,
Random processes that do not satisfy Eq,
(2.124) are called nonstationary processes,
2.2.7.1 Introduction
,2,1
),,,,;,,,(),,;,,( 21212121
n
tttxxxFtttxxxF nnnnnn
(2.124)
Examples of stationary processes in our
daily life:
– variations of voltage due to thermal noise,
– toss of ship,
– errors in measuring distances of moving
objects,
– vibrations at a certain location in geological
prospecting,
– voltage fluctuations in electrical light power
lines,
– all noises and interferences,
2.2.7.1 Introduction
A stationary random process is said to be
ergodic if the statistical characteristics of
the time average of its any one sample are
identical to those of the set average of the
process,
Many stationary random processes
encountered in engineering are of ergodism,
2.2.7.1 Introduction
1,Mean value,mean square value
and variance
For an ergodic process x(t),its mean
value is defined as
where
E[x],mathematical expectation of the
variable x,
x(t),sample function;
T,time duration for observation,
2.2.7.2 characteristic parameters for random process
dttxTxE TTx )(1lim][ 0? (2.125)
The variance of a random signal σ2x
Relationship between μx,σ2x and ψ2x,
The square root of ψ2x is called the root of
mean square value xrms.
When μx=0,then
2.2.7.2 characteristic parameters for random process
dttxT XTTx 202 ])([1lim (2.127)
222 xxx (2.128)
22 xx
rm sx x
Estimate formula for mean value,variance
and mean square value,
2.2.7.2 characteristic parameters for random process
dttxT Tx )(1? 0?
dttxT Tx )(1 202?
dttxT xTx 202 ]?)([1
(2.129)
(2.130)
(2.131)
2,Probability density function and
probability distribution function
In the observation time T,the total sum
of the time durations when the
instantaneous value of a random signal x(t)
falls into the interval (x,x+Δx) is
2.2.7.2 characteristic parameters for random process
N
i
inx ttttT
1
21?
(2.132)
Fig,2.49 Physical interpretation of probability density function
Probability of the random signal x(t):
Probability density function:
– Gaussian process or normal process:
The probability density function of a random variable
is of the form of a Gaussian function,
2.2.7.2 characteristic parameters for random process
T
TxxtxxP x
T lim])([
(2.133)
x
xxtxxPxp
x?
])([lim)(
0
x
T
T x
T
x?
0
lim
(2.134)
(2.135)
xxxp
x
x
x
]2 )(e x p [21)( 2
2
(2.136)
Probability distribution function,
where Tx’ is the total time when x(t) is lower
than or equal to x,
2.2.7.2 characteristic parameters for random process
T
TxtxPxP x /lim])([)(
(2.137)
Fig,2.50 (a) The probability density function and (b) the probability
distribution function of a normal process
many random phenomena obey or
approximately obey a normal distribution,
– Examples:
heights of the male adults in a certain geological
region,
errors in measuring the lengths of components,
heights of ocean waves,
electrical currents and voltages due to thermal noises
in semiconductor components,etc,
Probability density function can be used to
identify different random processes.
2.2.7.2 characteristic parameters for random process
2.2.7.2 characteristic parameters for random process
Fig,2.51 Graphs of probability functions of four random signals
3,Correlation analysis
1) Definition of correlation
2.2.7.2 characteristic parameters for random process
Fig,2.52 Different situations of correlation for variables x and y
The classical method for evaluating the
degree of linear correlation between two
random variables is to calculate their
covariance σxy and correlation coefficient
ρxy,
where E is the mathematical expectation,
where σx and σy are the standard deviations
of x and y respectively,
2.2.7.2 characteristic parameters for random process
N
i
yixiN
yxxy
yx
N
yxE
1
1lim
(2.142)
11 xy
yx
xy
xy
(2.143)
The variances σ2x and σ2y of x and y are
Using Cauchy-Schwarz inequality
we get |ρxy|≤1.
