Spectral Analysis AOE March 2011 Lowe 1

Announcements Lectures on both Monday, March 28 th, and Wednesday, March 30 th. – Fracture Testing – Aerodynamic Testing Prepare for the Spectral Analysis sessions for next week: anual/inst4/index.html 2

What is spectral analysis Seeks to answer the question: “What frequencies are present in a signal?” Gives quantitative information to answer this question: – The “power (or energy) spectral density” Power/energy: Amplitude squared ~V 2 Spectral: Refers to frequency (e.g. wave spectra) Spectral density: Population per unit frequency ~1/Hz Units of PSD: V 2 /Hz – The phase of each frequency component How much of the power is sine versus cosine 3

Spectral analysis/time analysis Given spectral analysis (power spectral density + phase), then we can reconstruct the signal at any and all frequencies: 4

Mathematics: Fourier Transforms The Fourier transform is a linear transform – Projects the signal onto the orthogonal functions, sine and cosine: Two functions are orthogonal if their inner product is zero: 5

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Fourier Transform We have chosen the functions of interest, now we design the transform: – The Fourier transform works by correlating the signals of interest to sines and cosines. – Since there are two orthogonal functions that will fully describe the periodic signal (why?), then a succinct representation is complex algebra. Note: 7

Complex Trigonometry 8 Note that time and frequency are called conjugate variables: one is the inverse of the other.

Fourier transform 9 Generally, the second moment, or ‘correlation’, of two periodic variables may be written as: Does this look familiar? A correlation among periodic signals is the inner product of those signals! The Fourier transform is a correlation of a signal with all sines and cosines:

Fourier transform 10

Conclusions from cos(t) 11

Properties of the Fourier transform 12

Digital signals Of course, we rarely are so lucky as to have an analytic function for our signal More often, we sample, a signal We can write the Fourier transform in a discrete manner (i.e., carry out the integration at discrete times/frequencies). The Discrete Fourier Transform is 13

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Multiply: 15

Raw Discrete Fourier Transform Results 16

FFT and PSD The Fast Fourier Transform is an algorithm used to compute the Discrete Fourier Transform based upon Beware of scaling: – There are many scalings out there for discrete Fourier Transforms – There is one easy way to solve this, though, compute the power spectral density and signal phase. 17

PSD Definitions and Signal Phase Double-sided spectrum: Single-sided spectrum: Phase: 18