Correlation, discrete Fourier transforms and the power spectral density

Transcription

1 Correlation, discrete Fourier transforms and the power spectral density visuals to accompany lectures, notes and m-files by Tak Igusa Department of Civil Engineering Johns Hopkins University December 3, 8

2 Effect of filtering on correlation () = () ( ) g t () original function or data of duration g% t = g t = original function with reversed time Ggg ( τ ) = g g% = covariance of g in terms of time lag T g t = g h = filtered data ( h = filter) f G ( ) ( ) g covaria f g τ = g f f g% f= Ggg h h% = nce of g f T T τ The covariance of the filtered data is the convolution of the covariance of the original data with h h h % h % is triangular for the box filter and Gaussian for the Gaussian filter

3 Effect of filtering on the correlation.5 Uncorrelated artificial data y Correlation for k =, is -. Correlation for k = 3, is y j+ y j y j y j 3

4 Correlation of the uncorrelated data Correlation for k =, is -. 3 Correlation for k =, is Correlation for k = 3, is.7 3 y j+k y j+k y j+k.5.5 y j.5.5 y j.5.5 Correlation for k = 4, is Correlation for k = 5, is.39 3 y j Correlation for k = 6, is.56 3 y j+k y j+k y j+k y j y j y j correlation function correlation.5 spike at time lag = time lag t 4

5 Effect of filtering on the correlation.. Box-filtered data y time Correlation for k =, is.84. Correlation for k = 3, is.56. y j+ y j y j y j 5

6 Correlation of the filtered data. Correlation for k =, is.84. Correlation for k =, is.67. Correlation for k = 3, is.56 y j+k y j+k y j+k.9.. y j.9.. y j y j+k. Correlation for k = 4, is y j+k Correlation for k = 5, is y j+k..9 y j Correlation for k = 6, is y j y j y j correlation.5 correlation function triangular shape agrees with theory time lag t 6

7 Alternate interpreation of correlation: image alignment scale v scale v scale v 3 y i+ (x) y i (x) 7

8 Image matching scale v scale v scale v 3 y i+ (x) y i (x) product y i+ (x) y i (x) 8

9 Image matching definition of correlation ρ = Area of product ( Area )( Area ) i i+ y i+ (x) y i (x) product y i+ (x) y i (x) y i+ ( x) yi ( x) 9

10 Image matching definition of correlation ρ y ( x) y ( x) i i+ = yi ( x) ds y ( ) i+ dx x dx y i+ (x) y i (x) product y i+ (x) y i (x) y i+ ( x) yi ( x)

11 Image matching scale v scale v scale v 3 y i+ (x) y i (x) perfectly aligned highest correlation product y i+ (x) y i (x)

12 Image matching scale v scale v scale v 3 y i+ (x) y i (x) product y i+ (x) y i (x) correlation, ρ v v v 3 slopes

13 Effect of noise on image matching signal, y (x) noise, e (x) data, y (x) + e(x) correlation, ρ maximum correlation y ( x) ds ρ ( ) ds + ( x) y x e ds slopes 3

14 Box filtering viewed as a weighted average data filter weighted data average filter weighted data average filtered data 4

15 Gaussian filtering viewed as a weighted average data filter weighted data average filter weighted data average filtered data 5

16 Correlation of data in terms of image matching uncorrelated data data shifted data product shifted data product correlation = normalized area of the product spike at tau = 6

17 Correlation of data in terms of image matching box-filtered data data shifted data product shifted data product correlation = normalized area of the product triangular shape 7

18 Correlation of data in terms of image matching Gaussian-filtered data data shifted data product shifted data product correlation = normalized area of the product Gaussian shape 8

19 FFTintro.m a quick application of the DFT n = ; dt =.3; ts = (:n-)*dt; f = ; T = max(ts); df = /T; fs = (:n-)*df; y = cos(*pi*f*ts); Y = fft(y); Y = Y/n; % number of data % time increment, delta t % time % frequency ( Hz) % time interval % frequency increment % frequencies % cosine data % fft % DFT (mult by dt for FT) figure(), set(gcf,'name','ft demo') subplot(,,), plot( ts, y ) set(gca,'fontsize',) xlabel('time'), ylabel('y(t)'), title('original function') subplot(,,), plot( fs, real(y), 'b-', fs, imag(y), 'r-' ) set(gca,'fontsize',) xlabel('frequency'), ylabel('y(f)'), title('dft') 9

