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6.003: Signals and Systems Lecture 16 November 5, 200916.003: Signals and SystemsFourier SeriesNovember 5, 2009Last Time: Describing Signals by Frequency ContentHarmonic content is natural way to describe some kinds of signals.Ex: musical instruments (http://theremin.music.uiowa.edu/MIS)pianotpianokviolintviolinkbassoontbassoonkLast Time: Fourier SeriesDetermining harmonic components of a periodic signal.ak=1TZTx(t)e−j2πTktdt (“analysis” equation)x(t)= x(t + T ) =∞Xk=−∞akej2πTkt(“synthesis” equation)Separating harmonic componentsUnderlying properties.1. Multiplying two harmonics produces a new harmonic with thesame fundamental frequency:ejkω0t× ejlω0t= ej(k+l)ω0t.Closure: the set of harmonics is closed under multiplication.2. The integral of a harmonic over any time interval with lengthequal to a period T is zero unless the harmonic is at DC:Zt0+Tt0ejkω0tdt ≡ZTejkω0tdt =0, k 6= 0T, k = 0= T δ[k]Separating harmonic componentsUnderlying properties.1. Multiplying two harmonics produces a new harmonic with thesame fundamental frequency:ejkω0t× ejlω0t= ej(k+l)ω0t.Closure: the set of harmonics is closed under multiplication.2. The integral of a harmonic over any time interval with lengthequal to a period T is zero unless the harmonic is at DC:Zt0+Tt0ejkω0tdt ≡ZTejkω0tdt =0, k 6= 0T, k = 0= T δ[k]Orthogonality: harmonics are orthogonal (⊥) to each other.Fourier Series as Orthogonal DecompositionsAnalogy with vectors in 3-space.Let ˆx, ˆy, and ˆz represent direction vectors in 3-space.Vector ¯r can be expressed as sum of components xˆx + yˆy + z ˆz wherex = ¯r · ˆxy = ¯r · ˆyz = ¯r · ˆzSimilarly for Fourier series (where basis functions are φk(t) = ej2πTkt),a signal can be expressed as a sum of orthogonal components:x(t) =∞Xk=−∞akφk(t)where the coefficient of each component is a dot productak= x(t) · φk(t) ≡1TZTx(t)φ∗k(t)dt6.003: Signals and Systems Lecture 16 November 5, 20092Check YourselfHow many of the following pairs of functions are orthog-onal (⊥) in T = 3?1. cos 2πt ⊥ sin 2πt ?2. cos 2πt ⊥ cos 4πt ?3. cos 2πt ⊥ sin πt ?4. cos 2πt ⊥ ej2πt?SpeechVowel sounds are quasi-periodic.battbaittbettbeettbittbitetboughttboattbuttboottSpeechHarmonic content is natural way to describe vowel sounds.batkbaitkbetkbeetkbitkbitekboughtkboatkbutkbootkSpeechHarmonic content is natural way to describe vowel sounds.battbatkbeettbeetkboottbootkSpeech ProductionSpeech is generated by the passage of air from the lungs, throughthe vocal cords, mouth, and nasal cavity.Adapted from T.F. WeissLipsNasalcavityHard palateTongueSoft palate(velum)PharynxVocal cords(glottis)EsophogusEpiglottisLungsStomachLarynxTracheaSpeech ProductionControlled by complicated muscles, the vocal cords are set into vi-brational motion by the passage of air from the lungs.Looking down the throat:Gray's Anatomy Adapted from T.F. WeissGlottisVocalcordsVocal cords openVocal cords closed6.003: Signals and Systems Lecture 16 November 5, 20093Speech ProductionVibrations of the vocal cords are “filtered” by the mouth and nasalcavities to generate speech.FilteringNotion of a filter.LTI systems• cannot create new frequencies.• can only scale magnitudes and shift phases of existing components.Example: Low-Pass Filtering with an RC circuit+−vi+vo−RCLowpass FilterCalculate the frequency response of an RC circuit.+−vi+vo−RCKVL: vi(t) = Ri(t) + vo(t)C: i(t) = C ˙vo(t)Solving: vi(t) = RC ˙vo(t) + vo(t)Vi(s) = (1 + sRC)Vo(s)H(s) =Vo(s)Vi(s)=11 + sRC0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringLet the input be a square wave.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|X(jω)|0−π20.01 0.1 1 10 100ω1/RC∠X(jω)|Lowpass FilteringLow frequency square wave: ω0<< 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringHigher frequency square wave: ω0< 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|6.003: Signals and Systems Lecture 16 November 5, 20094Lowpass FilteringStill higher frequency square wave: ω0= 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Lowpass FilteringHigh frequency square wave: ω0> 1/RC.t12−120Tx(t) =Xk odd1jπkejω0kt; ω0=2πT0.010.110.01 0.1 1 10 100ω1/RC|H(jω)|0−π20.01 0.1 1 10 100ω1/RC∠H(jω)|Source-Filter Model of Speech ProductionVibrations of the vocal cords are “filtered” by the mouth and nasalcavities to generate speech.buzz fromvocal cordsspeechthroat andnasal cavitiesSpeech ProductionX-ray movie showing speech in production.DemonstrationArtificial larynx.buzz fromvocal cordsspeechthroat andnasal cavitiesFormantsResonant frequencies of the vocal tract.frequencyamplitudeF1F2F3Formant heed head had hod haw’d who’dMen F1 270 530 660 730 570 300F2 2290 1840 1720 1090 840 870F3 3010 2480 2410 2440 2410 2240Women F1 310 610 860 850 590 370F2 2790 2330 2050 1220 920 950F3 3310 2990 2850 2810 2710 2670Children F1 370 690 1010 1030 680 430F2 3200 2610 2320 1370 1060 1170F3 3730 3570 3320 3170 3180 3260http://www.sfu.ca/sonic-studio/handbook/Formant.html6.003: Signals and Systems Lecture 16 November 5, 20095Speech ProductionSame glottis signal + different formants → different vowels.glottis signal vocal tract filter vowel soundakakbkbkWe detect changes in the filter function to recognize vowels.SingingWe detect changes in the filter function to recognize vowels ... atleast sometimes.Demonstration.“la” scale.“lore” scale.“loo” scale.“ler” scale.“lee” scale.Low Frequency: “la” “lore” “loo” “ler” “lee”.High Frequency: “la” “lore” “loo” “ler” “lee”.http://www.phys.unsw.edu.au/jw/soprane.htmlSpeech ProductionWe detect changes in the filter function to recognize vowels ... atleast sometimes.lowintermediatehighSpeech ProductionWe detect changes in the filter function to recognize vowels ... atleast sometimes.lowintermediatehighContinuous-Time Fourier Series: SummaryFourier series represent signals by their frequency content.Representing a signal by its frequency content is useful for manysignals, e.g., music.Fourier series motivate a new representation of a system as a filter.Representing a system as a filter is


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