Fourier series and frequency analysis

Integrate numerically: Trapezoid rule

Orthogonality of sines, cosines

integers are mn, when 0)2sin()2cos(

21)2cos()2cos(

integers are mn, and mn when 0)2cos()2cos(

21)2sin()2sin(

integers are mn, and mn when 0)2sin()2sin(

1

0

1

0

1

0

1

0

1

0

dttmtn

dttntn

dttmtn

dttntn

dttmtn

Sines and cosines areorthogonal functions

• In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός (orthos), meaning "straight", and γωνία (gonia), meaning "angle". (wikipedia)

MATLAB Demo

Fourier series

• Any periodic signal can be decomposed to a sum of sines and cosines.

1 to0on t periodic is f(t) Where

)2cos()2sin()(1

0

nnn ntbntabtf

If f(t) is given, what are a’s and b’s?

10 )2cos()2sin()(

nnn ntbntabtf

Use orthogonality

dtttfa

dttftbtbtb

tatatabt

dttfntbntabmt

tfntbntab

nnn

nnn

1

0

1

321

32101

0

10

1

0

10

)2sin()(...000...0002

0

)(...)6cos()4cos()2cos(...)6sin()4sin()2sin(

)2sin(

1mfor compute

)()2cos()2sin()2sin(

)()2cos()2sin(

Coefficients

dtnttfb

dtnttfa

dttfb

n

n

o

1

0

1

0

1

0

)2cos()(2

)2sin()(2

)(

Orthogonal functions

product)(dot product inner theis

basis orthogonal are ,,

)(

^^^

^

^^^^

^^^

kji

iFa

Fkcjbiai

Fkcjbia

productinner theis

basis orthogonal are ...)4cos(),2cos( , )6sin(),4sin(),2sin(

)()2sin(2

)(...)23cos()22cos()2cos(...)23sin()22sin()2sin(

)2sin(

)(...)6cos()4cos()2cos(....)6sin()4sin()2sin(

1

0

1

0

1

321

3211

0

321

3210

dt

ttttt

dttfta

dttftbtbtb

tatatat

tftbtbtbtatatab

“Normal” vector space

F is known vector

Function space

F(t) is the known function

FFT

• Fast Fourier Transform• N log(N) operations rather than N2-matters

when N is big• Classic algorithms requires 2n points• Uses complex notation• Often use windowing on real data

FFT Demo

Questions?

Project

• Fits within the theme “The natural world”• Teams of 4 (nominally)• All students must be in the same section (class

time will be used to work)• You will submit project preferences, we will

form the teams. • Class after spring break will be focused on

project, still may be some lectures.

Other things

• May use “raw” sensors and build your own circuits, or you may buy commercial sensors.

• May leverage labs we have already done. • May focus more on the experiment, the data, the

analysis –or- more on building something interesting. • May focus more on making a finished “product” (though

must stay within spirit of the class).• Each team will get a budget. • Option of doing a PCB if needed (can share with another

team).

The natural world

• Environment • Weather balloon – atmospheric temperature profile• Lakes - (you can rent boats/canoes for Lake Waban).• Weather station (wind, sun, rain).• Underwater ROV• …..

• Bio-instrumentation• EMG – like EKG but senses muscle activity. • EEG (Brain waves)• Pulse oximeter – all analog, calibrated. • Biomechanics (accelerometers on yourself)• Wii balance board/force plate• Heart rate by measuring your weight change• …

Possible examples: