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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 The DiscreteTime Unit Step Function Definition < = , , 1 ] [ n n n u The DiscreteTime Unit Step Function The ContinuousTime Unit Step Function Definition < = , , 1 ) ( n t t u Figure 1.38 (p. 44) Continuoustime version of the unitstep 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 DiscreteTime Unit Impulse Function Definition = = , , 1 ] [ n n n Aka, Kronecker delta function Figure 1.41 (p. 46) Discretetime 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 discretetime unit impulse is the first difference of the discretetime unit step The discretetime unit step is the running sum of the discrete unit impulse (or, the superposition of delayed discretetime unit impulses) The ContinuousTime 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 ContinuousTime 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 MichiganDearborn.
 Spring '09
 DianguangMa

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