When ρxy =1,all data points will be on a line:
x and y are linearly correlated,
When ρxy =0,x and y are totally uncorrelated.
2.2.7.2 characteristic parameters for random process
22 xx xE (2.144)
22 yy xE (2.145)
222 yxyx yExEyxE
(2.146)
)( xy xmy
2) Crosscorrelation function and
autocorrelation function
A covariance coefficient Cxy(τ) for time
variables x and y is defined as:
where
Rxy(τ) is called the crosscorrelation function
of x(t) and y(t),where τ is the time-shift,
2.2.7.2 characteristic parameters for random process
yxxy
T
yx
T
yxxy
R
dttytx
T
tytxEC
0
1
l i m (2.147)
TTxy dttytxTR 01lim
(2.148)
When y(t)≡x(t),the autocovariance
function,Cx(τ) is:
where
Rx(τ) is called the autocerrelation
function of x(t),
2.2.7.2 characteristic parameters for random process
2
0
1lim
xx
T
xxTx
R
dttxtx
T
C
(2.149)
TTx dttxtxTR 01lim (2.150)
Properties:
a.
b.
A crosscorrelation function takes on its
maxinmum value usually at τ≠0,
2.2.7.2 characteristic parameters for random process
xx RR
)()( yxxy RR?
yxxy RR
(2.151)
(2.152)
222m a x0 xxxxx RR (2.153)
c.
d.
e.
correlation coefficient function,
and |ρxy|≤1 holds for all values of τ,
2.2.7.2 characteristic parameters for random process
2222 xxxxx R (2.154)
yxyxxyyxyx R (2.155)
2xxR (2.156)
yxxyR (2.157)
00 yxxy RRR
(2.158)
22 00
00
yyxx
yxxy
yx
xy
xy
RR
R
CC
C
(2.159)
f,The autocorrelation function of a periodic
function is still a periodic one,and has the
same frequency as that of the original periodic
function but contains no information on the
phase,
Crosscorrelation function for two signals
having same periodic component of a certain
frequency will contain the periodic component
of this frequency and never converges even if
τ→∞.
Two signals having periodic components
of different frequencies are not correlated.
2.2.7.2 characteristic parameters for random process
2.2.7.2 characteristic parameters for random process
Fig,2.53 (a) Autocorrelation function
(b) Corsscorrelation function
Example 1:
Find the autocorrelation function of the
sinusoidal function x(t)=Asin(ωt+υ),
Solution:
where
2.2.7.2 characteristic parameters for random process
0
0
0
0
])(s i n [)s i n (
1
)()(
1
lim)(
T
T
T
x
dtttA
T
dttxtx
T
R
2
0?T
Letting ωt+υ=θ,then dt=dθ/ω,so that
Rx(τ) keeps the information on both
amplitude and frequency of the original
signal,but loses the phase information,
2.2.7.2 characteristic parameters for random process
c o s2)s i n (s i n2)(
22
0
2 A
dAR x
Autocorrelation function can be used to
detect periodic components embedded in a
random process,
2.2.7.2 characteristic parameters for random process
Fig,2.54 (a) A harmonic signal embedded in a random noise
(b) its autocorrelation function
Example 2:
Two periodic signals
where θ — initial phase angle of x(t)
φ — the phase difference of x(t) and
y(t)
Find their crosscorrelation function Rxy(τ).
