6 003 Signals and Systems Lecture 11 6 003 Signals and Systems October 20 2009 Mid term Examination 2 Wednesday October 28 7 30 9 30pm Walker Memorial Frequency Response No recitations on the day of the exam Coverage cumulative with more emphasis on recent material lectures 1 12 homeworks 1 7 Homework 7 will include practice problems for mid term 2 However it will not collected or graded Solutions will be posted Closed book 2 page of notes 8 12 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions during open office hours October 20 2009 Conflict Contact freeman mit edu by Friday October 23 5pm Review Microscope Last time we saw how a linear time invariant LTI system can be characterized by its unit sample impulse response Blurring can be represented by convolving the image with the optical point spread function 3D impulse response DT y n x h n X x k h n k k CT y t x h t Z x h t d target Characterizing a system by its unit sample impulse response is especially insightful for some systems image Blurring is inversely related to the diameter of the lens Hubble Space Telescope Frequency Response Hubble images before and after upgrading the optics Today we will investigate a different way to characterize a system the frequency response Many systems are naturally described by their responses to sinusoids Example audio systems before after http hubblesite org 1 6 003 Signals and Systems Lecture 11 Check Yourself October 20 2009 Frequency Response Preview If the input to a linear time invariant system is an eternal sinusoid then the output is also an eternal sinusoid same frequency possibly different amplitude and possibly different phase angle How were frequencies modified in following music clips HF high frequencies LF low frequencies 1 2 3 4 5 increased decreased clip 1 clip 2 HF HF LF LF HF LF LF HF none of the above x t cos t y t M cos t t LTI system t The frequency response is a plot of the magnitude M and angle as a function of frequency Demonstration Frequency Response Measure the frequency response of a mass spring dashpot system Calculate the frequency response x t Methods solve differential equation find particular solution for x t cos 0 t find impulse response of system convolve with x t cos 0 t y t New method use eigenfunctions and eigenvalues Eigenfunctions Check Yourself Eigenfunctions If the output signal is a scalar multiple of the input signal we refer to the signal as an eigenfunction and the scale multiplier as the eigenvalue Consider the system described by y t 2y t x t Determine if each of the following functions is an eigenfunction of this system If it is find its eigenvalue eigenvalue x t system x t 1 2 3 4 5 eigenfunction 2 e t for all time et for all time ejt for all time cos t for all time u t for all time 6 003 Signals and Systems Lecture 11 October 20 2009 Complex Exponentials Rational System Functions Complex exponentials are eigenfunctions of LTI systems If a system is represented by a linear differential equation with constant coefficients then its system function is a ratio of polynomials in s If x t est and h t is the impulse response then y t h x t Z h es t d est LTI h t est Z h e s d H s est Example y t 3y t 4y t 2x t 7x t 8x t H s est Then H s Furthermore the eigenvalue is H s 2s2 7s 8 N s D s s2 3s 4 Vector Diagrams Vector Diagrams The value of H s at a point s s0 can be determined graphically using vectorial analysis Example Find the response of the system described by 1 H s s 2 Factor the numerator and denominator of the system function to make poles and zeros explicit to the input x t e2jt for all time s plane s0 z0 s0 z1 s0 z2 H s0 K s0 p0 s0 p1 s0 p2 s0 p0 s plane s0 2 s0 s0 z0 z0 z0 The denominator of H s s 2j is 2j 2 a vector with length 2 2 and angle 4 Therefore the response of the system is j 1 y t H 2j e2jt e 4 e2jt 2 2 Each factor in the numerator denominator corresponds to a vector from a zero pole here z0 to s0 the point of interest in the s plane Vector Diagrams Frequency Response The value of H s at a point s s0 can be determined by combining the contributions of the vectors associated with each of the poles and zeros Response to eternal sinusoids H s0 K s0 2j Let x t cos 0 t for all time Then 1 j 0 t x t e e j 0 t 2 s0 z0 s0 z1 s0 z2 s0 p0 s0 p1 s0 p2 and the response to a sum is the sum of the responses 1 y t H j 0 ej 0 t H j 0 e j 0 t 2 The magnitude is determined by the product of the magnitudes s0 z0 s0 z1 s0 z2 H s0 K s0 p0 s0 p1 s0 p2 The angle is determined by the sum of the angles H s0 K s0 z0 s0 z1 s0 p0 s0 p1 3 6 003 Signals and Systems Lecture 11 October 20 2009 Conjugate Symmetry Frequency Response The complex conjugate of H j is H j Response to eternal sinusoids The system function is the Laplace transform of the impulse response Z h t e st dt H s Let x t cos 0 t for all time which can be written as 1 j 0 t x t e e j 0 t 2 The response to a sum is the sum of the responses 1 y t H j 0 ej 0 t H j 0 e j 0 t 2 n o Re H j 0 ej 0 t n o Re H j 0 ej H j 0 ej 0 t n o H j 0 Re ej 0 t j H j 0 where h t is a real valued function of t for physical systems H j Z H j h t e j t dt Z h t ej t dt H j y t H j 0 cos 0 t H j 0 Frequency Response Vector Diagrams The magnitude and phase of the response of a system to an eternal cosine signal is the magnitude and phase of the system function evaluated at s j cos t H j cos t H j H s H j 5 H s s z1 5 s plane 5 5 5 0 H j 2 5 5 Vector Diagrams H s 5 0 2 H j 5 s z1 s p1 5 s plane 5 5 H j 2 5 5 H s 3 s plane 5 5 …
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