lect10 - CONTINUOUS TIME SYSTEMS A continuous time system...

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CONTINUOUS TIME SYSTEMS A continuous time system is one which takes as input a continuous time signal and generates a continuous time signal as output. Just as in the discrete time case, the systems of most interest to us in this course are the Linear, Time invariant systems. Let L denote the operator mapping the input of the system to its output, i.e. )} ( { ) ( t x L t y = . The system is Linear if, for any complex numbers, 1 a and 2 a , and any inputs, ) ( 1 t x and ) ( 2 t x , we have } ) ( { )} ( { } ) ( ) ( { 2 2 1 1 2 2 1 1 t x L t x L t x t x L a a a a + = + . (10.1) In other words, if ) ( 1 t y is the output due to ) ( 1 t x , and ) ( 2 t y is the output due to ) ( 2 t x , then the input ) ( ) ( 2 2 1 1 t x t x a a + results in the output ) ( ) ( 2 2 1 1 t y t y a a + . It easily follows from this that for a Linear system, arbitrary linear combinations of inputs give rise to the same linear combinations of the respective outputs, as shown in the diagram below. x(t) y(t) S ) ( t x i i a ) ( t y i i a L
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The system is called Time Invariant if, for any input, ) ( t x , and corresponding output, ) ( t y , an arbitrary time shift of the input, say ) ( t - t x , yields the correspondingly time shifted output ) ( t - t y , as depicted below. A system that is both Linear and Time Invariant is called a Linear, Time Invariant (LTI) system. IMPULSE RESPONSE The Impulse Response of an LTI system is the response of the system to the input ) ( t d , and is usually denoted by ) ( t h . The diagram below illustrates the situation. We will see next that the impulse response completely determines the behavior of an LTI system. ) ( t - t x ) ( t - t y L ) ( t d ) ( t h L
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THE CONVOLUTION INTEGRAL FOR LTI SYSTEMS Using nothing more than the linearity and time invariance properties of an LTI system, we can easily show that the output is the convolution of the input and the impulse response. Indeed, an arbitrary input, ) ( t x , can be written as - - = t t d t d t x t x ) ( ) ( ) ( . (10.2) Since the response to ) ( t d - t is ) ( t - t h (by time invariance), and the ) ( t x can be thought of as weighting coefficients, the output becomes (using Linearity) - - = t t t d t h x t y ) ( ) ( ) ( . (10.3) The integral in (10.3) is known as the convolution integral, and is usually denoted by ) ( ) ( t h t x . Thus - - t t t d t h x t h t x ) ( ) ( ) ( ) ( . The diagram below illustrates the situation. ) ( t x ) ( ) ( ) ( t h t x t y = h(t)
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Note that once the impulse response, ) ( t h , of an LTI system is known, the response to any arbitrary input, ) ( t x , can be found by convolving ) ( t x with ) ( t h . In this sense, the impulse response completely determines an LTI system, and it is fairly common to label the LTI system block with the impulse response, as above. The convolution integral can be written alternatively in the form
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lect10 - CONTINUOUS TIME SYSTEMS A continuous time system...

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