2.2.7.2 characteristic parameters for random process
)s in ()( tAtx
)s in ()( tBty
Solution:
The result shows that for two periodic
signals with the same frequency,their
crosscorrelation function maintains their
common frequency ω,their respective
amplitude A and B,and their phase
difference υ,
2.2.7.2 characteristic parameters for random process
)c o s (
2
1
))(s i n ()s i n (
1
)()(
1
lim)(
0
0
AB
dttBtA
T
dttytx
T
R
oT
o
T
T
xy
Estimates of an autocorrelation function
and a crosscorrelation function
2.2.7.2 characteristic parameters for random process
Tx dttxtxTR 01
Txy dttytxTR 01
(2.160)
(2.161)
3) Applications of correlation functions
a,Identification of different signals
2.2.7.2 characteristic parameters for random process
Fig,2.55 Autocorrelation functions of several typical signals
b,Velocity and displacement measurements
2.2.7.2 characteristic parameters for random process
Fig,2.56 Sound propagation distance measurement using correlation
– The task of the correlator:
adjust the time τ,τ=T,the delayed signal μ1(t-τ) is
then equal to the signal μ2 =μ1(t-τ)
– The correlator must make the mean square of
difference of the two signals a minimum:
2.2.7.2 characteristic parameters for random process
Fig,2.58 Fluid flow measurement
M i nTttEttEE 2112212 )()())()(( (2.164)
– For a steady-state signal,ideally
when τ=T,the correlation function Rμ1μ2(τ=T)
has the maximum,The correlator finds the
time τmax,T= τmax,By determining the
distance s between the two transducers,the
velocity can be calculated as v=s/T.
2.2.7.2 characteristic parameters for random process
c o n s tTtEtE )()( 2121
)()()( 2111 TRTttE
(2.165)
(2.166)
4) Power spectrum analysis
a,Auto-power spectral density
Its inverse Fourier transform is then
Sx(f),the auto-power spectral density function,
or simply the auto-power density spectrum of
x(t),
Eqs,(2.167) and (2.168) are called the
Wiener-Khintchine equations,
2.2.7.2 characteristic parameters for random process
deRfS fjxx 2 (2.167)
dfefSR fjxx 2 (2.168)
)()( fSR xFT
I F T
x
When τ=0,
Sx(f) represents the frequency distribution of
the random signal power,and it is therefore also
called the power spectrum.
2.2.7.2 characteristic parameters for random process
Fig,2.59 One-sided power spectrum and two-sided power spectrum
dffSdttxTR xT T
Tx
22
2
1l i m0 (2.169)
b,Parseval’s theorem
According to the frequency convolution
theorem,
and
letting k=0,thus
Again letting h(t)=x(t),we obtain
2.2.7.2 characteristic parameters for random process
)(*)()()( fHfXthtx?
dffkHfXdtethtx ktj )()()()( 2?
dffHfXdtthtx )()()()(
dffXfXdttx )()()(2
Since x(t) is a real function,then X(-f)=X*(f),
that is,X(-f) is the conjugate of X(f),thus
Eq,(2.170) is known as the Parseval’s theorem,
|X(f)|2 is often referred to as the energy-density
spectrum of signal x(t),and Eq,(2.170) is also
called the signal energy equality,
2.2.7.2 characteristic parameters for random process
dffXdffXfXdttx 2*2 (2.170)
The average power of signal over the whole time
axis:
We often use one-sided power spectrum,
2.2.7.2 characteristic parameters for random process
dffxTdttxTP T
T
TT
22
2
2 1lim1lim
(2.171)
21lim fXTfS Tx (2.172)
0 dffGdffSP xx (2.173)
0)(2)( ffSfG xx (2.174)
c,Cross-power spectral density function
Sxy(f) is also known as the cross-power density
spectrum or cross-power spectrum of x(t) and
y(t),
Fourier transform pair:
The inverse Fourier transform of Sxy(f),
The cross-power of x(t) and y(t),
2.2.7.2 characteristic parameters for random process
dfeRfS ftjxyxy 2 (2.175)
)()( fSR xyFTI F Txy
(2.176)
dfefSR fjxyxy 2
(2.177)
dffXfY
T
dttytx
T
P
T
T
TT
*
2
2
1
lim
1
lim
(2.178)
So
where
One-sided cross-power spectrum Gxy(f)
2.2.7.2 characteristic parameters for random process
fXfYTfS Txy *1lim
(2.178)
fSfSfS yxxyxy * (2.179)
)()(1lim * fYfXTfS txy
02 ffSfG xyxy
(2.180)
where Cxy(f) = cospectrum
Qxy(f) = quad spectrum
2.2.7.2 characteristic parameters for random process
fjQfCfG xyxyxy
(2.181)
fC
fQ
ar c t gf
fQfCfG
efGfG
xy
xy
xy
xyxyxy
fj
xyxy
xy
22
(2.182)
4,Estimates of auto-spectrum and
cross-spectrum
Estimate of auto-spectrum,
Estimate of cross-spectrum,
2.2.7.2 characteristic parameters for random process
21? fXTfS x?
