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Lesson_13 - Lesson Title Impulse Invariant FIR Lesson...

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Lesson Title: Impulse Invariant FIR Lesson Number: 13 (Section 6-1 to 6-2) Background: In Chapter 5, elementary FIR filters were discussed. The Chapter began by establishing the rules and purpose of convolution and ended with a discussion of some of the basic properties of FIRs supported with selected demonstrations. In Chapter 6, the behavior of such systems are explored in the frequency domain. Discrete-Time System Frequency Response It has been established that the response of an LTI consists of transient and steady- state responses. The focus of initial Chapter 6 studies is the analysis of the steady state responses of LTI systems in the frequency domain. This analysis gives rise to the concept of the steady-state frequency response of the LTI system. The development of a steady state frequency response framework can begin with the now familiar convolution sum: [ ] [ ] k n x h n y N k k - = = 0 1. Consider the special case where the input is the discrete-time signal x[n] is given by: [ ] ( 29 π ω π ϖ ω φ ω < - = = + : / ; s n j T Ae n x 2. which represents a complex exponential signal of magnitude A, normalize frequency ω , and phase offset φ . In continuous time, the signal is assumed to satisfy: ( 29 ( 29 φ ϖ + = t j Ae t x 3. Upon substituting Equation 2 into Equation 1, the response of an (N+1)-order FIR to a complex exponential signal is given by: [ ] ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 n j j n j j k j N k k k j n j j N k k k n j N k k e Ae e Ae e h e e Ae h Ae h n y ω φ ω φ ω ω ω φ φ ω ω = = = = - = - = + - = 0 0 0 4. where the operator is called the frequency response function or simply frequency response . Specifically, the frequency response can be defined to be: ( 29 ( 29 ( 29 C = = - = k j N k k j e h e H ω ω ω 0 5
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which is a complex function of the normalized frequency ω . It should be noted that Equation 4 is restricted to the case where the input is a complex exponential [ ] ( 29 φ ω + = n j Ae n x . The exponential form of the input also suggests (using Euler’s equation) that Equation 5 can expand the support the study of a system’s steady-state sinusoidal response. Whether the input is an exponential or sinusoid, the frequency response function of the filter can be represented in magnitude/phase form or real and
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