20 original function y(t) time DFT.5 Y(f) result from MATLAB fft: scale by /n to get the DFT scale by /(n*df) = dt to get the FT also need to change the frequency axis frequency

21 y(t) original function cos4πt = exp πi t + exp πi t ( ) ( ) Y(f) time DFT.5 Nyquist frequency (halfway) frequency

22 frequency Illustration of Fourier series convergence sine functions multiplied by Fourier coefficients Fourier coefficients sinc function frequency original function, g(t) Fourier amplitudes 3 4 frequency partial sums

23 frequency Illustration of Fourier series convergence sine functions multiplied by Fourier coefficients Fourier coefficients Gauss function frequency original function, g(t) Fourier amplitudes 3 4 frequency partial sums 3

24 frequency Illustration of Fourier series convergence sine functions multiplied by Fourier coefficients Fourier coefficients original function, g(t) Fourier amplitudes: Gauss fn, noise partial sums frequency 3 4 frequency 4

25 IFT.m, an interactive tool to explore the DFT G(f) f = Gauss = Gauss with σ =.8 4π ( ) ( πσ f ) G f frequency, f (Hz) 6 () g t t t = Gauss 5 Gauss πσ σ.8 4 g(t) time, t (sec) 5

26 Data from the AIRS instrument (on the NASA Aqua satellite) Relative Humidity (RH) in the tropics (5S-5N) RH (/-/3) longitude/ day

27 Relative Humidity (RH) in the tropics (5S-5N) 35 RH (/-/3) longitude/ 5 selected -dimensional time sequence of data day 7

28 Relative Humidity (RH) in the tropics (5S-5N, E) RH (minus avg) - time (days) original function 6 Fourier amplitude of the RH 5 FT of RH /45 /8 /9 /6 /4 days 8

29 Effect of a Gaussian window RH (minus avg) - time (days) Window type: Gaussian, sigma =., area =.5 6 Fourier amplitude of the RH 5 FT of RH /45 /8 /9 /6 /4 days 9

30 Box-filtered, zero-padded data, filter width: 9 days RH (minus avg) - FT of h time (days) Time-domain: Box filter, total width:. Fourier amplitude of the box filter FT of RH Fourier amplitude of the RH attenuation of the original Fourier amplitude due to filtering /45 /8 /9 /6 /4 days 3

31 Gaussian-filtered, zero-padded data, filter st dev: days RH (minus avg) - FT of h time (days) Time-domain: Gaussian filter, sigma:. Fourier amplitude of the Gaussian filter FT of RH Fourier amplitude of the RH attenuation of the original Fourier amplitude due to filtering /45 /8 /9 /6 /4 days 3

32 Data from Prof. Tony Dalrymple s wave tank.5 wave height, channel height wave height, channel (zoomed in) height

33 Autocorrelation The cosine shape is expected, and very little additional information can be found from these plots. Se we turn to the PSD. Channel, wave amplitude:.33 correlation correlation Channel, wave amplitude: time lag 33

34 Power spectral density psd, channel psd, channel integrate this peak to get the mean-square of the dominant /3 Hz wave higher harmonics of the dominant wave (at /3 Hz intervals) that indicate that the waves are periodic (at /3 Hz) but not purely sinusoidal psd - psd - random surface motion /3 Hz frequency 4 /3 Hz frequency 3/3 Hz 4/3 Hz 34

35 Data from AIRS (longitude versus DJF days, -5) RH OLR U PV V time (days)

36 Cross-correlation with respect to data at 3E D corr OLR, longitude fixed at D corr RH 35 D corr U longitude tau 35 D corr V 35 D corr PV longitude tau tau examine in more detail in the next slide 36

37 Cross-correlation of U wind with respect to data at 3E D Gaussian shape at large scales with additional small scale component correlation tau longitude 37

38 Relative humidity, DJF 5, near the equator and near the tropopause Atmospheric data 35 3 Raw data: The randomness difficult to describe 5 longitude time (days) 38

39 Relative humidity, DJF 5, near the equator and near the tropopause FFT of atmospheric data 8 6 Same data after D Fourier transform Some definitive patterns emerge.9.8 wave number (/longitude) frequency (/day) 39

40 Effect of windowing and sampling (DFT vs FT, see notes) 5 original function (t) original FT (f) windowed function (t) sampled & windowed function (t) inverse FT after sampling in frequency (t) FT after time window (f) FT after time window & sampling (f) sampling in frequency (f)