(2.183)
fYfXTfS xy *1?
(2.184)
fXfYTfS yx *1? (2.185)
Measurement Technology
(3)
Prof,Wang Boxiong
1,Unit impulse function
Assuming a rectangular pulse pΔ(t) of
a width Δ and an amplitude 1/Δ,its area is
equal to 1,As Δ→0,the limit of pΔ(t) is
called the unit impulse function or delta
function denoted by δ(t),
2.2.6.4 Fourier transforms of power signals
Fig,2.36 Rectangular pulse function and delta function δ(t)
δ(t) is a pulse with unbounded amplitude and
zero time duration,This impulse function must
be treated as a so-called generalized function.
Properties:
1)
2)
The two properties for the impulse function can
be conveniently summarized into one defining
equation for δ(t),
provided x(t) is continuous at t=t0,
2.2.6.4 Fourier transforms of power signals
0,0
0,)(
t
tt? (2.96)
1)()(
tdt?
(2.97)
)()()( 00 txdttttx? (2.99)
The Fourier transform of the impulse
function δ(t):
Fourier transform pair:
2.2.6.4 Fourier transforms of power signals
1)()()( dtettFX tj (2.100)
1)(?t?
(2.101)
Fig,2.37 δ(t) and its Fourier transform
2.2.6.4 Fourier transforms of power signals
0)( 0 tjett
(2.102)
Fig,2.38 δ(t-t0) and its Fourier transform
)(2 00tje
Using the symmetry property,we can derive
the transform pairs:
(2.103)
2.2.6.4 Fourier transforms of power signals
)(21
(2.104)
Fig,2.39 The unity and its Fourier transform
Furthermore,we have the following
relation,
2.2.6.4 Fourier transforms of power signals
)()()()()( txtxtttx (2.105)
)(
)()(
)()()()(
tx
dtx
txtttx
)()()()()( 000 ttxtxtttttx (2.106)
Fig,2.40 Convolution of an arbitrary function with a unit impulse
2,Sinusoidal functions
Using the transform pair
we see
Similarly,
2.2.6.4 Fourier transforms of power signals
2c o s
00
0
tjtj ee
t
(2.109)
)(2 00tje
)()(c o s 000t (2.110)
)()(s in 000 jt (2.111)
2.2.6.4 Fourier transforms of power signals
Fig,2.42 Sinusoidal functions and their spectra
3,The Signum Function
The signum function,denoted by sgn(t),
is defined as
If
then
Suppose we differentiate the signum
function,Its derivative is 2δ(t):
2.2.6.4 Fourier transforms of power signals
0,1
0,0
0,1
)s g n (
t
t
t
t
(2.112)
)()(?Xtx?
)()( Xjdt tdx?
)(2)s g n ( ttdtd
The transform of
where
Thus the transform of sgn(t) is
The transform pair:
2.2.6.4 Fourier transforms of power signals
)sgn( tdtd
)()s g n ( Xjtdtd?
2)(2)( tFXj
jX
2)(?
jt
2)s g n (? (2.113)
2.2.6.4 Fourier transforms of power signals
Fig,2.43 sgn(t) and its spectra
5,Periodic functions
Periodic functions can be represented as
a sum of complex exponentials; because
we can transform complex exponentials by
means of Eq.(2.103),we should be able to
represent a periodic function using the
Fourier integral,Assuming x(t) is periodic of
period T,then
2.2.6.4 Fourier transforms of power signals
n
tjn
n eCtx
0)(?
