6 003 Signals and Systems Lecture 16 6 003 Signals and Systems November 5 2009 Last Time Describing Signals by Frequency Content Harmonic content is natural way to describe some kinds of signals Fourier Series Ex musical instruments http theremin music uiowa edu MIS piano piano t k violin violin t k bassoon bassoon t November 5 2009 k Last Time Fourier Series Separating harmonic components Determining harmonic components of a periodic signal Underlying properties ak Z 2 1 x t e j T kt dt T T x t x t T X k ak ej 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency analysis equation 2 kt T e jk 0 t e jl 0 t e j k l 0 t synthesis equation Closure the set of harmonics is closed under multiplication 2 The integral of a harmonic over any time interval with length equal to a period T is zero unless the harmonic is at DC Z t0 T Z 0 k 6 0 e jk 0 t dt e jk 0 t dt T k 0 t0 T T k Separating harmonic components Fourier Series as Orthogonal Decompositions Underlying properties Analogy with vectors in 3 space Let x y and z represent direction vectors in 3 space 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency Vector r can be expressed as sum of components xx y y z z where x r x e jk 0 t e jl 0 t e j k l 0 t y r y Closure the set of harmonics is closed under multiplication z r z 2 The integral of a harmonic over any time interval with length equal to a period T is zero unless the harmonic is at DC Z t0 T Z 0 k 6 0 e jk 0 t dt e jk 0 t dt T k 0 t0 T 2 Similarly for Fourier series where basis functions are k t e j T kt a signal can be expressed as a sum of orthogonal components X x t ak k t k T k where the coefficient of each component is a dot product Z 1 ak x t k t x t k t dt T T Orthogonality harmonics are orthogonal to each other 1 6 003 Signals and Systems Lecture 16 Check Yourself November 5 2009 Speech Vowel sounds are quasi periodic How many of the following pairs of functions are orthogonal in T 3 bat bait bet t beet t t t 1 cos 2 t sin 2 t 2 cos 2 t cos 4 t bit bought bite boat 3 cos 2 t sin t 4 cos 2 t ej2 t t t but t t boot t t Speech Speech Harmonic content is natural way to describe vowel sounds Harmonic content is natural way to describe vowel sounds bat bait bet bat beet bat t k k bit k bought bite k k k boat k but k beet t k boot k beet k boot k boot t k Speech Production Speech Production Speech is generated by the passage of air from the lungs through the vocal cords mouth and nasal cavity Controlled by complicated muscles the vocal cords are set into vibrational motion by the passage of air from the lungs Lo o ki n g do w n t he t h r o at Vocal cords open Glottis Nasal cavity Hard palate Soft palate velum Lips Tongue Vocal cords closed Pharynx Epiglottis Vocal cords Larynx Vocal cords glottis Esophogus Trachea G r a y s A n a to m y Stomach Lungs Adapt ed f r om T F W ei s s 2 A d a p ted f rom T F Wei s s 6 003 Signals and Systems Lecture 16 November 5 2009 Speech Production Filtering Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech Notion 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 R vi Lowpass Filter Lowpass Filtering Calculate the frequency response of an RC circuit Let the input be a square wave R C i t C v o t H j 0 12 x t 0 0 01 0 1 1 10 100 1 RC 10 100 1 RC 2 0 01 0 1 1 t T X 1 j 0 kt e j k k odd 1 0 1 0 01 1 2 X j 1 H j Ri t vo t Solving vi t RC v o t vo t Vi s 1 sRC Vo s 1 Vo s H s Vi s 1 sRC vo C vi t 0 2 T 0 1 0 01 X j vi KVL vo C 0 0 01 0 1 1 10 100 1 RC 0 01 0 1 1 10 100 1 RC 2 Lowpass Filtering Lowpass Filtering Low frequency square wave 0 1 RC Higher frequency square wave 0 1 RC 1 2 0 12 0 0 12 2 T x t 0 1 0 1 j 0 kt e j k k odd 1 0 01 0 1 1 10 100 1 RC 10 100 1 RC 2 0 01 0 1 1 0 2 T 0 1 0 01 H j 0 01 t T X H j 1 j 0 kt e j k k odd 1 H j t T X H j x t 1 2 3 0 0 01 0 1 1 10 100 1 RC 0 01 0 1 1 10 100 1 RC 2 6 003 Signals and Systems Lecture 16 November 5 2009 Lowpass Filtering Lowpass Filtering Still higher frequency square wave 0 1 RC High frequency square wave 0 1 RC 1 2 0 12 t T X 0 2 T x t 1 j 0 kt e j k k odd 1 0 1 1 10 100 1 RC 10 100 1 RC 2 0 01 0 1 1 0 2 T 0 1 0 01 H j 0 0 01 t T X 0 1 0 01 H j 0 12 H j 1 j 0 kt e j k k odd 1 H j x t 1 2 0 0 01 0 1 1 10 100 1 RC 0 01 0 1 1 10 100 1 RC 2 Source Filter Model of Speech Production Speech Production Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech X ray movie showing speech in production buzz from vocal cords throat and nasal cavities speech Demonstration Formants Artificial larynx Resonant frequencies of the vocal tract amplitude F1 F2 F3 frequency Men Women Children buzz from vocal cords throat and nasal cavities speech Formant F1 F2 F3 F1 F2 F3 F1 F2 F3 heed 270 2290 3010 310 2790 3310 370 3200 3730 head 530 1840 2480 610 2330 2990 690 2610 3570 had 660 1720 2410 860 2050 2850 1010 2320 3320 hod 730 …
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