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Note that in general the Fourier transform of a function is a complex-valued function, so in general it is not only positive or negative. Roughly speaking, the magnitude of the Fourier transform says something about the presence of certain frequencies components in a signal, regardless of the phase (or sign, in the real-valued case). The phase determines ...


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For an LTI system with impulse response $h(t)$ and transfer function $H(s)=\mathcal{L}\{h(t)\}$, an input signal $x(t)=e^{st}$ (note: no multiplication with the unit step function $u(t)$), results in an output $$\begin{align}y(t)&=(x\star h)(t)\\&=\int_{-\infty}^{\infty}h(\tau)x(t-\tau)d\tau\\&=\int_{-\infty}^{\infty}h(\tau)e^{s(t-\tau)}d\tau\\&...


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