MAE 591 RANDOM DATA
C. W. Lee
Hilbert Transform1 While the Fourier transform moves the independent variable of a signal from the time to the frequency domain or vice versa, the Hilbert transform leaves the signal in the same domain. The Hilbert transform
Tutorial paper, July 98
DIRECTIONAL SPECTRUM ANALYSIS AND ITS APPLICATIONS TO ROTATING MACHINE DIAGNOSIS
Chong-Won Lee, and Yun-Sik Han Center for Noise and Vibration Control(NOVIC), Department of Mechanical Engineering KAIST, Science Town, Taejon, 305-70
2
(t )
1 0 -1 -2 -2 -1 0 1 2
t, second
2 w(t) 1 0 -1 -2 -2 1 Im[W (f)] 0.5 0 -0.5 -1 -10 -8 -6 -4 -2 0 2 4 6 8 f , Hz 10 -1 0 1 2
t, second
Figure 1. Haar scaling function and Haar wavelet with its Fourier transform
MAE 591 RANDOM DATA
C. W. Lee
Fig.2 De
Time-frequency Map Short-time Fourier Transform (STFT) : Spectrogram, Sonogram A short data window centered at time t with time duration T has the frequency bandwidth of 1 approximately B . Thus STFT features that all spectral estimates have the same(cons
Example: Identification of a cuber in nonlinear system shown below.
x (t )
Cuber
A( f )
y3 (t)
The input x(t ) is normal distributed with zero mean and x = 2.7 as shown in Fig.A(a). Now we consider three different cases for the linear system A( f ) .
i)
i
Nonlinear System Analysis and Identification1 1. SISO cubic nonlinear system subject to a Gaussian stationary random input 1.1 Output moments through a squarer with sign The general problem to be analyzed for wave forces on small-diameter fixed structures
Energy Source Identification
m1 x1 (t ) x 2 (t ) m2
w1 w2
H 11 ( f ) H 22 ( f )
u1 u2
H1y ( f ) H 2y ( f )
v1
n(t )
v2
y (t )
mq x q (t ) Measured Inputs
wq
H qq ( f )
uq
H qy ( f )
vq Measured Output
Transducers
Sources
Input/Output System
Consider the m
Conditioned Spectral Analysis Again, for simplicity, consider the two input/single output system, where the two inputs x1 (t ) and x 2 (t ) are not necessarily uncorrelated. One of the inputs, say x 2 (t ) , can be decomposed into the sum of two uncorrela
Multiple Input / Single Output Relationships Consider the multiple input-multiple output system shown below. Here xi (t ), i = 1,2, , q are assumed to be the stationary inputs with zero mean which are uncorrelated with the uncorrelated extraneous output n
MAE591 RANDOM DATA
Single Input-Multiple Output Relations1 Consider the single input-multiple output system shown below. Here x(t ) is assumed to be the stationary input with zero mean which is uncorrelated with the uncorrelated extraneous output noises w
Coherence, Measurement Noise and System Identification: SISO 1. Coherence Function(COH)1 Assume S xx ( f ) and S yy ( f ) are both different from zero, meaning that both spectra are
rich, and do not contain delta functions, meaning that both spectra have
MAE591 RANDOM DATA
Single-input/single-output relationships (Random Vibration) Response of a linear time-invariant discrete dynamic system :
x (t )
h ( ) H( f )
h( ) x (t )d = h( ) x (t )d
0
y (t )
y (t ) =
For stationary processes: Mean value
y = E[ y
MAE 591 RANDOM DATA
Overlapping of Hanning window: Effective power weighting1
n 1 1 2 t kT 2 2 weff ;n (t ) = w 2 t where w (t ) = 1 cos 4 T n k =0 4t 3 1 for n = 2 4 + 4 cos T 2 weff ;n (t ) = 3n for n = 3,4,5,. 8
2
percentage overlap
advantage
disadvan
MAE 591 RANDOM DATA
Time Windows Windowing is multiplying the time series by a data window, which is equivalent to applying a convolution operation to the raw Fourier transform. The purpose of windowing or tapering is to suppress large side lobes in the e
MAE 591 RANDOM DATA
Zoom Transform Let y (t ) be the bandpass filtered record of the original data record x (t ) , i.e. B B x (t ) ; f 0 f f 0 + y (t ) = 2 2 0 ; otherwise j 2 f 1t and v (t ) = y (t )e , which is the modulated record of y (t ) with the mo
MAE 591 RANDOM DATA
Decimation1 1. Downward decimation: Downward decimation discards unwanted or redundant information with a corresponding compaction of the data. Filtering is usually performed in conjunction with decimation, and usually a low pass filte
MAE 591 RANDOM DATA
Digital Filter Analog linear filter can be expressed as:
y ( t ) = h ( ) x ( t ) d
where x(t) and y(t) are the filter input and output, and h() is the weighting function of filter. And the frequency response of the filter becomes
H ( f
MAE591 Random Data
Correlation Function Estimate
1. Correlation (Covariance) and Convolution : Direct Computation Auto-correlation defines the degree to which a function is correlated with itself as a function of time delay, while cross-correlation define
MAE591 Random Data
FFT
It is convenient to start with n = 0 to define the finite Fourier transform of sampled data, i.e.
X ( f , T ) = h x( n ) exp( j 2fnh )
n =0 N 1
At the discrete frequencies of f k = transform (DFT) is often defined as1
Xk =
k k = , k
MAE 591 RANDOM DATA
C. W. Lee
Probability Distribution Function
x x with z = 0 and 2 = 1 . z x
z
1. Normal distribution for standardized variable z =
p( z ) = 100 percentage point :
1 2
e
z2 2
, P( z ) =
1 2
e
u2 2
du
P ( z ) = p( z )dz = Pr[ z z ] = 1
MAE 591 RANDOM DATA
Data Acquisition and Processing 1 1. Data acquisition: Type of data 1. Continuous (analog) data 2. Originally digital data: stock market, labour statistics, annual rainfall - all physical quantities Problems associated with continuous
MAE 591 RANDOM DATA
Spectral Density Functions i) Power in electric circuits: P = Ri 2 1T 1T 2 Case of A.C. : Pav = Ri 2 dt = R( I a sin 2 ft ) dt = RI 2 T0 T0 Ia 1 I= ,f = T 2 1T Arbitrary current : Pav = lim Ri 2 dt = RI 2 T T 0 1T I 2 = lim i 2 dt T T
MAE 591 RANDOM DATA
Stationary Process 1. Stationary in the strict sense: strongly stationary A process is stationary if its statistics are not affected by a shift at time scale, i.e. x(t ), x(t + ), > 0 , have the same statistics for every, e.g. p( x(t )
MAE 591 RANDOM DATA
Probability Density Function Estimate and Errors Consider N data values cfw_xn , n = 1,2 ,3, . , N , from a transformed record x ( t ) that is stationary with x = 0 . The probability density function of x ( t ) can be estimated by Nx w
MAE 591 RANDOM DATA
Normal (Gauss) Distribution The normal probability density function is expressed by
p( x ) = (x )2 1 x exp 2 2x 2x
The normal probability distribution function is
P ( x) = 1 2x ( ) 2 exp 22x d x
x
N-dimensional normal distribution:
MAE 591 RANDOM DATA
Central Limit Theorem Let x i ( t ), i = 1,2, . , N , be N statistically independent random variables with respective probability densities p i ( x ) . Let i and i2 be the mean value and variance of each random variable x i ( t ) . Con
MAE 591 RANDOM DATA
Stochastic Processes A stochastic process x ( i ) ( t ) is statistically determined if the N-th order probability distribution function P ( x1 ( t1 ), x 2 ( t2 ), , x N ( t N ) ) = Prcfw_x ( t1 ) x1 ( t1 ) x ( t2 ) x 2 ( t2 ) x ( t N )
Probability Definition: event: outcomes of experiments and their collections set: a collection of objects; A, B elements: the objects of a set; ai , bi , i = 1, 2, . k subset: a part of a larger set space: the largest set; S empty set or null set: a set o
Impulse response and FRF Delta Function:
t = 0 Properties: ( t ) = , 0 t 0 Fourier transform:
( t ) dt = 1 ,
x(t ) (t t 0 )dt = x(t 0 )
(t ) e j 2ft dt = 1 for all f
Inverse Fourier transform :
e j 2ft df = (t )
Impulse Response Function of a linear s