convolution_06 - Convolution Impulse Response Of A System...

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Convolution Impulse Response Of A System Discrete-Time Systems Linear Time-Varying Consider the response of the system when the input is a unit impulse function . In general, if the input is delayed or advanced, the system response will change. At the input of the system apply an impulse signal, , delayed by k units. Denote the ( ) nk δ output to this signal . Here the variable is n and the delay is a fixed value k . ( ) k hn Symbolically we can represent this as This information can be used to determine the response to any input signal, x(n) . To understand how this can be done, consider the following sum, () () ( ) k xn xk n k =−∞ =− This is just the sifting sum discussed previously. Its plausibility was also demonstrated. Now it will be demonstrated in a different way. Discrete-time signals have been considered to be functions of an integer variable. But they can also be viewed as an ordered sequence of numbers. This view is very useful at times. The following development is based on this idea. Consider the construction on the following page. This construction depends on the idea that sequences of numbers add like vectors, and a sequence can be multiplied by a constant just as a vector is multiplied by a scalar.
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3 It is clear that all of the weighted impulse sequences (discrete-time signals) add up to the signal x(n) . This point of view it is sometimes called the superposition sum. Still a third view of this sum is that it is an expansion of an arbitrary signal x(n) into a weighted sum of impulses. For the purposes of system analysis this view is also very useful. Using the linearity property ( ) ( ) ( ) ( ) k xk n k xkh n δ −→ Each value of k corresponds to a different input and output. For Linear Systems Sum of inputs Sum of responses to each input Thus () ( ) () () k kk ∞∞ =−∞ =−∞ ∑∑ But the left-side is x(n) and so the right-side is the system response, i.e., it must be y ( n ), that is () () () k k y nx k h n =−∞ = This shows the output can be determined from a knowledge of the input and the impulse responses . ( ) k hn Unfortunately, for sequences of infinite length, there are an infinite number of responses that must be determined to find the solution using this procedure. ( ) k
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4 Time-Invariance to the Rescue For a system that is both linear and time-invariant (LTI), denote the system response to the unit impulse function h(n) , that is ( ) ( ) nh n δ Then by definition of time-invariance, () () ( ) ()( ) and because of linearity (homogeneity) nk hnk xk n k xkhn k −→ Again using linearity (superposition) ( ) ( ) ( ) ( ) kk xn yn ∞∞ =−∞ =−∞ =− = ∑∑ This shows that knowledge of the unit impulse response of a LTI, discrete-time system can be used to determine the response of the system to any input. That is, () ()( ) k y nx k h n k =−∞ This sum is called a convolution .
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convolution_06 - Convolution Impulse Response Of A System...

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