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Does this mean we cannot map high-pass analog filters to high-pass digital filters using impulse invariance? In my opinion this is not correct albeit it is stated everywhere regarding this topic. The reason this is not correct is due to the lemma of riemann-lebegue which states that any fourier transform converges to zero towards infinity (for L1 functions ...

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This JAES paper gives a close form of the Fourier transform of the synchronized swept-sine (SSS) signal which has the same form as the ESS $$x(t) = \sin \big\{2\pi f_1L \big[\exp(t/L) -1 \big]\big\}$$ where $$L = \frac{1}{f_1} \mathrm{round}\left[\frac{\hat{T}f_1}{\ln(f_2/f_1)}\right]$$ and $\hat{T}$ is the approximate time length of $x(t)$. The authors ...

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This is a well-known result with no specific name. In signal processing, we usually call the kernel impulse response (as correctly mentioned in a comment by Marcus Müller). The response to a unit step signal is called step response. So what you've found is how to compute the impulse response from the step response. If $h(t)$ denotes the impulse response of a ...

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The Z-transform of the transfer function is $H(z) = Y(z)/X(z)$: the ratio of output to input Z-transforms. Matlab provides a ztrans() method for computing Z-transforms symbolically (I didn't know this existed until now). I'm assuming this is a homework problem, so I won't provide a complete solution.

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