MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 003 Signals and Systems Spring 2004 Tutorial 3 Monday February 23 and Tuesday February 24 2004 Announcements Problem set 3 is due this Friday Today s Agenda Choosing a Basis Eigenfunctions of LTI Systems CT Fourier Series Dirichlet conditions Properties of Fourier series DT Fourier Series Parseval s Relation 41 1 Choosing a Basis So far we have thought of signals as functions of time x t and x n However it is sometimes more convenient to express signals in terms of another basis Let s de ne a discrete basis to be a set of signals indexed by k called k t in CT and k n in DT so that there exists coe cients ak such that we can express a large class of signals in terms of the basis signals as superposition sums of the basis signals x t ak k t k x n ak k n k Then given a basis the coe cients ak are just as good at specifying the signal as the original x t and x n In DT it is easy to identify a default basis Since x n x k n k k we see that we can choose the basis signals to be k n n k so that ak x k We do this all the time in other areas The coe cients ak are boldfaced for emphasis Whole numbers Represent a general number by their decimal coe cients so that the basis is 10k 32618 3 104 2 103 6 102 1 101 8 100 Now let s change to the 16k basis and write 32618 in base 16 32618 7 163 15 162 6 161 10 160 We see that if we know the basis the set of numbers 3 2 6 1 8 and 7 15 6 10 both identify the number 32618 Vectors Represent any vector by orthogonal components v 3 1 2 3 1 0 0 1 0 1 0 2 0 0 1 Let s rotate our coordinate system so that we can express v in terms of new basis vectors 3 3 1 3 1 1 v 3 1 2 3 1 0 0 1 0 3 0 2 2 2 2 2 2 Currency Represent an amount by more elementary denominations 34 92 1 20 dollar bill 1 10 dollar bill 4 1 dollar bill 3 quarter 1 dime 1 nickel 2 penny 42 Of course we can break down 32 41 several ways and can even write it terms of foreign currency Taylor series more speci cally the MacLaurin series Represent a CT signal f t with continuous derivatives as a polynomial the basis is tk f t f 0 t0 f 0 t f 0 t2 2 f 0 t3 3 f k 0 k 0 k tk What s the most appropriate basis for representing signals that are the inputs and outputs of LTI systems To answer this question we need to determine the eigenfunctions of LTI systems 2 Eigenfunctions of LTI Systems We saw how to determine the output y t y n of any CT DT LTI system given the input x t x n and the impulse response h t h n through the use of the convolution integral sum In general the output looked nothing like the input and the calculation was rather tedious Convolution came from expressing signals as superpositions of shifted scaled impulses Perhaps another set of basis signals would provide more insight on LTI system behavior We showed in lecture that a certain set of input signals namely complex exponentials of the form x t est x n z n are eigenfunctions of LTI systems i e the corresponding outputs are simply scaled versions of inputs of this form and this scaling factor is the eigenvalue We showed that the outputs of CT and DT LTI systems in response to x t est and x n z n are y t H s est H s x t and y n H z z n H z x n respectively where the eigenvalues H s and H z associated with the given eigenfunctions are H s H z h e s d h k z k k where h t and h n are the impulse responses of the systems CT LTI x t est y t H s est h t DT LTI x n z n x t H z est h n Because of the superposition property of LTI systems this suggests another way of writing signals Instead of expressing the input signal as the linear combination of scaled and shifted impulses we can also express the input as the linear combination of complex exponentials Then the output is the same linear combination of the exponentials scaled by the appropriate eigenvalue So if the inputs are 43 x t ak esk t k x n ak zkn k then the outputs are y t ak H sk esk t k y n ak H zk zkn k If you are familiar with linear algebra from 18 03 or 18 06 you ve seen the concepts of eigenvectors and eigenvalues of matrices This is exactly the same idea We think of a matrix A as a linear transformation system from one input vector x into another output vector y y Ax Let vk be the eigenvectors of A with corresponding eigenvalues k so that when A is applied to an eigenvector the result is the same eigenvector scaled by the eigenvalue k vk Avk This is powerful because if we express an arbitrary input vector x as the superposition sum of eigenvectors ak vk x k then we can express the output vector as the same linear combination with each component scaled by the corresponding eigenvalue y Ax A ak vk k ak Avk k y ak k vk k Thus for the linear transformation A vk is the most natural basis for representing vectors We are now see a need to express signals in terms of complex exponentials But how do we do so This question motivates us to spend this tutorial studying the Fourier series which expresses periodic signals in that form 44 3 CT Fourier Series Let us begin by representing periodic CT signals as the linear combination of complex exponentials using the Fourier series representation We will then do the same with periodic DT signals Later we will do the same with aperiod CT and DT signals using the Fourier transform Say x t is periodic with fundamental period T and fundamental frequency 0 2 T Then we can write x t as the sum of the harmonically related components k t ejk 0 t all with period T and the coe cients are unique They are related by CT Fourier Series x t ak 1 T ak ejk 0 t Synthesis equation k x t e jk 0 t dt Analysis equation T The coe cients ak are known as Fourier series coe cients We can extract ak by exploiting the orthogonality of the k t components that is 1 …
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