Taking the LaPlaceLaplace transform of a system given by a differential equation yields its transfer function H(s)$H(s)$. The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is stable. Then the region of convergence will include the imaginary axis. We know that the imaginary axis corresponds to the Fourier transform of the impulse response h(t)$h(t)$ of H(s)$H(s)$. Does this imply that the Fourier transform will also yield a causal h(t)$h(t)$?
In my book it says we can continue the LaPlaceLaplace transform analyticalyanalytically in the complex plane such that the whole LaPlaceLaplace plane is included in the region of convergence except for the poles. This would imply that I can take the Fourier transform of unstable systems by substituting s=jw$s=j\omega$ . How does that even make sense? Im so confused! And how is that related to the impulse response being causal then?
How is causality in LaPlaceLaplace transform related to Fourier transform?
How is causality in LaPlace transform related to Fourier transform?
Taking the LaPlace transform of a system given by a differential equation yields its transfer function H(s). The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is stable. Then the region of convergence will include the imaginary axis. We know that the imaginary axis corresponds to the Fourier transform of the impulse response h(t) of H(s). Does this imply that the Fourier transform will also yield a causal h(t)?
In my book it says we can continue the LaPlace transform analyticaly in the complex plane such that the whole LaPlace plane is included in the region of convergence except for the poles. This would imply that I can take the Fourier transform of unstable systems by substituting s=jw . How does that even make sense? Im so confused! And how is that related to the impulse response being causal then?
How is causality in Laplace transform related to Fourier transform?
Taking the Laplace transform of a system given by a differential equation yields its transfer function $H(s)$. The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is stable. Then the region of convergence will include the imaginary axis. We know that the imaginary axis corresponds to the Fourier transform of the impulse response $h(t)$ of $H(s)$. Does this imply that the Fourier transform will also yield a causal $h(t)$?
In my book it says we can continue the Laplace transform analytically in the complex plane such that the whole Laplace plane is included in the region of convergence except for the poles. This would imply that I can take the Fourier transform of unstable systems by substituting $s=j\omega$ . How does that even make sense? Im so confused! And how is that related to the impulse response being causal then?
How is causality in LaPlace transform related to Fourier transform?
Taking the LaPlace transform of a system given by a differential equation yields its transfer function H(s). The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is stable. Then the region of convergence will include the imaginary axis. We know that the imaginary axis corresponds to the Fourier transform of the impulse response h(t) of H(s). Does this imply that the Fourier transform will also yield a causal h(t)?
In my book it says we can continue the LaPlace transform analyticaly in the complex plane such that the whole LaPlace plane is included in the region of convergence except for the poles. This would imply that I can take the Fourier transform of unstable systems by substituting s=jw . How does that even make sense? Im so confused! And how is that related to the impulse response being causal then?