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How do I find the transfer function (using the bilateral z-transform) of the problem below.

A stable LTI system with input x[n] and output y[n] is modeled by the difference equations

c[n] + (L/6-4)c[n+1] + (2L/3+6)c[n+2] + (L/6-4)c[n+3] + c[n+4] = x[n+2] - 2x[n+3] + x[n+4]

and

y[n] = x[n] - c[n-2] + 2c[n-1] - c[n]

Here c is an intermediate variable and L is a non-negative parameter.

My first thought was to fine a way to eliminate the intermediate variable, but I don't think that is possible. My other thought was to find the transfer function H in terms of both X and C. I would do this by following Y[z] = H[z]X[z], so H[z] = Y[z]/X[z]. I just don't know what to do with the C terms...

Insight into this would be great. Also, if this is not the right place to ask the question, help in directing it to the right place would be appreciated as well. Thanks!

1 Answers 1

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Although the question is old, I would like to suggest a solution:

First find the transfer function relationship between c[n] and x[n] using the first equation. Let assume that $C[z]=G[z]X[z]$.

Also convert the second equation into the z-domain: $Y[z]=X[z]+f(z)C[z] \rightarrow Y[z]=X[z]+f(z)G[z]X[z] \rightarrow Y[z]=(1+f(z)G[z])X[z]$.

Thus the asked transfer function becomes $H[z]=1+f(z)G[z]$.

Let me correct if there are errors in my derivation.

Regards,