The Fourier transform of x(t),
where
2.2.6.4 Fourier transforms of power signals
n
n
n
tjn
n
n
tjn
n
nC
eFC
eCF
txFX
)(2
][
][
)]([)(
0
0
0
(2.117)
dtetxTC tjn
T
Tn
0)(1 2
2
Example 12:
A special kind of periodic function is the unit
impulse train shown in Figure 2.47,This
function is useful in applications involving
sampling of time waveforms,
2.2.6.4 Fourier transforms of power signals
Fig,2.47 Periodic impulse sequence
Solution,The function has the form:
Expand it in a Fourier series as
where
the Fourier series representation of the unit
impulse train is
2.2.6.4 Fourier transforms of power signals
n
kTttx )()(?
(2.118)
tjnn eCtx 0)(?
TdtetTdtetxTC tjn
T
T
tjn
n
1)(1)(1 00 2
2
n
tjne
Ttx 0
1)(? (2.119)
Take the Fourier transform on both sides of
(2.119),
That is,
2.2.6.4 Fourier transforms of power signals
]1[)( 0
n
tjne
TFX
n T
nTX 2),(2)( 00
n n
nkTt )()( 00
(2.120)
Periodic impulse sequence and its frequency spectrum
A signal is said to be random if it depends on
probabilistic laws,Such signals:
– have unpredictable instantaneous values;
– cannot be desired by analytical time models;
– can be characterized by their statistical and
spectral properties,
An observed random signal must be seen as
a particular experimental realization of a set
(referred to as the ensemble) of similar
signals that can all be produced by the same
phenomenon or stochastic process,
2.2.7 Random signal description
2.2.7.1 Introduction
Sample function,the record of every
long-time observation for the time process
of a random signal,denoted by x(t)(Fig,
2.48),
2.2.7.1 Introduction
Fig,2.48 Random process and sample function
Sample record,a sample function in a
finite time interval,
Random process,the set (or ensemble)
of all sample functions under the same
experimental conditions,that is,
2.2.7.1 Introduction
),(,),(),()( 21 txtxtxtx i? (2.121)
The commonly used statistical characteristic
parameters:
– mean value,mean square value,variance,
probability density function,probability
distribution function,and power spectral density
function,etc,
– They are deduced in terms of set average,
They are calculated by averaging the simultaneous
observations for all sample functions at a certain time
instant ti(set average or ensemble average).
E.g.,mean value μx(t1) and mean square value ψ2x
(t1),
2.2.7.1 Introduction
N
i
iNx txNt
1
11 )(
1lim)(? (2.122)
N
i
iNx txNt
1
1
2
1
2 )(1lim)(?
(2.123)
A random process is said to be stationary if
all its statistical properties are time-invariant.
Strictly speaking,if the n-dimensional
distribution function of a random process,
{x(t)},for the n values t1,t2,…,t n of time and
any real number,satisfies the following
relationship,
{x(t)} is called a stationary random
process or stationary process,
Random processes that do not satisfy Eq,
(2.124) are called nonstationary processes,
2.2.7.1 Introduction
,2,1
),,,,;,,,(),,;,,( 21212121
n
tttxxxFtttxxxF nnnnnn
(2.124)
Examples of stationary processes in our
daily life:
– variations of voltage due to thermal noise,
– toss of ship,
– errors in measuring distances of moving
objects,
– vibrations at a certain location in geological
prospecting,
– voltage fluctuations in electrical light power
lines,
– all noises and interferences,
2.2.7.1 Introduction
A stationary random process is said to be
ergodic if the statistical characteristics of
the time average of its any one sample are
identical to those of the set average of the
process,
Many stationary random processes
encountered in engineering are of ergodism,
2.2.7.1 Introduction
1,Mean value,mean square value
and variance
For an ergodic process x(t),its mean
value is defined as
where
E[x],mathematical expectation of the
variable x,
x(t),sample function;
T,time duration for observation,
2.2.7.2 characteristic parameters for random process
dttxTxE TTx )(1lim][ 0? (2.125)
The variance of a random signal σ2x
Relationship between μx,σ2x and ψ2x,
The square root of ψ2x is called the root of
mean square value xrms.
