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Ve216LectureNotesChapter1Part2 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009
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The Discrete-Time Unit Step Function Definition < = 0 , 0 0 , 1 ] [ n n n u
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The Discrete-Time Unit Step Function
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The Continuous-Time Unit Step Function Definition < = 0 , 0 0 , 1 ) ( n t t u
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Figure 1.38 (p. 44) Continuous-time version of the unit-step function of unit amplitude.
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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 ½ and the other shifted to the right by ½; 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 ).
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The Discrete-Time Unit Impulse Function Definition = = 0 , 0 0 , 1 ] [ n n n δ Aka, Kronecker delta function
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Figure 1.41 (p. 46) Discrete-time form of impulse.
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Sampling Property of Unit Impulse The unit impulse can be used to sample the value of a signal at n=0 ] [ ] 0 [ ] [ ] [ n x n n x δ δ = More generally ] [ ] [ ] [ ] [ 0 0 0 n n n x n n n x - = - δ δ
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Relationship between Step and Impulse = -∞ = - = = - - = 0 ] [ ] [ ] [ ] 1 [ ] [ ] [ k n m k n m n u n u n u n δ δ δ 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)
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The Continuous-Time Unit Impulse Function Definition 1 ) ( 0 , 0 ) ( = = - dt t t t δ δ Aka, Dirac delta function
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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 Δ of a rectangular pulse of area a to approach zero. ) ( lim ) ( 0 t x t = δ
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The Continuous-Time Unit Step Function We view the unit impulse as the limiting form of any pulse x Δ (t) that is an even function of time t with duration Δ and unit area. ) ( lim ) ( 0 t x t = δ
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Sampling Property of Impulse The unit impulse can be used to sample the value of a signal at t=0 ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( 0 0 0 t t t x t t t x t x t t x - = - = δ δ δ δ
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Sifting Property of Impulse The integral sifts out the value of x(t) at time t=t 0 ) ( ) ( ) ( 0 0 t x dt t t t x = - - δ
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