# How to perform model fitting for system identification

I am having a really hard time in understanding how to formulate a model say linear AR model to represent a communication channel or maybe any motion. I have the experimental data representing the kinematics of robot movement. In the case of a communication channel, it will have a medium, transmitter and receiver. So, will the medium be represented by AR model or do we represent the transmitter and receiver by these models? How to decide the order of the system to be chosen initially. I am not from the area of signal processing but need to use this for a part of my research work. I have gone through literature review and the background of Kalman filter, Least square method, recursive least square for estimation purpose. But I cannot find a good example which shows how the parameters are estimated from the time series or motion data or speech modeling. Can somebody explain or point out the starting point and how to go about it? I know this is a very broad topic but an explaination with a small example will be really an eye opener for a beginner.

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In any signal processing problem, there are usually two components: the signal model and the channel model.

The signal model is the mathematical description of how your ideal signal, call it $s(t)$ is generated.

The channel model is the mathematical description of how your channel corrupts or alters the signal.

One aim in telecommunications is to look at the received signal $r(t)$ --- $s(t)$ corrupted by the channel --- and try to undo the effects of the channel.

For example, suppose $$s(t) = A \sin(\omega_0 t + \phi)$$ and suppose the channel is: $$r(t) = h(t)*s(t) + n(t)$$ where $h(t)$ is the impulse response of the channel, and $n(t)$ is some additive noise.

In this context, your "AR" filter is the $h(t)$.

If the context is in robotics or kinematics, then things get a little more complicated.

Did you open the "Algorithm" tab on the link in your comment? It's pretty straighforward:

The polyfit MATLAB file forms the Vandermonde matrix, $\bf V$, whose elements are powers of $x$:
$v_{i,j} = x_i^{n - j}$
It then uses the backslash operator, $\backslash$, to solve the least squares problem ${\bf V} p \approx y$.

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Thank you for this reply. I am stuck in how to fit a data with a polynomial of order n as is shown here mathworks.com/help/matlab/ref/polyfit.html . I cannot understand the theory behind all those complex functions so as to solve by hand as it is a research work and simply calling these inbuilt functions do not help.Moreover, reading the documentation makes it more complicated. – SKM Feb 13 '13 at 21:43
So it uses the LMS algorithm for estimation?And what is the reason for choosing this Vandermonde matrix? But system identification needs to use prediction as well, so don't we need linear predictors?In general, given an experimental data, how would one begin modeling using mathematical techniques actually pertaining to signal processing without using MATLAB functions? – SKM Feb 13 '13 at 22:37
Well, it solves a least squares problem. Be careful: "system identification" is one thing, but the problem you describe is another. I suggest you pose the question you really want answered. There are not enough specifics here to answer comprehensively. – Peter K. Feb 13 '13 at 22:50