When μx=0,then
2.2.7.2 characteristic parameters for random process
dttxT XTTx 202 ])([1lim (2.127)
222 xxx (2.128)
22 xx
rm sx x
Estimate formula for mean value,variance
and mean square value,
2.2.7.2 characteristic parameters for random process
dttxT Tx )(1? 0?
dttxT Tx )(1 202?
dttxT xTx 202 ]?)([1
(2.129)
(2.130)
(2.131)
2,Probability density function and
probability distribution function
In the observation time T,the total sum
of the time durations when the
instantaneous value of a random signal x(t)
falls into the interval (x,x+Δx) is
2.2.7.2 characteristic parameters for random process
N
i
inx ttttT
1
21?
(2.132)
Fig,2.49 Physical interpretation of probability density function
Probability of the random signal x(t):
Probability density function:
– Gaussian process or normal process:
The probability density function of a random variable
is of the form of a Gaussian function,
2.2.7.2 characteristic parameters for random process
T
TxxtxxP x
T lim])([
(2.133)
x
xxtxxPxp
x?
])([lim)(
0
x
T
T x
T
x?
0
lim
(2.134)
(2.135)
xxxp
x
x
x
]2 )(e x p [21)( 2
2
(2.136)
Probability distribution function,
where Tx’ is the total time when x(t) is lower
than or equal to x,
2.2.7.2 characteristic parameters for random process
T
TxtxPxP x /lim])([)(
(2.137)
Fig,2.50 (a) The probability density function and (b) the probability
distribution function of a normal process
many random phenomena obey or
approximately obey a normal distribution,
– Examples:
heights of the male adults in a certain geological
region,
errors in measuring the lengths of components,
heights of ocean waves,
electrical currents and voltages due to thermal noises
in semiconductor components,etc,
Probability density function can be used to
identify different random processes.
2.2.7.2 characteristic parameters for random process
2.2.7.2 characteristic parameters for random process
Fig,2.51 Graphs of probability functions of four random signals
3,Correlation analysis
1) Definition of correlation
2.2.7.2 characteristic parameters for random process
Fig,2.52 Different situations of correlation for variables x and y
The classical method for evaluating the
degree of linear correlation between two
random variables is to calculate their
covariance σxy and correlation coefficient
ρxy,
where E is the mathematical expectation,
where σx and σy are the standard deviations
of x and y respectively,
2.2.7.2 characteristic parameters for random process
N
i
yixiN
yxxy
yx
N
yxE
1
1lim
(2.142)
11 xy
yx
xy
xy
(2.143)
The variances σ2x and σ2y of x and y are
Using Cauchy-Schwarz inequality
we get |ρxy|≤1.
When ρxy =1,all data points will be on a line:
x and y are linearly correlated,
When ρxy =0,x and y are totally uncorrelated.
2.2.7.2 characteristic parameters for random process
22 xx xE (2.144)
22 yy xE (2.145)
222 yxyx yExEyxE
(2.146)
)( xy xmy
2) Crosscorrelation function and
autocorrelation function
A covariance coefficient Cxy(τ) for time
variables x and y is defined as:
where
Rxy(τ) is called the crosscorrelation function
of x(t) and y(t),where τ is the time-shift,
2.2.7.2 characteristic parameters for random process
yxxy
T
yx
T
yxxy
R
dttytx
T
tytxEC
0
1
l i m (2.147)
TTxy dttytxTR 01lim
(2.148)
When y(t)≡x(t),the autocovariance
function,Cx(τ) is:
where
Rx(τ) is called the autocerrelation
function of x(t),
2.2.7.2 characteristic parameters for random process
2
0
1lim
xx
T
xxTx
R
dttxtx
T
C
(2.149)
TTx dttxtxTR 01lim (2.150)
Properties:
a.
b.
