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Unformatted text preview: DiscreteTime Linear, Time Invariant Systems and zTransforms Linear, time invariant systems Continuoustime, linear, time invariant systems refer to circuits or processors that take one input signal and produce one output signal with the following properties. Both the input and output are continuoustime signals. The system is linear. This means that if the input signals x 1 ( t ) and x 2 ( t ) generate the output signals y 1 ( t ) and y 2 ( t ), respectively, and if a 1 and a 2 are constants, then the input signal a 1 x 1 ( t ) + a 2 x 2 ( t ) generates the output signal a 1 y 1 ( t ) + a 2 y 2 ( t ). The system is time invariant. This means that if the input signal x ( t ) generates the output signal y ( t ), then, for each real number s , the time shifted input signal x ( t ) = x ( t- s ) generates the time shifted output signal y ( t ) = y ( t- s ). Example 1 A simple example of a continuoustime, linear, time invariant system is the RC lowpass filter that is used, for example in amplifiers, to suppress the high frequency parts of signals. This is the electrical circuit +- x ( t ) R C i ( t ) +- y ( t ) with the voltage source x ( t ) viewed as the input signal and the voltage y ( t ) across the capacitor viewed as the output signal. If i ( t ) denotes the current in the circuit, the voltage across the resistor is Ri ( t ). If q ( t ) denotes the charge on the capacitor, then the voltage across the capacitor is y ( t ) = q ( t ) C . So by Kirchhoffs voltage law, x ( t ) = Ri ( t ) + y ( t ) Since i ( t ) = dq dt ( t ) = C dy dt ( t ), we have that RC dy dt ( t ) + y ( t ) = x ( t ) dy dt ( t ) + 1 RC y ( t ) = 1 RC x ( t ) This first order constant coefficient linear ODE is easily solved by multiplying it by the integrating factor (1) e t/RC . e t/RC dy dt ( t ) + 1 RC e t/RC y ( t ) = 1 RC x ( t ) e t/RC d dt ( e t/RC y ( t ) ) = 1 RC x ( t ) e t/RC Change t to in this equation and integrate both sides from =- to = t . Assuming that e t/RC y ( t ) tends to zero as t - , e t/RC y ( t ) = Z t- 1 RC x ( ) e /RC d y ( t ) = Z t- 1 RC x ( ) e ( - t ) /RC d For each possible input signal x ( t ), this determines the corresponding output signal y ( t ). This rule is obviously linear. To see that it is also time invariant, observe that the output signal that corresponds to the (1) In general, multiplying the equation y ( t ) + p ( t ) y ( t ) = g ( t ) by the integrating factor e R p ( t ) dt turns the left hand side into a perfect derivative. April 4, 2007 DiscreteTime Linear, Time Invariant Systems and zTransforms 1 input signal x ( t- s ) is Z t- 1 RC x ( - s ) e ( - t ) /RC d = - s = Z t- s- 1 RC x ( ) e ( + s- t ) /RC d = y ( t- s ) as desired....
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This note was uploaded on 01/27/2011 for the course MATH 267 taught by Professor Yilmaz during the Spring '10 term at The University of British Columbia.
- Spring '10