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

2. 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

3. 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)

4. 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

5. 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

6. 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:

7. 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 = i i i i i i
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.

8. 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 = i i i i i i
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)

9. 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’

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

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

12. Intro to Signal Processing Toolbox
Cautions for Fourier Space - Aliasing

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

14. 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:

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

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

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

18. Intro to Signal Processing Toolbox Signal Processing Toolbox
Example: Windowing

19. Intro to Signal Processing Toolbox Signal Processing Toolbox
Example: Windowing

20. 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.

21. 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