A crosscorrelation function takes on its
maxinmum value usually at τ≠0,
2.2.7.2 characteristic parameters for random process
xx RR
)()( yxxy RR?
yxxy RR
(2.151)
(2.152)
222m a x0 xxxxx RR (2.153)
c.
d.
e.
correlation coefficient function,
and |ρxy|≤1 holds for all values of τ,
2.2.7.2 characteristic parameters for random process
2222 xxxxx R (2.154)
yxyxxyyxyx R (2.155)
2xxR (2.156)
yxxyR (2.157)
00 yxxy RRR
(2.158)
22 00
00
yyxx
yxxy
yx
xy
xy
RR
R
CC
C
(2.159)
f,The autocorrelation function of a periodic
function is still a periodic one,and has the
same frequency as that of the original periodic
function but contains no information on the
phase,
Crosscorrelation function for two signals
having same periodic component of a certain
frequency will contain the periodic component
of this frequency and never converges even if
τ→∞.
Two signals having periodic components
of different frequencies are not correlated.
2.2.7.2 characteristic parameters for random process
2.2.7.2 characteristic parameters for random process
Fig,2.53 (a) Autocorrelation function
(b) Corsscorrelation function
Example 1:
Find the autocorrelation function of the
sinusoidal function x(t)=Asin(ωt+υ),
Solution:
where
2.2.7.2 characteristic parameters for random process
0
0
0
0
])(s i n [)s i n (
1
)()(
1
lim)(
T
T
T
x
dtttA
T
dttxtx
T
R
2
0?T
Letting ωt+υ=θ,then dt=dθ/ω,so that
Rx(τ) keeps the information on both
amplitude and frequency of the original
signal,but loses the phase information,
2.2.7.2 characteristic parameters for random process
c o s2)s i n (s i n2)(
22
0
2 A
dAR x
Autocorrelation function can be used to
detect periodic components embedded in a
random process,
2.2.7.2 characteristic parameters for random process
Fig,2.54 (a) A harmonic signal embedded in a random noise
(b) its autocorrelation function
Example 2:
Two periodic signals
where θ — initial phase angle of x(t)
φ — the phase difference of x(t) and
y(t)
Find their crosscorrelation function Rxy(τ).
2.2.7.2 characteristic parameters for random process
)s in ()( tAtx
)s in ()( tBty
Solution:
The result shows that for two periodic
signals with the same frequency,their
crosscorrelation function maintains their
common frequency ω,their respective
amplitude A and B,and their phase
difference υ,
2.2.7.2 characteristic parameters for random process
)c o s (
2
1
))(s i n ()s i n (
1
)()(
1
lim)(
0
0
AB
dttBtA
T
dttytx
T
R
oT
o
T
T
xy
Estimates of an autocorrelation function
and a crosscorrelation function
2.2.7.2 characteristic parameters for random process
Tx dttxtxTR 01
Txy dttytxTR 01
(2.160)
(2.161)
3) Applications of correlation functions
a,Identification of different signals
2.2.7.2 characteristic parameters for random process
Fig,2.55 Autocorrelation functions of several typical signals
b,Velocity and displacement measurements
2.2.7.2 characteristic parameters for random process
Fig,2.56 Sound propagation distance measurement using correlation
– The task of the correlator:
adjust the time τ,τ=T,the delayed signal μ1(t-τ) is
then equal to the signal μ2 =μ1(t-τ)
– The correlator must make the mean square of
difference of the two signals a minimum:
2.2.7.2 characteristic parameters for random process
Fig,2.58 Fluid flow measurement
M i nTttEttEE 2112212 )()())()(( (2.164)
– For a steady-state signal,ideally
when τ=T,the correlation function Rμ1μ2(τ=T)
has the maximum,The correlator finds the
time τmax,T= τmax,By determining the
distance s between the two transducers,the
velocity can be calculated as v=s/T.
