1. Spectral Analysis AOE 3054 23 March 2011 Lowe

2. Announcements • Lectures on both Monday, March 28th, and Wednesday, March 30th. • Fracture Testing • Aerodynamic Testing • Prepare for the Spectral Analysis sessions for next week: http://www.aoe.vt.edu/~aborgolt/aoe3054/manual/inst4/index.html

3. 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 ~V2 • Spectral: Refers to frequency (e.g. wave spectra) • Spectral density: Population per unit frequency ~1/Hz • Units of PSD: V2/Hz • The phase of each frequency component • How much of the power is sine versus cosine

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

5. 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:

7. 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:

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

9. Fourier transform 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:

10. Fourier transform • We immediately note: • It yields an answer that is only a function of frequency • It is very closely related to the inner product of the sine/cosine set of • If the indefinite integration of the kernel is a function time, the realizability of answers requires a non-infinite limit at ( • E

11. Conclusions from cos(t) • Remember, the Dirac delta function is non-zero only when its input is zero. • So, is • Zero for all not equal to the angular frequency of cos(t) • 1 for =1 • What is the amplitude of cos(t)?

12. Properties of the Fourier transform • It is linear: the sum of the FT is the FT of the sum. • The convolution of two signals in time is the product of those signals in the Fourier domain • Likewise, Fourier domain convolution is equivalent to time-domain multiplication • The Fourier transform of the derivative of a signal may be determined by multiplying the transform by • The Fourier transform of the derivative of a signal may be determined by dividing the transform by

13. 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

14. Example: cos(2)

15. Multiply:

16. Raw Discrete Fourier Transform Results

17. 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.

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