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Unformatted text preview: Review of Key Concepts from ECSE Review of Key Concepts from ECSE303 303 009 . R. Chen 1/2/2 L. Review of key concepts from ECSE Review of key concepts from ECSE303 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 009 • Laplace transform Bibliography . R. Chen 1/2/2 • Chapters 14, 6, 9 in Signals and Systems by Oppenheim et. al. • Course notes from ECSE303 (B. Boulet) L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 2 Course notes from ECSE 303 (B. Boulet) Discretetime signals • A discretetime 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 009 . R. Chen 1/2/2 L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 3 adapted from B. Boulet Periodic Signals • A signal is periodic if there exists a value T or N such that ) ( ) ( t x T t x = + ) ( ) ( t x T t x + ] [ ] [ n x N n x = + The smallest value of T or N for which the above holds is known as the fundamental period . 009 . R. Chen 1/2/2 L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 4 Exponential Signals • Continuous time: ) exp( ) ( at C t x = where C = Cexp( j θ ) and a = r + j ω • Using Euler’s relation, ( ) ( ) θ ω θ ω + + + = t e C j t e C t x rt rt sin cos ) ( 009 • The exponential decays if r < 0 and grows if r > 0. It is periodic or constant if r = 0. . R. Chen 1/2/2 L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 5 Exponential Signals • Discrete time: n C n x α = ] [ where C = Cexp( j θ ) and α =  α exp( j ω ) • Using Euler’s relation, ( ) ( ) θ ω α θ ω α + + + = n C j n C n x n n sin cos ] [ 009 • The exponential decays if  α  < 1 and grows if  α  > 1 and has constant amplitude if  α  = 1. . R. Chen 1/2/2 • It is periodic if  α  = 1 and there exists integers m and N so that f m = = ω L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 6 2 f N = = π Real Exponential Signals • Discrete time: n C n x α = ] [ where C = Cexp( j θ ) and α is real • There are 6 cases to consider, depending on the value of α 009 . R. Chen 1/2/2 L. Review of key concepts from ECSE Review of key concepts from ECSE303 303 7 adapted from B. Boulet Real Exponential Signals • Case 1: α = 1 ⇒ constant signal • Case 2: α > 1 ⇒ positive signal with exponential growth 009 • Case 3: 0 < α < 1 ⇒ positive signal with exponential decay . R. Chen 1/2/2 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.
 Spring '11
 ChenandBacsy

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