Difference equation with variable coefficients in MATLAB

Question

Given the following difference equation

$$y[n]-\left(\dfrac{n}{n+1}\right)y[n-1] = n\cdot x[n]$$

How can we use MATLAB to solve it?

I know if the coefficients are constant we can simply use filter(b, a,x), but not sure how to do it with variable coefficients.

A loop would be a good idea... I mean, you can find $y[n]$ iteratively. — GKH, Jan 31 at 5:36
Well, in Octave I would do this: y(1) = -1; x = randn(1,100); for n = 2:100 y(n) = n*x(n) + (n/(n+1))*y(n-1); end — GKH, Jan 31 at 21:03

jomegaA · Accepted Answer · 2020-01-31 07:58:07Z

1

When $n=0$ you may need to know $y[-1]$. The current output $y(n)$ depends on the current input $x(n)$ and previous output $y(n-1)$ scaled.

answered Jan 31 at 7:58

jomegaA

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$\begingroup$ Suppose I know that, How would I go about solving the system. I think that is my question. $\endgroup$ – Morcus Jan 31 at 19:45
$\begingroup$ The input sequence $x[n]$ is known then $y[n]$ can be computed with $$y[n]= n\cdot x[n] + \left(\dfrac{n}{n+1}\right)y[n-1]$$ when $n=0$ use $y[-1]$ if available otherwise $y[-1]=0$ Do you need a MATLAB code example to solve above problem? But, I guess its not that complicated to implement. $\endgroup$ – jomegaA Jan 31 at 19:51
$\begingroup$ Well for $n=0;\, y[0] = 0$ nevertheless. $\endgroup$ – jomegaA Jan 31 at 19:55

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