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Review_of_ECSE_303

# Review_of_ECSE_303 - L R Chen Review Review of Key Concepts...

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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 2010 • Laplace transform Bibliography . R. Chen 8/31/ • Chapters 1-4, 6, 9 in Signals and Systems by Oppenheim et. al. • Course notes from ECSE-303 (B. Boulet) L. 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 • A signal is periodic if there exists a value T or N such that ) ( ) ( 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 . 2010 Note that N must be an integer. . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 4
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 θ ω + + + = t e C j t e C t x rt 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 C n x α = ] [ where C = |C|exp( j θ ) and α = | α |exp( j ω 0 ) • Using Euler’s relation, ( ) ( ) θ ω α θ ω α + + + = n C j n C n x n n 0 0 sin cos ] [ 2010 The exponential decays if | α | < 1 and grows if | α | > 1 and has constant amplitude if | α | = 1. . R. Chen 8/31/ It is periodic if | α | = 1 and there exists integers m and N so that 0 0 f m = = ω L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 6 2 N π

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Real Exponential Signals • Discrete time: n C n x α = ] [ where C = |C|exp( j θ ) and α is real • There are 6 cases to consider, depending on the value of α 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 • Case 1: α = 1 constant signal • Case 2: α > 1 positive signal with exponential growth Case 3: 0 < < 1 positive signal with exponential decay 2010 • Case 3: 0 < α . R. Chen 8/31/ L. Review of key concepts from ECSE Review of key concepts from ECSE-303 303 8 adapted from B. Boulet
Real Exponential Signals • Case 4: α < - 1 signal alternates between positive and negative values with exponential growth • Case 5: 1 signal alternates between + C and C 2010 α = -1 and - . R. Chen 8/31/ L.

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