Ve216LectureNotesChapter1Part2

Ve216LectureNotesChapter1Part2 - Ve216 Lecture Notes...

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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 The Discrete-Time Unit Step Function Definition < = , , 1 ] [ n n n u The Discrete-Time Unit Step Function The Continuous-Time Unit Step Function Definition < = , , 1 ) ( n t t u Figure 1.38 (p. 44) Continuous-time version of the unit-step function of unit amplitude. Figure 1.39 (p. 44) (a) Rectangular pulse x ( t ) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x ( t ) as the difference of two step functions of amplitude A , with one step function shifted to the left by and the other shifted to the right by ; the two shifted signals are denoted by x 1 ( t ) and x 2 ( t ), respectively. Note that x ( t ) = x 1 ( t ) x 2 ( t ). The Discrete-Time Unit Impulse Function Definition = = , , 1 ] [ n n n Aka, Kronecker delta function Figure 1.41 (p. 46) Discrete-time form of impulse. Sampling Property of Unit Impulse The unit impulse can be used to sample the value of a signal at n=0 ] [ ] [ ] [ ] [ n x n n x = More generally ] [ ] [ ] [ ] [ n n n x n n n x- =- Relationship between Step and Impulse =- =- = =-- = ] [ ] [ ] [ ] 1 [ ] [ ] [ k n m k n m n u n u n u n The discrete-time unit impulse is the first difference of the discrete-time unit step The discrete-time unit step is the running sum of the discrete unit impulse (or, the superposition of delayed discrete-time unit impulses) The Continuous-Time Unit Impulse Function Definition 1 ) ( , ) ( = = - dt t t t Aka, Dirac delta function Figure 1.42 (p. 46) (a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration of a rectangular pulse of area a to approach zero. ) ( lim ) ( t x t = The Continuous-Time Unit Step Function We view the unit impulse as the limiting form of any pulse x (t) that is an even function of time t with duration and unit area....
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This note was uploaded on 10/26/2009 for the course EECS EECS 216 taught by Professor Dianguangma during the Spring '09 term at University of Michigan-Dearborn.

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Ve216LectureNotesChapter1Part2 - Ve216 Lecture Notes...

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