113_1_convolution_continuous_domain

113_1_convolution_continuous_domain - Chapter 4 Convolution...

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Chapter 4 Convolution Outline Convolution approach to LTI system analysis Evaluation of the convolution integral Properties of convolution Determining the impulse response General property of convolution 4.1 Convolution Approach to LTI System Analysis The convolution approach is a time-domain method for analyzing LTI systems that fully exploits the linearity and time-invariance properties. 4.1.1 General Approach The convolution approach to LTI systems analysis is based on the general approach described in Section 3.4. The family of elementary signals or test signals is taken to be the set of delayed impulses. An outline of the approach is as follows: Test signal e ( t )= δ ( t ). Family of test signals e α ( t δ ( t ατ ). Express x ( t ) as a weighted sum of e α ( t ) x ( t ··· + x 1 δ ( t + τ )+ x 0 δ ( t x 1 δ ( t τ x 2 δ ( t 2 τ = X α x α δ ( t ατ ) The critical issue here is the method for determining the values of the coefficients { x α } . In examining this issue, we will ±nd that for a general signal x ( t ), the sum
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CHAPTER 4. CONVOLUTION 60 must proceed in the limit to an integral. The test signal parameter will become the continuous delay variable τ and x ( t ) must be represented as the integral x ( t )= Z α x α δ ( t τ ) Determine the response of the LTI system H to δ ( t ): δ ( t ) H h ( t ) The response, denoted by h ( t ), is known, understandably, as the impulse response of the system. Apply linearity and time invariance to determine the response to x ( t ): x 0 δ ( t ) H −→ x 0 h ( t ) x 1 δ ( t τ ) H x 1 h ( t τ ) . . . x ( t X α x α δ ( t ατ ) H y ( t X α x α h ( t ατ ) In the general case, we will Fnd that the representation of x ( t ) requires that the sum must become an integral. Thus, the representation of y ( t ) will become an integral over all weighted, delayed impulse responses x ( t Z α x α δ ( t τ ) H y ( t Z α x α h ( t τ ) To develop this approach, it is necessary to determine how to represent an input signal by delayed impulses, how to determine the impulse response h ( t ), and how to evaluate a response y ( t ). The signal representation problem is described in the next section, the eval- uation of the response is given in the following section, and Fnally, methods are considered for determining the impulse response of a system. 4.1.2 Representation of Signals by Impulses We will be considering the use of the impulse δ ( t ) as the test signal e ( t ). We need to determine how to express an input x ( t ) as a weighted sum of impulses. ±irst choose a narrow pulse p ( t ) as an approximation to the test signal e ( t ):
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CHAPTER 4. CONVOLUTION 61 Approximate x ( t ) by a series of these pulses: The pulse centered at t α = α ∆is x ( α ∆) p ( t α ∆) The approximation is: x ( t ) X α = −∞ x ( α ∆) p ( t α ∆) This approximation improves as ∆ gets smaller and more pulses are used in the approxima- tion. Now, let ∆ 0: x ( t ) = lim 0 X α = −∞ x ( α ∆) p ( t α ∆) = lim 0 X α = −∞ x ( α ∆) ± 1 p ( t α ∆) ² , As ∆ 0: α
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113_1_convolution_continuous_domain - Chapter 4 Convolution...

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