Review_of_ECSE_303

Review_of_ECSE_303 - L. R. Chen 8/31/2010 Review Review of...

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Review of Key Concepts from ECSE Review of Key Concepts from ECSE-303 303 2010 . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 1 Content • CT and DT signals and systems • LTI systems, differential and difference LTI systems • Fourier series representation of periodic CT signals • Fourier series representation of periodic, CT signals • Fourier transform of CT signals • Laplace transform Bibliography • Chapters 1-4, 6, 9 in Signals and Systems by Oppenheim et. al. • Course notes from ECSE-303 (B. Boulet) Review of key concepts from ECSE Review of key concepts from ECSE-303 303 2 Course notes from ECSE 303 (B. Boulet)
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Discrete-time signals A discrete-time signal is a function of a discrete variable - it is only defined for integer values of the independent variable (time steps) - denoted by x [ n ] - example: the value of a stock at the end of the month 2010 . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 3 adapted from B. Boulet Periodic Signals •As igna lis periodic if there exists a value T or N such that ) ( ) ( t t x x + ] [ ] n n x = + The smallest value of T or N for which the above holds is known as the . 2010 Note that . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303
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Exponential Signals • Continuous time: ) exp( ) ( at C t x = where C = |C|exp( j θ ) and a = r + j ω 0 • Using Euler’s relation, ( ) ( ) θ ω + + + = e j rt 0 0 sin cos ) ( 2010 The exponential decays if r < 0 and grows if r > 0. It is periodic or constant if r = 0. . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 5 Exponential Signals • Discrete time: n α = ] [ where C = |C|exp( j θ ) and = | |exp( j ω 0 ) • Using Euler’s relation, ( ) ( ) + + + = 0 0 sin cos ] [ The exponential decays if | | < 1 and grows if | | > 1 and has constant amplitude if | | = 1. It is periodic if | | = 1 and there exists integers m and N so that 0 f m = = Review of key concepts from ECSE Review of key concepts from ECSE-303 303 6 0 2 N π
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Real Exponential Signals • Discrete time: n C x α = ] [ where C = |C|exp( j θ ) and is real • There are 6 cases to consider, depending on the value of 2010 . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 7 adapted from B. Boulet Real Exponential Signals •Case1: = 1 constant signal •Case 2: α > 1 positive signal with exponential growth • Case 3: 0 < < 1 positive signal with exponential decay Review of key concepts from ECSE Review of key concepts from ECSE-303 303 8 adapted from B. Boulet
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Real Exponential Signals •Case4: α < - 1 signal alternates between positive and negative values with exponential growth • Case 5 1 signal alternates between + C and C 2010 • Case 5: α = -1 and - . R. Chen 8/31/ L.
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This note was uploaded on 07/09/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Review_of_ECSE_303 - L. R. Chen 8/31/2010 Review Review of...

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