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LCS_9_notes

# LCS_9_notes - 1 LINEAR CIRCUITS AND SIGNALS Some Basic...

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1 LINEAR CIRCUITS AND SIGNALS Some Basic System Theoretic Concepts Department of Electrical and Computer Engineering University of Miami I. P RELIMINARIES Recall that u ( t ) U ( ω ) -→ h ( t ) H ( ω ) -→ y ( t ) = ( h * u )( t ) Y ( ω ) = H ( ω ) U ( ω ) . (1) The zero-state response y ( t ) of an LTI system with frequency response H ( ω ) to an arbitrary input u ( t ) is given by the convolution formula y ( t ) = ( h * u )( t ) . Here h ( t ) H ( ω ) . As we demonstrate later, this convolution formula turns out to be more general than the FT formula Y ( ω ) = H ( ω ) U ( ω ) . Indeed, while the FT H ( ω ) of h ( t ) may not exist, the convolution formula is always valid for a LTI system. We will also discuss the notions of causality and stability and how they are related to the impulse response of a LTI system. II. I MPULSE R ESPONSE AND THE Z ERO -S TATE R ESPONSE A. Measuring the Impulse Response Two methods of determining the impulse response of a system in a laboratory setting: (a) Note that ( δ * h )( t ) = h ( t ) ⇐⇒ lim 0 ( p * h )( t ) = h ( t ) , where p ( t ) = 1 rect t . (2) So, application of p with decreasing should result in an output that converges to h ( t ) . If convergence is not forthcoming (usually because of the presence of an impulsive signal in the impulse response), then the next method can be used instead. (b) Take the unit step response and use the following property applicable to LTI systems: 1( t ) -→ LTI -→ ( h * 1)( t ) = d dt 1( t ) -→ LTI -→ ( h * δ )( t ) = h ( t ) . (3)

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2 Example Suppose the unit step response of a LTI system is y ( t ) = e - t 1( t ) . To determine the system’s impulse response, let us use approach (b) described above. h ( t ) = y (1) ( t ) = e - t δ ( t ) - e - t 1( t ) = δ ( t ) - e - t 1( t ) . Can we use approach (a) instead? Note that ( p * h )( t ) = [ p * ( δ ( t ) - e - t 1( t ))]( t ) = p ( t ) - p * e - t 1( t ) . So lim 0 ( p * h )( t ) = lim 0 p ( t ) - ( δ ( t ) * e - t 1( t )) = lim 0 p ( t ) - e - t 1( t ) . In a laboratory setting, it will be impossible to determine the area corresponding to p ( t ) as 0 , and therefore approach (a) will not be of much use in this scenario. B. Response of an LTI System to Arbitrary Inputs We now demonstrate the convolution formula related to the zero-state response of an LTI system without recourse to frequency domain notions.
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LCS_9_notes - 1 LINEAR CIRCUITS AND SIGNALS Some Basic...

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