6 003 Signals and Systems Fourier Series November 5 2009 Last Time Describing Signals by Frequency Content Harmonic content is natural way to describe some kinds of signals Ex musical instruments http theremin music uiowa edu MIS piano piano t k violin violin t k bassoon bassoon t k Last Time Fourier Series Determining harmonic components of a periodic signal ak Z 2 1 x t e j T kt dt T T x t x t T X k ak ej analysis equation 2 kt T synthesis equation Separating harmonic components Underlying properties 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency e jk 0 t e jl 0 t e j k l 0 t 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 jk 0 t jk 0 t e dt e dt T k 0 t0 T T k Separating harmonic components Underlying properties 1 Multiplying two harmonics produces a new harmonic with the same fundamental frequency e jk 0 t e jl 0 t e j k l 0 t 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 jk 0 t jk 0 t e dt e dt T k 0 t0 T T k Orthogonality harmonics are orthogonal to each other Fourier Series as Orthogonal Decompositions Analogy with vectors in 3 space Let x y and z represent direction vectors in 3 space Vector r can be expressed as sum of components xx y y z z where x r x y r y z r z 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 where the coefficient of each component is a dot product Z 1 ak x t k t x t k t dt T T Check Yourself How many of the following pairs of functions are orthogonal in T 3 1 cos 2 t sin 2 t 2 cos 2 t cos 4 t 3 cos 2 t sin t 4 cos 2 t ej2 t Check Yourself How many of the following are orthogonal in T 3 cos 2 t sin 2 t cos 2 t t 1 2 3 1 2 3 2 3 sin 2 t t cos 2 t sin 2 t t 1 Z 3 dt 0 therefore YES 0 Check Yourself How many of the following are orthogonal in T 3 cos 2 t cos 4 t cos 2 t t 1 2 3 1 2 3 2 3 cos 4 t t cos 2 t cos 4 t t 1 Z 3 dt 0 therefore YES 0 Check Yourself How many of the following are orthogonal in T 3 cos 2 t cos t cos 2 t t 1 2 3 1 2 3 1 2 3 sin t t cos 2 t sin t t Z 3 dt 6 0 therefore NO 0 Check Yourself How many of the following are orthogonal in T 3 cos 2 t e2 t e2 t cos 2 t j sin 2 t cos 2 t sin 2 t but not cos 2 t Therefore NO Check Yourself How many of the following pairs of functions are orthogonal in T 3 2 1 cos 2 t sin 2 t 2 cos 2 t cos 4 t 3 cos 2 t sin t X 4 cos 2 t ej2 t X Speech Vowel sounds are quasi periodic bat bait t bit bet t t bought bite t beet t but boat t boot t t t t Speech Harmonic content is natural way to describe vowel sounds bat bait k bit bet k k bought bite k beet k but boat k boot k k k k Speech Harmonic content is natural way to describe vowel sounds bat bat t k beet beet t k boot boot t k Speech Production Speech is generated by the passage of air from the lungs through the vocal cords mouth and nasal cavity Nasal cavity Hard palate Soft palate velum Lips Tongue Pharynx Epiglottis Larynx Vocal cords glottis Esophogus Trachea Stomach Lungs Adapted fro m T F Wei s s Speech Production Controlled by complicated muscles the vocal cords are set into vibrational motion by the passage of air from the lungs L o ok i n g d o wn t h e t h ro a t Vocal cords open Glottis Vocal cords closed Vocal cords G r a y s An a tom y Adapt ed from T F We iss Speech Production Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech Filtering 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 C vo Lowpass Filter Calculate the frequency response of an RC circuit R vi t Ri t vo t C i t C v o t Solving vi t vo C RC v o t vo t Vi s 1 sRC Vo s 1 Vo s H s Vi s 1 sRC 1 H j 0 1 0 01 H j vi KVL 0 01 0 1 1 10 100 1 RC 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Let the input be a square wave 1 2 0 21 1 j 0 kt e j k k odd 1 X j X 0 2 T 0 1 0 01 X j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Low frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Higher frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 1 RC 0 2 0 01 0 1 1 Lowpass Filtering Still higher frequency square wave 0 1 RC 1 2 0 21 1 j 0 kt e j k k odd 1 H j X 0 2 T 0 1 0 01 H j x t t T 0 01 0 1 1 10 1 RC 100 10 100 …
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