Transcription of Lecture 4: Convolution
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Lecture 4: Convolution
4 ConvolutionIn Lecture 3 we introduced and defined a variety of system properties towhich we will make frequent reference throughout the course. Of particularimportance are the properties of linearity and time invariance, both becausesystems with these properties represent a very broad and useful class and be-cause with just these two properties it is possible to develop some extremelypowerful tools for system analysis and linear system has the property that the response to a linear combina-tion of inputs is the same linear combination of the individual responses. Theproperty of time invariance states that, in effect, the system is not sensitive tothe time origin. More specifically, if the input is shifted in time by someamount, then the output is simply shifted by the same importance of linearity derives from the basic notion that for a linearsystem if the system inputs can be decomposed as a linear combination ofsome basic inputs and the system response is known for each of the basic in-puts, then the response can be constructed as the same linear combination ofthe responses to each of the basic inputs.
=f xT) hT(t) dr If Time-Invariant: hkj t) = ho(t -kA) h,(t) = he (t - r) LTI: 56(t - kA) A hk(t) A +01 v(t) f x(r) h(t-7) dr-1 -0 Convolution Integral TRANSPARENCY 4.7 Interpretation of the convolution integral as a superposition of the responses from each of the rectangular pulses in the representation of the input. x(t) 0 t ti x (0) x(A) x(kA ...
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