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Spectral Analysis AOE 3054 23 March 2011 Lowe 1
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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: http://www.aoe.vt.edu/~aborgolt/aoe3054/m anual/inst4/index.html 2
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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
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Spectral analysis/time analysis Given spectral analysis (power spectral density + phase), then we can reconstruct the signal at any and all frequencies: 4
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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
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Complex Trigonometry 8 Note that time and frequency are called conjugate variables: one is the inverse of the other.
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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:
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Fourier transform 10
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Conclusions from cos(t) 11
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Properties of the Fourier transform 12
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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
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Raw Discrete Fourier Transform Results 16
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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
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PSD Definitions and Signal Phase Double-sided spectrum: Single-sided spectrum: Phase: 18
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