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Short answer: You can't. If an attacker can insert a signal that covers the whole bandwidth (e.g. a white signal, or at least one that has no spectral zeros) into the system (and he can do that over an arbitrarily long time, or add up observations), they will get an output, and can through the magic of correlation get the impulse response.


4

The system $$y[n] = x[n] \star (u[n]-u[n-2])$$ where $u[n]$ is the unit step function, has memory. Indeed the system is equivalent to $$y[n] = x[n] \star ( \delta[n] + \delta[n-1] ) \implies y[n] = x[n] + x[n-1]$$ and as it's clear from the given I/O relationship, the current value of the output $y[n]$, depends on the values input $x[n]$ at other times ...


3

Long answer: Let's model the information flow from your "hidden" IIR $X$ to your observable output $Y$ as $$ X \longrightarrow Y$$ Then, we call the amount of information you get per observation the *mutual information $I(X;Y)$; that information is the reduction of uncertainty about $X$ to be achieved by observing $Y$. We call the expected uncertainty ...


3

It is very strange phenomena that one object is completely dropped out of attention of researchers. It is Urysohn operator. First of all Urysohn is equivalent to multiple parallel Hammersteins and Urysohn followed by static nonlinearity is a model of any deterministic dynamic object, it maps any given input to any provided output. I obtained Ph.D. in ...


2

You can use pyvib to do frequency based subspace identification. Beware that there is no estimation of the initial state. It is possible to do optimization of the identified model, if the data is not perfectly linear. See the implemenentation, maybe you can use it, in case you want to do your own implementation. Somewhat incomplete example. Take a look at ...


2

The easiest is Urysohn adaptive filter: http://www.ezcodesample.com/UAF/UAF.html It can build nonlinear model by few lines of code. The theoretical details can be found here http://www.ezcodesample.com/NAF/index.html The site has downloadable coding sample. Besides UAF, the other common methods are: Kernel LMS, Voltera LMS, Neural networks, Point cloud. ...


1

Looks like you've done a lot of work on your projects. As @MarcusMüller said, by far the majority of people start with ReLU and go from there. It doesn't have the "vanishing gradient" problem that tanh has for example. All your questions are open ended but common for designing neural networks. There are so many "nobs to turn" to try and make your network be ...


1

True that a chirp signal helps to get the FRF, but every time we change the frequency we can't reach the steady state, so this will cause bias in the estimation. As an advice try to use the multisine excitation, they are more suitable for such cases.


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SIDPAC is a freely available program from software.nasa.gov. It is targeted toward aircraft system id problems however the underlying methods are applicable to other problem types.


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I believe the point of feeding white noise into the system is for the filter to adapt its coefficients before actually generating the signal $x[k]$. This would mean there are two "operating modes" for the system: coefficient adapting mode (in which white noise, a broadband signal, is used to adapt the filter to the feedback path), and performing mode (where ...


1

You do get a time-invariant response. Your code produces the same output for all three signals. In particular, it produces the same output for $y(\sigma^T\{x(t)\}))$ as for $\sigma^T\{y(x(t))\}$ (plots 2 and 3 in your code). It is hard to see in your case because you have shifted the signal $3\cdot 2\pi$ in time. Whats a cosine shifted by $6\pi$? The same ...


1

The chosen cost function is the mean squared error, i.e., the integral over a squared magnitude of the difference between frequency responses. The function $$E(e^{j\omega})=H(e^{j\omega})-\frac{B(e^{j\omega})}{A(e^{j \omega})}\tag{1}$$ depends on frequency, so you can't minimize it directly, unless you want to minimize it for exactly one frequency $\omega$,...


1

First of all, I want to thank Fat32 and Hilmar for answering the question. I'm sorry I couldn't answer before, because I had some external problems to solve. Finally I was able to solve the problem by using the reverse time property of the Zeta-Transform (despite having some difficulties with the initial conditions). Consider a non-minimum phase system ...


1

Here is one way to think about it: Let's say you have an unstable transfer function, which is unstable because of a single pole outside the unit circle. We can write this as $$H_0(z)=H_{stable}(z) \cdot \frac{1}{1-z^{-1} \cdot p}$$ That's a cascade of the stable and unstable part. Now we can multiply this with a unity filter that has a pole and zero at the ...


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An unstable system can be stablized by all-pass decomposition for exampe, but simply constructing the anti-causal impulse respone extension will not provide the same output; so I think it's not possible, but I'm not rigorous at this point. Following just shows why. Assume that your unstable and causal IIR filter has the impulse response $h_+[n]$ and by ...


1

Convolution is a linear operator. As such, it can be, at least theoretically, inverted. But it is infinite in length, and in coefficient amplitude precision. Which, in real world practice, cannot be reached. So the balance resides in what you call "protecting", and there might be some "privacy by design" possibilities: if the algorithm is a mere ...


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