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. Chang Department of Mechanical Engineering NCKU

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

§ 6.1 Time and Frequency Domain Analysis(2) (2) Frequency domain Continuous spectrum for transient signal By spectrum analyzer Discrete spectrum for periodic signal By eq.

§ 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 Fourier’s analysis Finite transient signal- continuous energy spectrum Periodic signal- discrete line spectrum Energy 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 signal 4. 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 domain Note: 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 N 2 - 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

§ 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 the features of weighting W to reduce computational time by machine.

§ 6.2 Fourier Transform in Discrete Form(8) (1) Weighting of phase W n

§ 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) 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 error Note: B e 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

§ 6.3 Spectrum Estimator(7) (2) Estimation algorithms (a) Narrow-band filtering method Narrow-band filter with B e : Filtered output data for frequency f k ( f k =k/Nh )

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

§ 6.3 Spectrum Estimator(9) Discrete approximation Autocorrelation 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 Algorithm D.O.FStandard error Correlation Method Fourier Method Narrow-band Filter Method : 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

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

§ 6.4 Practical Considerations(3) (b) Finite-time signal Leakage 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