square wave fourier analysis + + = adding sines with multiple frequencies we can reproduce any shape
Post on 22-Dec-2015
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Square waveSquare wave
Fourier AnalysisFourier Analysis
+
+
=
sin(2 )f t
1sin(2 3 )
3f t
1sin(2 5 )
5f t
1sin(2 ) sin(2 2 )
31
sin(2 5 )5
f t f t
f t
Adding sines with Adding sines with multiple frequencies we multiple frequencies we
can reproduce ANY can reproduce ANY shapeshape
Joseph Fourier (1768-1830)Joseph Fourier (1768-1830)
1 1 2 2 3 3( ) sin(2 ) sin(2 2 ) sin(2 3 ) ...P t A f t A f t A f t
ANYANYperiodic periodic functionfunction
fundamentalfundamental 11stst harmonic harmonic 22ndnd harmonic harmonic
integer multiples of integer multiples of fundamental frequencyfundamental frequency
AAii and and i i wave shapewave shape
timbretimbre
Ohm’s lawOhm’s law
We (We (pretty muchpretty much) can’t hear the phases) can’t hear the phases
sin(2 ) 0.5sin(2 2 )
0.2sin(2 3 ) sin(2 4 )
f t f t
f t f t
sin(2 ) 0.5sin(2 2 )
0.2sin(2 3 ) sin(2 4 )4
f t f t
f t f t
AA i i timbretimbrebut not but not ii
Fourier spectrumFourier spectrum
same same information information (except the (except the
phases)phases)
ff
AA
Examples of Fourier spectraExamples of Fourier spectra
Clarinet nowClarinet now
It is now a great time to read It is now a great time to read chapter 4 of Berg & Storkchapter 4 of Berg & Stork
Roughly …Roughly …
Amplitude LoudnessAmplitude Loudness
Frequency PitchFrequency Pitch
Wave shape TimbreWave shape Timbre
But …But …
Tone qualityTone quality
other things contributing to other things contributing to timbre besides the waveform of timbre besides the waveform of
the steady tonethe steady tone
Roughly …Roughly …
Amplitude LoudnessAmplitude Loudness
Frequency PitchFrequency Pitch
Wave shape TimbreWave shape Timbre
It is the wave shape of the It is the wave shape of the whole sound that matters, whole sound that matters,
not only of the “steady not only of the “steady state”state”
Regarding timbre …Regarding timbre …
Attack and decay transientsAttack and decay transients
Spectrum decaySpectrum decay
InharmonicitiesInharmonicities
sin(2 ) sin(2 2.01 ) sin(2 3.1 )f t f t f t
sin(2 ) sin(2 2 ) sin(2 3 )f t f t f t
higher harmonics higher harmonics slightly off the slightly off the
integer x f valueinteger x f value
VibratoVibrato
TremoloTremolo
oscillation in oscillation in frequencyfrequency
oscillation in oscillation in amplitudeamplitude
FormantsFormants
Chorus effectChorus effect
[let us all sing together][let us all sing together]
Some real examplesSome real examples
but first, but first, spectrographsspectrographs