Spectral Analysis AOE 3054 - PowerPoint Presentation

Slide1

Spectral Analysis

AOE 305423 March 2011Lowe

1Slide2

Announcements

Lectures on both Monday, March 28th, and Wednesday, March 30

th

.

Fracture TestingAerodynamic TestingPrepare for the Spectral Analysis sessions for next week: http://www.aoe.vt.edu/~aborgolt/aoe3054/manual/inst4/index.html

2Slide3

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/HzUnits of PSD: V2/HzThe phase of each frequency componentHow much of the power is sine versus cosine

3Slide4

Spectral analysis/time analysis

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

4Slide5

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

:

5Slide6

6Slide7

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:

7Slide8

Complex Trigonometry

8

Note that time and frequency are called

conjugate

variables: one is the inverse of the other.Slide9

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:

 Slide10

Fourier transform

10

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

 Slide11

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)?

11

 Slide12

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

12Slide13

Digital signals

Of course, we rarely are so lucky as to have an analytic function for our signalMore 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

 Slide14

Example:

cos(2

)

14

 Slide15

Multiply:

15Slide16

Raw Discrete Fourier Transform Results

16Slide17

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 TransformsThere is one easy way to solve this, though, compute the power spectral density and signal phase.

17Slide18

PSD Definitions and Signal Phase

Double-sided spectrum:Single-sided spectrum:

Phase:

18