Suppose (for the sake of this problem) we know that if the injected current input to...

  

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Suppose (for the sake of this problem) we know that if the injected current input to an LTI system model of a nerve membrane is r(t) = et u(t) then the output membrane voltage is determined to be y(t) = [e-t-e-] u(t). Do the following: a. Using Laplace Transform analysis, find the system transfer function in the Laplace domain for this membrane and reduce it to its simplest form. b. Determine what the region of convergence (ROC) would be for this system to be causal. If the system operates within this ROC, is it stable (i.e. does the Fourier Transform exist ?) c. You should be able to surmise that this system can also be characterized by a differ- ential equation in the time domain. Without solving for this, what is the order of this LTI system? Suppose (for the sake of this problem) we know that if the injected current input to an LTI system model of a nerve membrane is r(t) = et u(t) then the output membrane voltage is determined to be y(t) = [e-t-e-] u(t). Do the following: a. Using Laplace Transform analysis, find the system transfer function in the Laplace domain for this membrane and reduce it to its simplest form. b. Determine what the region of convergence (ROC) would be for this system to be causal. If the system operates within this ROC, is it stable (i.e. does the Fourier Transform exist ?) c. You should be able to surmise that this system can also be characterized by a differ- ential equation in the time domain. Without solving for this, what is the order of this LTI system?

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