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Frequency Space

Sine waves can be mixed with DC signals, or with other sine

waves to produce new waveforms. Here is one example of a

complex waveform:

V(t) = Ao + A1sin1t + A2sin 2t + A3sin 3t + … + Ansin nt--- in this case---V(t) = Ao + A1sin1t

Ao

A1

Fourier Analysis

Just an AC component superimposed on aDC component

More dramatic results are obtained by mixing a sine wave of a particular frequency

with exact multiples of the same frequency. We are adding harmonics

to the fundamental frequency. For example, take the fundamental frequency and add 3rd

harmonic (3 times the fundamental frequency) at reduced amplitude, and subsequently add

its 5th, 7th and 9th harmonics:

Fourier Analysis, cont’d

the waveform begins to look more and more like a square wave.

This result illustrates a general principle first formulated by the

French mathematician Joseph Fourier, namely that any complex waveform

can be built up from a pure sine waves plus particular harmonics of the

fundamental frequency. Square waves, triangular waves and sawtooth waves

can all be produced in this way.

...)7sin(7

1)5sin(

5

1)3sin(

3

1)sin(

1

1)(

,

tttttf

thatshownbecanitwavesquarethefor

oooo

(try plotting this using Excel)

Fourier Analysis, cont’d

Spectral Analysis• Spectral analysis means determining the

frequency content of the data signal• Important in experiment design for

determining sample rate, fs - sampling rate theorem states: fs max fsignal to avoid aliasing

• Important in post-experiment analysis- Frequency content is often a primary experiment result. Experiment examples:

- determining the vibrational frequencies of structures

- reducing noise of machines- Developing voice recognition software

Spectral analysis key points

Any function of time can be made up by adding sine andcosine function of different amplitudes, frequencies, and phases.

These sines and cosines are called frequency components or harmonics.

Any waveform other than a simple sine or cosine has more than one frequency component.

Example Waveform• 1000 Hz sawtooth, amplitude 2 Volts

1000 Hz Sawtooth

-3

-2

-1

0

1

2

3

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

time

am

pli

tud

e (

volt

s)

tnbtbtbb

tnatatatfgeneralIn

n

nn

002010

0201

cos...2coscos

sin...2sinsin)(,

Fundamental frequency term Harmonic terms

b0 is the average value of thefunction over period, T

Period, T = .001 sec

Fourier Coefficients, an and bn

• These coefficients are simply the amplitude at each component frequency

• For odd functions [f(t)=-f(-t)], all bn= 0, and have a series of sine terms (sine is an odd function)

• For even functions [f(t)=f(-t)], all an= 0, and have a series of cosine terms (cosine is an even function)

• For arbitrary functions, have an and bn terms.

• Coefficients are calculated as follows:

functionsoddfordttntfT

b

functionsevenfordttntfT

a

T

n

T

n

0cos)(2

0sin)(2

0

0

0

0

An odd function (sine wave)

More odd functions

Fundamentalor First Harmonic

Third HarmonicSine series orPure imaginary amplitudes

An even function (cosine wave)

More even functions

Fundamentalor First Harmonic

Second HarmonicCosine series orPure real amplitudes

Periodic, but neither even nor odd

Cosine and sine series orComplex amplitudes

Sawtooth Fourier Coefficients• Odd function so:

• Using direct integration or numerical integration we find the first seven an’s to be:

• We can plot these coefficients in frequency space:

0

sin)(2

0

0

n

T

n

b

dttntfT

a

0000.0

0331.1801.

000.00000.0

0648.6211.1

4

73

62

51

a

aa

aa

aa

Fourier Coefficients

-0.5

0

0.5

1

1.5

2

0

1000

2000

3000

4000

5000

6000

7000

8000

frequency

ampl

itude

Our sawtooth wave is an ________ function. Therefore all ____ = 0

Start with a sine wave...

Add an odd harmonic (#3) ...

Add another (#5)...

And still another (#7)...

Let’s transform a “Sharper” sawtooth

sec10.05.660)(

sec05..060)(

sec1./1

sec,/83622,10 .

tforttf

tforttf

fT

radfHzf

Sharp Sawtooth

-4

-2

0

2

4

0 0.02 0.04 0.06 0.08 0.1

time

f(t)

6366.)83.623sin()660()83.623sin(601.

23sin)(

2

9549.)83.622sin()660()83.622sin(601.

22sin)(

2

9098.1)83.62sin()660()83.62sin(601.

21sin)(

2

05.

0

1.

05.0

03

05.

0

1.

05.0

02

05.

0

1.

05.0

01

dtttdtttdtttfT

a

dtttdtttdtttfT

a

dtttdtttdtttfT

a

T

T

T

Even or odd?

Frequency Domain Plot of Fourier Coefficients

Sharp Sawtooth Fourier Coefficients

-2-10

123

0 5 10 15 20 25 30 35

Frequency

Ampli

tude

Get “powerspectrum”by squaringFouriercoefficients

"Power Spectrum"

0

1

2

3

4

0 10 20 30 40

Frequency

Relat

ive Po

wer

Construction of Sharp Sawtooth by Adding 1st, 2nd, 3rd

Harmonic

Third Harmonic

First HarmonicSecond Harmonic

Spectral Analysis of Arbitrary Functions

• In general, there is no requirement that f(t) be a periodic function

• We can force a function to be periodic simply by duplicating the function in time (text fig 5.10)

• We can transform any waveform to determine it’s Fourier spectrum

• Computer software has been developed to do this as a matter of routine. - One such technique is called “Fast Fourier Transform” or FFT- Excel has an FFT routine built in

Voice Recognition

The “ee” sound

Voice Recognition (continued)

The “eh” sound

Voice Recognition (continued)

The “ah” sound

Voice Recognition (continued)

The “oh” sound

Voice Recognition (continued)

The “oo” sound