Chapter 6 Spectrum Estimation 6.1 Time and Frequency Domain Analysis

6.2 Fourier Transform in Discrete Form

6.3 Spectrum Estimator

6.4 Practical Considerations

R. J. ChangDepartment of Mechanical EngineeringNCKU

6.1 Time and Frequency Domain Analysis(1)1.Signal Representation (1)Time domain Transient Periodic function By oscilloscope By eq. Physical signal Mathematical signal

6.1 Time and Frequency Domain Analysis(2)(2) Frequency domainContinuous spectrum for transient signalBy spectrum analyzerDiscrete spectrum for periodic signalBy eq.

Amplitude

6.1 Time and Frequency Domain Analysis(3) (2) Stochastic stationary signal Modern Fourier analysis by Khintchine and Wiener Generalized harmonic signal- power spectrum

(3) Fourier transform pair 2. Fourier Analysis (1) Deterministic signal Classical Fouriers analysis Finite transient signal- continuous energy spectrum Periodic signal- discrete line spectrumEnergy conservation:

6.1 Time and Frequency Domain Analysis(4)3. Evolution of Fourier Analysis(2) Discrete data analysis Fourier Transform Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT)(1) Mathematical analysis Fourier Series Fourier Transform Generalized Fourier Analysis

6.1 Time and Frequency Domain Analysis(5) Uncertainty Principle Finite cutoff in time (T*) and frequency domain (B*) cannot violate the equation:

Meaning: Except for zero signal, a signal cannot be both band-limited and time-limited simultaneously.

Note: Band-limited signal- Smooth signal Band-unlimited signal- Rough signal4. Fundamental principle

6.1 Time and Frequency Domain Analysis(6)5. Continuous and Discrete Spectral Analysis

6.2 Fourier Transform in Discrete Form(1)

1. Discrete Fourier Transform (1) Discrete formulation (a) Discrete time Discrete in time domain Periodic in freq. domain

6.2 Fourier Transform in Discrete Form(2)(b) Discrete frequency Discrete in freq. domain Periodic in time domainNote: 1. k is a time-domain index, k=0,1,2,N-1 n is a frequency-domain index, n=0,1,2,,N-1 2. For DFT, one has discrete and periodic in both time and frequency domain.

6.2 Fourier Transform in Discrete Form(3)(2) Matrix Formulation

6.2 Fourier Transform in Discrete Form(4) (3) Important Properties(a) Symmetric matrix

(b) Inversion

(c) Computational requirement For N-sample, one requires N2- complex multiplication

6.2 Fourier Transform in Discrete Form(5)2. Discretization and Reconstruction Discrete signal processing:Continuous time signal Discrete time signal Discrete frequency signal Continuous frequency signal

N=9

6.2 Fourier Transform in Discrete Form(6)Ex:

6.2 Fourier Transform in Discrete Form(7)3. Fast Fourier Transform By Cooley and Turkey in 1965 FFT algorithm is a recursive form based on thefeatures of weighting W to reduce computational timeby machine.

6.2 Fourier Transform in Discrete Form(8)(1) Weighting of phase Wn

6.2 Fourier Transform in Discrete Form(9)(2) Factorization The number of complex multiply-add operations for the DFT becomes AB(A+B).

6.2 Fourier Transform in Discrete Form(10)(3) Speed ratio (a) For p factors in factorization (b) For power of 2 (smallest prime factor)

6.2 Fourier Transform in Discrete Form(11)

6.3 Spectrum Estimator(1)Existence of Spectrum Representation Wold Decomposition Theorem: For wide sense stationary process

6.3 Spectrum Estimator(2)2. Continuous Analysis (1) Estimation procedure (a) Average power(b) Power spectral density

6.3 Spectrum Estimator(3)(c) General procedure

6.3 Spectrum Estimator(4)(2) Estimator error (a) Biased error

(b) Random errorNote: Be is optimally traded off for minimum error

6.3 Spectrum Estimator(5)(3) Various errors

6.3 Spectrum Estimator(6)3. Discrete estimation (1) General procedure

PhysicalSignalSource

Anti-AliasingFilter

S/H andData Recording

Wild-point Editing

Trend Removal

StationaryQualification

Window Selection

Discrete Esitimator

Trend model

Window function

% of C.I.

Run test

Data smoothing

MSB LSB

* Random error* Biased error* Discrete resolution* Amplitude error

Display

6.3 Spectrum Estimator(7)(2) Estimation algorithms (a) Narrow-band filtering method Narrow-band filter with Be:

Filtered output data for frequency fk ( fk =k/Nh )

6.3 Spectrum Estimator(8)(b)Indirect method (Blackman-Turkey method)

6.3 Spectrum Estimator(9)Discrete approximationAutocorrelation function is estimated by(Unbiased)(Biased)or

6.3 Spectrum Estimator(10)(c) Direct method (DFT method)Resolution is

6.3 Spectrum Estimator(11)(3) Random error

: Number of autocorrelation lags

: Number of complex Fourier components

: Bandwidth of narrow-band filter : data record length : Total number of data points

6.4 Practical Considerations(1)1. Error Types and Sources

Window selection

Bias

Amplitude

Resolution

Source

Discretization

Fault data (Wild point)

Type

Estimation algorithm

Random

6.4 Practical Considerations(2)2. Window selection (1) Leakage problem (a)Infinite-time signal

6.4 Practical Considerations(3)(b) Finite-time signalLeakage problem can be reduced by using proper window function

6.4 Practical Considerations(4)(2) Window functions (a)Rectangular window

6.4 Practical Considerations(5)(b) Hanning window

6.4 Practical Considerations(6) (c) Hamming window

6.4 Practical Considerations(7)3. Data Partition and Average