6 003 Signals and Systems Fourier Series April 1 2010 Mid term Examination 2 Wednesday April 7 7 30 9 30pm 34 101 No recitations on the day of the exam Coverage Lectures 1 15 Recitations 1 15 Homeworks 1 8 Homework 8 will not collected or graded Solutions will be posted Closed book 2 pages of notes 8 21 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions during open office hours Conflict Contact freeman mit edu before Friday April 2 5pm 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 e j analysis equation 2 kt T synthesis equation 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 ak e j analysis equation 2 kt T synthesis equation k We can think of Fourier series as an orthogonal decomposition Orthogonal Decompositions Vector representation of 3 space let r represent a vector with components x y and z in the x y and z directions respectively x r x y r y z r z analysis equations r xx y y z z synthesis equation Z 2 1 ak x t e j T kt dt T T x t x t T X k ak e j analysis equation 2 kt T synthesis equation Orthogonal Decompositions Vector representation of 3 space let r represent a vector with components x y and z in the x y and z directions respectively x r x y r y z r z analysis equations r xx y y z z synthesis equation Fourier series let x t represent a signal with harmonic components a0 a1 ak for harmonics e j0t e j Z 2 1 x t e j T kt dt ak T T x t x t T X k ak e j 2 t T e j 2 kt T respectively analysis equation 2 kt T synthesis equation Orthogonal Decompositions Vector representation of 3 space let r represent a vector with components x y and z in the x y and z directions respectively x r x y r y z r z analysis equations r xx y y z z synthesis equation Fourier series let x t represent a signal with harmonic components a0 a1 ak for harmonics e j0t e j Z 2 1 x t e j T kt dt ak T T x t x t T X k ak e j 2 t T e j 2 kt T respectively analysis equation 2 kt T synthesis equation Orthogonal Decompositions Vector representation of 3 space let r represent a vector with components x y and z in the x y and z directions respectively x r x y r y z r z analysis equations r xx y y z z synthesis equation Fourier series let x t represent a signal with harmonic components a0 a1 ak for harmonics e j0t e j Z 2 1 x t e j T kt dt ak T T x t x t T X k ak e j 2 t T e j 2 kt T respectively analysis equation 2 kt T synthesis equation Orthogonal Decompositions Integrating over a period sifts out the k th component of the series Sifting as a dot product x r x r x cos Sifting as an inner product Z 2 1 j 2 kt T ak e x t x t e j T kt dt T T where Z 1 a t b t a t b t dt T T Orthogonal Decompositions Integrating over a period sifts out the k th component of the series Sifting as a dot product x r x r x cos Sifting as an inner product Z 2 1 j 2 kt T ak e x t x t e j T kt dt T T where Z 1 a t b t a t b t dt T T The complex conjugate makes the inner product of the k th and mth components equal to 1 iff k m Z Z 2 2 kt j 2 mt 1 1 1 if k m j 2 kt j mt j e T e T dt e T e T dt T T T T 0 otherwise 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 e j2 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 e j2 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 …
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