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2

You give a transfer function in $z^{-1}$, but for the purposes of making a state-space model, you need a transfer function in $z$ (more on that later). So your $H$ becomes $$H(z) = \frac{b_0 z + b_1}{z^2 + a_1 z + a_2}$$ When you invert that it's improper, which shows that you need to know information one time step into the future. That's not possible (or, ...


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It is dealt by calibration which requires having a good characterization of the filter, as Stanley mentioned. You asked about isolation: There are different implementations of how saturation is avoided during transmission in pulse-Doppler radar, however they virtually all involve physically isolating the receiver. The time while transmitting is your ...


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The delay is dealt with by knowing it. It is calculated and measured and any residual error is minuscule compared with any other uncertainty. It’s dealt with design and calibration.


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Piecewise defined convolution integrals are a distinguishing part of practical engineering work. Those piecewise intervals indicate on-off switching of various devices such as relays, capacitors, heaters, motors, transistors, etc. And they make the integrals tedius to evaluate (assuming the antiderivate is already available) unlike the case where the limits ...


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Once you understand the intuition of convolution using the link in the comments, you can work on deriving the equations. These problems are a bit tedious at first and can cause confusion if you try to do too much at once. I'm uploading my hand written notes on how to break the problem down. The key is to think about what happens when there is no overlap, ...


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The easiest way to solve the problem is using the knowledge of eigenfunctions of LTI system and the consequence that an LTI system's response to a sinusoidal input $x(t)=A\cos(\omega_0t+\phi)$ is given by $$y(t)=A\big|H(\omega_0)\big|\cos\big(\omega_0t+\phi+\angle H(\omega_0)\big)\tag{1}$$ where $H(\omega)$ is the system's frequency response. This is ...


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Apply the eigenfunction property of the LTI system after decomposing the sinusoidal input by Euler identity. The eigenfunction property of the LTI system states that $$ x(t) = e^{j \omega_0 t} \implies y(t) = H(\omega_0) e^{j \omega_0 t} $$ where $H(w)$ is the frequency response of the LTI system. Euler identity states that : $$ x(t) = \cos(t) = 0.5 \{...


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Actually, the question is not clear. But the answers carified what you've asked for. You can build a system of linear algebraic equations as some people advice, that is correct, but the matrix built on known signal is so-called poorly conditioned. That means when you try to invert it, the truncation errors kill solution and you receive random numbers in ...


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In general, one method to handle the issue that generalizes substantially to a problem of extracting two or more components is to take the spectra G¹, G² ⋯, Gⁿ of signals #1, #2, ..., #n, tabulate the total square Γ(ν) = |G¹(ν)|² + |G²(ν)|² + ⋯ + |Gⁿ(ν)|² at each frequency ν, and normalize G₁(ν) ≡ G¹(ν)* / Γ(ν), G₂(ν) ≡ G²(ν)* / Γ(ν), ..., G_n(ν) ≡ Gⁿ(ν)* / ...


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For those who prefer an approach using limits instead of integrals, it follows like this: Consider the following definition of the unit-impulse located at the origin $t=0$ : $$ \delta(t) = \lim_{\Delta \to 0} \delta_{\Delta}(t) \tag{1} $$ where the classical function $\delta_{\Delta}(t)$ is defined as $$ \delta_{\Delta}(t) = \begin{cases} {\frac{1}{\...


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[While commenting on Matt's answer, I tried to find a different path. I somehow failed to do so, but it is written, so] A folk (and false) interpretation of the Dirac $\delta(t)$ is that this is would be a function (false, in the classic sense, it cannot be evaluated; it should be understood as an application or operator on other functions, and called ...


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