Lesson_260 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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EEL 3135: Dr. Fred J. Taylor, Professor Lesson Title: Continuous-Time Signal and Systems Lesson Number: 26 (Section 9-5 to 9-10) Background: Continuous-time signals where introduced in Chapter 9. The concept of linear convolution was also defined and claimed to have the fundamental importance to continuous time signal processing as it held in the study of discrete-time signals and systems. Along with the linear convolution, several other important continuous-time signals were introduced including unit impulses ( δ (t)) and unit step (u(t)). Convolution Discrete-time convolution was a fundamentally important to the study of discrete-time linear systems. A completely complementary operation, called continuous-time convolution exists for continuous-time signals and systems. Specifically, continuous-time convolution is defined by: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 τ d h t x d t h x t x t h t y - - - = - = = 1. where h(t) is the linear system’s impulse response, x(t) the system input, and y(t) is the system’s output. Continuous-time convolution is associative, commutative, and distributive. Example Simulate the convolution of two 1 second pulses p(t)=u(t)-u(t-1). The outcome is shown in Figure 1. >> t=0:.01:2-.01; {200 clock ‘ticks’ over 2 seconds} >> p=[ones(100,1);zeros(100,1)]; {1 second pulse} >> subplot(1,2,1); >> plot(t,p) >> y=conv(p,p); {convolution} >> t=0:.01:4-.02; {400 clock ‘ticks’ over 4 seconds} >> subplot(1,2,2); >> plot(t,0.01*y); {note scale factor of 0.01} 1
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EEL 3135: Dr. Fred J. Taylor, Professor Figure 1: Convolution of 2 pulses. One second pulse shown over 2 second interval on left, convolution of 2 pulse shown over 4 second interval on right. End of example ----------------------------------
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Lesson_260 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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