2.2.7.2 characteristic parameters for random process
c o n s tTtEtE )()( 2121
)()()( 2111 TRTttE
(2.165)
(2.166)
4) Power spectrum analysis
a,Auto-power spectral density
Its inverse Fourier transform is then
Sx(f),the auto-power spectral density function,
or simply the auto-power density spectrum of
x(t),
Eqs,(2.167) and (2.168) are called the
Wiener-Khintchine equations,
2.2.7.2 characteristic parameters for random process
deRfS fjxx 2 (2.167)
dfefSR fjxx 2 (2.168)
)()( fSR xFT
I F T
x
When τ=0,
Sx(f) represents the frequency distribution of
the random signal power,and it is therefore also
called the power spectrum.
2.2.7.2 characteristic parameters for random process
Fig,2.59 One-sided power spectrum and two-sided power spectrum
dffSdttxTR xT T
Tx
22
2
1l i m0 (2.169)
b,Parseval’s theorem
According to the frequency convolution
theorem,
and
letting k=0,thus
Again letting h(t)=x(t),we obtain
2.2.7.2 characteristic parameters for random process
)(*)()()( fHfXthtx?
dffkHfXdtethtx ktj )()()()( 2?
dffHfXdtthtx )()()()(
dffXfXdttx )()()(2
Since x(t) is a real function,then X(-f)=X*(f),
that is,X(-f) is the conjugate of X(f),thus
Eq,(2.170) is known as the Parseval’s theorem,
|X(f)|2 is often referred to as the energy-density
spectrum of signal x(t),and Eq,(2.170) is also
called the signal energy equality,
2.2.7.2 characteristic parameters for random process
dffXdffXfXdttx 2*2 (2.170)
The average power of signal over the whole time
axis:
We often use one-sided power spectrum,
2.2.7.2 characteristic parameters for random process
dffxTdttxTP T
T
TT
22
2
2 1lim1lim
(2.171)
21lim fXTfS Tx (2.172)
0 dffGdffSP xx (2.173)
0)(2)( ffSfG xx (2.174)
c,Cross-power spectral density function
Sxy(f) is also known as the cross-power density
spectrum or cross-power spectrum of x(t) and
y(t),
Fourier transform pair:
The inverse Fourier transform of Sxy(f),
The cross-power of x(t) and y(t),
2.2.7.2 characteristic parameters for random process
dfeRfS ftjxyxy 2 (2.175)
)()( fSR xyFTI F Txy
(2.176)
dfefSR fjxyxy 2
(2.177)
dffXfY
T
dttytx
T
P
T
T
TT
*
2
2
1
lim
1
lim
(2.178)
So
where
One-sided cross-power spectrum Gxy(f)
2.2.7.2 characteristic parameters for random process
fXfYTfS Txy *1lim
(2.178)
fSfSfS yxxyxy * (2.179)
)()(1lim * fYfXTfS txy
02 ffSfG xyxy
(2.180)
where Cxy(f) = cospectrum
Qxy(f) = quad spectrum
2.2.7.2 characteristic parameters for random process
fjQfCfG xyxyxy
(2.181)
fC
fQ
ar c t gf
fQfCfG
efGfG
xy
xy
xy
xyxyxy
fj
xyxy
xy
22
(2.182)
4,Estimates of auto-spectrum and
cross-spectrum
Estimate of auto-spectrum,
Estimate of cross-spectrum,
2.2.7.2 characteristic parameters for random process
21? fXTfS x?
(2.183)
fYfXTfS xy *1?
(2.184)
fXfYTfS yx *1? (2.185)
