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**Unformatted text preview: **Digital Signal Processing Dr. Fred J. Taylor, Professor Lesson Title: Introduction to Frequency Response Lesson Number: [29] (Chapter 10) Background: In Chapter 9 the concept of continuous-time signals were claimed to be classified an extensions of discrete-time signals. Similarities include the concept of causality, signal type (e.g., unit step), delays, and so forth. It should be appreciated, however, that they are two distinctly different signal and system classes. One important differences is in the definition and properties of unit impulses as they apply to the study of continuous- and discrete-time signals. In the study of discrete-time signals, a Kronecker delta function was introduced and mathematically modeled as: K [k]={1 if k=0, 0 otherwise} 1. The continuous-time analog called the delta distribution of Dirac impulse distribution (often referred to incorrectly as a delta function) was given by: ( 29 , 1 = - d 2. This definition requires that an impulse be infinitely thin and infinitely tall about t=0. Even though this signal is physically impossible to create, whereas the Kronicker delta is physically realizable, it is unquestionably important to the mathematical study of continuous-time signals and systems. In Chapter 9 the sampling property was established to be: ( 29 ( 29 ( 29 x d x =- - 3 In Chapter 9, the output of a linear time-invariant (LTI) system was motivated by the study of discrete-time signals and systems. The output to an at-rest LTI system was defined in terms of a convolution operation and denoted y(t)=x(t)*h(t), where y(t) is the output, x(t) is the input, and h(t) is the systems impulse response. The convolution integral was given by: ( 29 ( 29 ( 29 ( 29 ( 29 d t x h d t h x t y - -- =- = 4 Later, it was shown that transform methods ( Laplace transform ) can be used to bypass the tedious details of performing the complicated integration using what was called the convolution theorem for Laplace transforms. Example An at-rest LTI system has a known impulse response given by h(t)= e- t , where =105.361. What is the systems step response? Chapter 10 - 1 Digital Signal Processing Dr. Fred J. Taylor, Professor From the definition of the convolution integral, one obtains: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ) ( 1 t u e e e e e e d e e d e d t h x t y t t t t t t t t t ------ -- =- = = = =- = where y(0)=0, y(0.1)~(1 ) and y( )=(1)....

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