Where we’re going Speed, Storage Issues Frequency Space.
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Transcript of Where we’re going Speed, Storage Issues 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
More odd functions
Fundamentalor First Harmonic
Third HarmonicSine series orPure imaginary amplitudes
More even functions
Fundamentalor First Harmonic
Second HarmonicCosine series orPure real 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
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