Spectral Analysis – Fourier DecompositionSpectral decomposition Fourier decompositionSine waves – one amplitude/ one frequencyClarinet spectrumMaking a triangle wave with a sum of harmonics. Adding in higher frequencies makes the triangle tips sharper and sharper.Sum of wavesWhat does a triangle wave sound like compared to the square wave and pure sine wave?Square waveWhich frequencies are added together?Periodic WavesSum of harmonicsLight spectrumSound spectrumSharp bends imply high frequencies Leaving out the high frequency components smoothes the curves High pass filter removes high frequencies – Makes the sound less shrill or brightFourier Transform in different time windows– Smooth Fourier TransformTaking a Fast Fourier Transform (FFT)Output of FFTAccuracy of FFTSpectral Analysis – Fourier DecompositionAdding together different sine wavesPHY103 image from http://hem.passagen.se/eriahl/e_flat.htmfSpectral decompositionFourier decomposition•Previous lectures we focused on a single sine wave.•With an amplitude and a frequency•Basic spectral unit ---- How do we take a complex signal and describe its frequency mix?We can take any function of time and describe it as a sum of sine waves each with different amplitudes and frequenciesSine waves – one amplitude/ one frequencySounds as a series of pressure or motion variations in air.Sounds as a sum of different amplitude signals each with a different frequency.Waveform vs Spectral view in AdobeAudition or Rave LiteClarinet spectrumClarinet spectrum with only the lowest harmonic remainingTime FrequencySpectral viewWaveform viewMaking a triangle wave with a sum of harmonics.Adding in higher frequencies makes the triangle tips sharper and sharper.From Berg and StorkSum of waves•Complex wave forms can be reproduced with a sum of different amplitude sine waves•Any waveform can be turned into a sum of different amplitude sine waves“Fourier decomposition - Fourier series”What does a triangle wave sound like compared to the square wave and pure sine wave?•(Done in lab and previously in class)•Function generators often carry sine, triangle and square waves (and often sawtooths too)If we keep the frequency the same the pitch of these three sounds is the same.However they sound different.Timbre --- that character of the note that enables us to identify different instruments from their sound.Timbre is related to the frequency spectrum.Square waveSame harmonics however the higher order harmonics are stronger. Square wave sounds shriller than the triangle which sounds shriller than the sine waveFrom Berg and StorkWhich frequencies are added together?To get a triangle or square wave we only add sine waves that fit exactly in one period. They cross zero at the beginning and end of the interval.These are harmonics.f frequency5f3fPeriodic Waves•Both the triangle and square wave cross zero at the beginning and end of the interval.•We can repeat the signalIs “Periodic”•Periodic waves can be decomposed into a sum of harmonics or sine waves with frequencies that are multiples of the biggest one that fits in the interval.Sum of harmonics•Also known as the Fourier series•Is a sum of sine and cosine waves which have frequencies f, 2f, 3f, 4f, 5f, ….•Any periodic wave can be decomposed in a Fourier seriesLight spectrumImage from http://scv.bu.edu/~aarondf/avgal.htmlSound spectrumf 3f 5f 7ffrequencyamplitudeTimeSharp bends imply high frequenciesLeaving out the high frequency components smoothes the curves High pass filter removes high frequencies – Makes the sound less shrill or brightFourier Transform in different time windows– Smooth Fourier Transform•We want to measure the spectrum at different times (different notes in a piece for example)•Fourier transform measured in different time intervals•You can adjust this interval (number of points used in the transform).Taking a Fast Fourier Transform (FFT)•Total interval P•Number of points N•Sampling dt•P=N*dt•Windowing function –entire interval is multiplied by a functiondTPOutput of FFT•Frequencies are computed at frequencies• f, 2f, 3f, 4f, ……Nf where 1/f=P is the length of the interval used to compute the FFT and N is the number of points•Difference between frequencies measured is set by the length of the whole interval P.•If P (or number of points N) is too small then precision of FFT is less.Accuracy of FFT•To get better frequency measurements you need a larger interval to measure in•You can’t make extremely fine frequency measurements over extremely small time intervals•Similar to a Heisenberg uncertainty relation•But if N is too big then all the notes will run together!•Trade off between precision in frequency measurement and time~ 1P fD