6 003 Signals and Systems Lecture 10 October 15 2009 6 003 Signals and Systems Multiple Representations of CT and DT Systems Convolution Verbal descriptions preserve the rationale Difference differential equations mathematically compact y n x n z0 y n 1 y t x t s0 y t Block diagrams illustrate signal flow paths X X Y z0 A Y s0 R Operator representations analyze systems as polynomials Y 1 Y A X 1 z0 R X 1 s0 A October 15 2009 Transforms representing diff equations with algebraic equations z 1 H z H s z z0 s s0 Convolution Responses to arbitrary signals Representing a system by a single signal Although we have focused on responses to simple signals n t we are generally interested in responses to more complicated signals How do we compute responses to a more complicated input signals No problem for difference equations block diagrams use step by step analysis Check Yourself Superposition Break input into additive parts and sum the responses to the parts x n What is y 3 X R n Y y n R n n n n x n y n n 1 0 1 2 3 4 5 n n 1 0 1 2 3 4 5 Superposition works if the system is linear 1 n n 6 003 Signals and Systems Lecture 10 October 15 2009 Linearity Superposition A system is linear if its response to a weighted sum of inputs is equal to the weighted sum of its responses to each of the inputs Break input into additive parts and sum the responses to the parts x n Given n x1 n system y n y1 n n and x2 n system y2 n the system is linear if 1 0 1 2 3 4 5 system n n x1 n x2 n n n n y1 n y2 n 1 0 1 2 3 4 5 is true for all and n Reponses to parts are easy to compute if system is time invariant Time Invariance Superposition A system is time invariant if delaying the input to the system simply delays the output by the same amount of time Break input into additive parts and sum the responses to the parts x n Given n x n system n the system is time invariant if x n n0 y n y n system n n n y n n0 is true for all n0 1 0 1 2 3 4 5 n n 1 0 1 2 3 4 5 n Superposition is easy if the system is linear and time invariant Structure of Superposition Convolution If a system is linear and time invariant LTI then its output is the sum of weighted and shifted unit sample responses Response of an LTI system to an arbitrary input n system x n h n y n n k system h n k x k n k system x k h n k x k n k system X LTI x k h n k x h n k This operation is called convolution x n X k X x k h n k k 2 y n 6 003 Signals and Systems Lecture 10 October 15 2009 Notation Notation Convolution is represented with an asterisk Do not be fooled by the confusing notation X Confusing but conventional notation X x k h n k x n h n x k h n k x h n k k It is customary but confusing to abbreviate this notation x n h n looks like an operation of samples but it is not x h n x n h n x 1 h 1 6 x h 1 Convolution operates on signals not samples Unambiguous notation X x k h n k x h n k The symbols x and h represent DT signals Convolving x with h generates a new DT signal x h Structure of Convolution y 2 Structure of Convolution X x k h 2 k y 3 k X x k h k x k h k k k h 2 k k 2 1 0 1 2 3 4 5 x k h 2 k k 2 1 k h 2 k X 0 1 2 3 4 5 x k h 3 k k k h 3 k h 3 k k k 2 1 0 1 2 3 4 5 x k h 3 k X k k 2 1 Check Yourself 0 1 2 3 4 5 k k Convolution Representing an LTI system by a single signal x n Which plot shows the result of the convolution above 1 2 3 4 h n y n Unit sample response h n is a complete description of an LTI system Given h n one can compute the response y n to any arbitrary input signal x n y n x h n X k 5 none of the above 3 x k h n k 6 003 Signals and Systems Lecture 10 October 15 2009 CT Convolution Convolution Convolution of CT signals is analogous to convolution of DT signals Convolution is an important computational tool DT y n x h n X Example characterizing LTI systems x k h n k k CT y t x h t Z x h t d Determine the unit sample response h n Calculate the output for an arbitrary input using convolution X y n x h n x k h n k Applications of Convolution Microscope Convolution is an important conceptual tool it provides an important new way to think about the behaviors of systems Images from even the best microscopes are blurred Example systems microscopes and telescopes Microscope Microscope Blurring can be represented by convolving the image with the optical point spread function 3D impulse response Measuring the impulse response of a microscope target Image diameter 6 times target diameter target impulse image Blurring is inversely related to the diameter of the lens 4 6 003 Signals and Systems Lecture 10 October 15 2009 Microscope Microscope Images at different focal planes can be assembled to form a threedimensional impulse response point spread function Blurring along the optical axis is better visualized by resampling the three dimensional impulse response Microscope Microscope Blurring is much greater along the optical axis than it is across the optical axis The point spread function 3D impulse response is a useful way to characterize a microscope It provides a direct measure of blurring which is an important figure of merit for optics Hubble Space Telescope Hubble Space Telescope Hubble Space Telescope 1990 Why build a space telescope Telescope images are blurred by the telescope lenses AND by atmospheric turbulence X X ha x y hd x y atmospheric blurring blur due to mirror size ht x y ha hd x y ground based telescope http hubblesite org 5 Y Y 6 003 Signals and Systems Lecture 10 October 15 2009 Hubble Space Telescope Hubble Space Telescope Telescope blur can be respresented by the convolution of blur due to atmospheric turbulence and blur due to mirror size The main optical components of the Hubble Space Telescope are two …
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