Discrete Fourier Transform and Applied Spectral Analysis

1. Discrete Fourier Transform and Applied Spectral Analysis

2. у у (1.11) Correlation Functions

3. Stationary Random processes Cross Correlation Functions From obvious expression: It could be derived:

4. Ergodic Random Processes Power Spectral Densities of the Ergodic Random Processes

5. Wiener-Khinchine Theorem

6. Discrete Fourier transform Let - periodicity interval of continuous function -the lowest frequency , then: ,where m - integer number. Fourier series: (1) Coefficients: (2) Continuous Fourier transform: Direct: (3) (4) Inverse:

7. The lowest frequency and increment of the angular frequency Direct Discrete Fourier transform (DFT): Current frequency “k” is the time domain index, “n” is the frequency domain index Relative frequency: Inverse Discrete Fourier transform (IDFT):

9. (10) Direct DFT: Inverse DFT : Properties of DFT: 1) symmetry property 2) skew-symmetry of the algorithm (11) (12)

10. Computation of DFT and IDFT. When k=N, Presentation of DFT in the form of IDFT:

11. FFT and IFFTAlgorithms If (1) Then (2) or (3) z(k) and y(k) are samples of the process in the time domain.

12. Then (1) can be rewritten as: (4) or N dimension In order to save memory cells, it is possible to save Y(n) and Z(n) in the same register: (5) 1 array of N dimension is transformed in (3) where (6)

13. 1 1 - - - - Taking into account: We receive: (7) “Butterfly” graph (algorithm)

14. 1 + - Example: transformation of two 4-points DFT in one 8-points DFT Example: Transformation of 2- and 4-pointed ДПФ in the 8-pointed one Common representation in graph form:

15. Transition from N/4-points to N/2-points DFT: Small Turn Factor

16. Transition from 2-points to 4-points DFT: Adjacency matrix

18. Power Spectral Density estimation Power or variance: Power Spectral Density:

19. Periodogram: Convolution in the Frequency Domain: -time domain: -frequency domain:

20. x(t) 2 1 t T -T Gibbs Phenomenon |X (T)| =|sinc(T)| X(T)=sinc(T)

21. Frequency Domain Representation of the Rectangular Window in the Logarithmic Scale

23. (3) GeneralizedHann –Hemming Window Blackman Window (4) (4) Flat Top Window (5)

25. (6) Kaiser Window

26. Spectral Resolution: (7) Bias of the Periodogram: (8) Variance of the Periodogram: (9)

27. Parseval's theorem: Estimations of cross spectra (10) (11) (12) (13) (14) The square of coherence coefficient: (15) (16) (17)

28. Error variances (18) Stochastic process generation. Rice-Pearson decomposition. (19) (20) (21)

29. References • MATLAB: “Signal Processing Toolbox. User’s Guide”.Math Works, 2011.-363 p. • В.П. Бабак, В.С. Хандецький, Е. Шрюфер. Обробка сигналів. Київ, «Либідь», 1999.- 495с.