Laboratory in Oceanography: Data and Methods Intro to the Signal Processing Toolbox MAR599, Spring 2009 Miles A. Sundermeyer

Intro to Signal Processing Toolbox Basics of Fourier Transforms Suppose we have time or space series data ... wish to quantify information content of signal wish to separate periodic component from random component

Intro to Signal Processing Toolbox Basics of Fourier Transforms Fourier Transform (cont’d) Basic assumptions x(t) is one realization from an ensemble of realizations x(t) has a mean and correlation function, x(t) is stationary mean and correlation function are independent of t (i.e., “weakly” stationary) make ergodic assumption – can replace an ensemble average with average over time of single realization (in general, don’t have multiple realizations)

Intro to Signal Processing Toolbox Basics of Fourier Transforms Fourier Transform (cont’d) Define a finite Fourier transform as: Define “Power Spectrum” as: where * denotes the complex conjugate The power spectrum quantifies the amount of energy contained in different frequencies in the time series. The “theoretical” power spectrum has the property: where k denotes realizations within an ensemble

Intro to Signal Processing Toolbox Basics of Fourier Transforms Fourier Transform (cont’d) Problems with this: have discrete data (digitized) not infinite time series only have one realization In practice, we thus perform Fourier analysis on our single realization: By doing this, implicitly assume our finite interval time series is periodic. T ... T -T 2T

Intro to Signal Processing Toolbox Basics of Fourier Transforms Fourier Transform (cont’d) Matlab uses Fourier transform equivalent to continuous integral transform on infinite domain: Discrete transform on finite domain:

Intro to Signal Processing Toolbox Basics of Fourier Transforms Example: simple fft >> x = 1+cos(2*pi*[0:7]/8) >> X = fft(x); % forward fft >> xnew = ifft(X); % inverse fft >> [x' fft(x)' xnew'] ans = 2.0000 8.0000 2.0000 1.7071 4.0000 + 0.0000i 1.7071 1.0000 0.0000 - 0.0000i 1.0000 0.2929 0 - 0.0000i 0.2929 0 0 0 0.2929 0 + 0.0000i 0.2929 1.0000 0.0000 + 0.0000i 1.0000 1.7071 4.0000 - 0.0000i 1.7071 Note: Imaginary parts are all zero - no sine component First fft value is freq (k-1) = 0, cos(0) = 1, => fft = (npts) * (mean(x)) 2nd & 7th fft values are same & real, represent cosine variability with 8 points, i.e., freq of 2p/8. Amp of cosine variability in orig signal = 2*X2/N Other terms are zero since zero energy at other freqs.

Intro to Signal Processing Toolbox Basics of Fourier Transforms Example: simple fft (cont’d) Add a sine component and repeat >> x = 1 + cos(2*pi*[0:7]/8) -2*sin(4*pi*[0:7]/8) >> X = fft(x); % forward fft >> xnew = ifft(X); % inverse fft >> [x' fft(x)' xnew'] ans = 2.0000 8.0000 2.0000 -0.2929 4.0000 + 0.0000i -0.2929 1.0000 0.0000 - 8.0000i 1.0000 2.2929 -0.0000 - 0.0000i 2.2929 0.0000 -0.0000 0.0000 -1.7071 -0.0000 + 0.0000i -1.7071 1.0000 0.0000 + 8.0000i 1.0000 3.7071 4.0000 - 0.0000i 3.7071 Note: X3 = 8i, X7 = -8i ... Xn and XN+2-n are complex conjugates Imag parts of X2 and X7 => sine w/ freq 2*2p/N has amp 2*X3/8 = 2. In General, frequencies represented by fft are: 2*pi(k-1)/N, k = 0:(N/2) zero freq (mean), 2*pi*(1/N) (lowest) ... 2*pi*((N/2 - 1)/N) (highest = Nyquist freq)

Intro to Signal Processing Toolbox Frequency Spectra Example: Muddy Creek, Chatham, MA stage data – fft/spectrum via 4 methods: Harmonic analysis 1/N X*X Matlab’s ‘spectrum’ Matlab’s ‘periodogram’

Intro to Signal Processing Toolbox Frequency Spectra Variance Preserving Form Variance preserving form: f · Pxx plotted on a semilogx axis

Intro to Signal Processing Toolbox Cautions for Fourier Space – Gibbs Phenomenon

Intro to Signal Processing Toolbox Cautions for Fourier Space - Aliasing

Intro to Signal Processing Toolbox Cautions for Fourier Space - Aliasing signal freq Nyquist freq

Intro to Signal Processing Toolbox Signal Processing Toolbox Convolution and filters The convolution of two functions is defined as:                                             where ∗ denotes the convolution operation. In Fourier space, the convolution is the product of the Fourier transforms of the functions:

Intro to Signal Processing Toolbox Signal Processing Toolbox Convolution and filters (cont’d) Matlab’s ‘fdesign’ function for filter building

Intro to Signal Processing Toolbox Signal Processing Toolbox Example: Low-Pass Filter

Intro to Signal Processing Toolbox Signal Processing Toolbox Example: Low-Pass Filter (cont’d)

Intro to Signal Processing Toolbox Signal Processing Toolbox Example: Windowing

Intro to Signal Processing Toolbox Signal Processing Toolbox Example: Windowing

Intro to Signal Processing Toolbox Signal Processing Toolbox Spectral Estimators in Matlab Spectral analysis includes three types of spectral estimators — power spectral density (PSD), mean-square spectrum (MSS) and pseudo spectrum. Power spectral density (psd) measures power per unit of frequency and has power/frequency units. Mean-square (power) spectrum (msspectrum) measures power at a specific frequency. Pseudospectrum (pseudospectrum) returns a pseudo spectrum that does not have any units.

Intro to Signal Processing Toolbox Signal Processing Toolbox Useful Tidbits: fft, ifft - compute forward and inverse fft spectrum - for computing various types of spectra spectrum.welch - for computing windowed spectra butter - for computing Butterworth filters freqz - for computing Fourier representations of filters filter, filtfilt - for time domain filtering Some References: Bendat, J. S., and A. G. Piersol: Random Data: Analysis and Measurement Procedures (1st Ed. 1971) Priestly, M. B.: Spectral Analysis and Time Series. 1983.