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What are the best approaches to characterize an unknown discrete time LTI system? I believe one of the approaches is to input a known input and measure the output to find a transfer function. What are some more approaches to better characterize this black box system?

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  • $\begingroup$ Giving an impulse input to your black box will give the impulse response of the system which can be used to calculate the output for an input x[n] $\endgroup$ – DSP Novice Apr 6 at 6:12
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    $\begingroup$ This is super broad. It's 90% of what the scientific field system identification is about. I know a library with literally multiple 100 kg of books on how to identify systems. You need to narrow things down, with a model of what it might be, and restrictions on what you can and can't do. $\endgroup$ – Marcus Müller Apr 6 at 7:13
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One option from the realms of adaptive filtering is the Least Mean Squares (LMS) filter depicted below:

enter image description here

The idea is you take the output of the unknown system, compare it with the output of your adaptive filter and minimize the difference by tweaking the filter coefficients, using a LMS algorithm. When the error $e(n)$ is zero (or more often, lower than a threshold you define) you found your unknown system's response.

There are countless resources where you can learn about LMS. In this case, given

the input vector: $$X(i) = [x(i) \space x(i-1) \ldots x(i-N+1)]^T$$ and the coefficients vector: $$W = [w_0 \space w_1 \ldots w_{N-1}]^T$$

The output of the adaptive filter is: $$\hat{y}(i)=X(i)^TW$$

Now define the cost function to be: $$e(i)^2 = \big(d(n)-\hat{y}(i)\big)^2$$ It can be shown that the gradient of the cost function with respect to the coefficients is:$$\hat{\nabla}(i) = \frac{\partial e(i)^2}{\partial W} = -2e(i)X(i)$$ Since the gradient points towards the maximum of the cost function you update the coefficients in the opposite direction: $$\hat{W}(i+1) = \hat{W}(i)-\mu\hat{\nabla}(i) = \hat{W}(i)+2\mu e(i)X(i)$$ where $\mu$ controls the adaptation speed. I've ignored a lot of details but this is the idea.

Note that the filter will produce a frequency response that is adapted to you input signal $x(n)$. It's not guaranteed for any other frequency. For example, in the response depicted below, my $x(n)$ is composed of 3 tones. You can see only at those particular frequencies does the filter (N=3,6,10) match the true response (H1(z)). So consider feeding it a wideband signal like white noise.

frequency response

If you want to learn more you can check for example:

Wikipedia article

B. Widrow, Adaptive Signal Processing, Prentice-Hall, 1985

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I think the best possible way to get the transfer function of a LTI blackbox would be giving complex exponential as input. Because complex exponential $Ae^{j\omega_0}$ are eigen functions of LTI systems. So output will be $$ Ae^{j\omega_0} \rightarrow H(e^{j\omega_0})Ae^{j\omega_0}. $$

If you have a reasonable idea of the LTI Blackbox, can restrict your $\omega_0$, to a range of frequencies. The magnitude response of output will be $$ M(\omega_0) = |H(e^{j\omega_0})|A $$ and the phase response will be $$ \angle(\omega_0) = \angle(H(e^{j\omega_0})). $$ If you have magnitude and phase response of output, you can easily remove the known input magnitude/phase to get the Black-box system's characteristics.

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