Given two inputs $\: x_1[n]\: x_2[n]\:$Is the system $\:y[n]=x_1[n]\times x_2[n]\:$ linear ?

My Approach:

$(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$

$(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$

$(ax_1\times x_2)[n]+(b x_3\times x_4)[n]= a S_1[n]+ b S_2 [n] \rightarrow a Y_1[n]+bY_2[n]$

If that is correct it is linear. But I am not sure about this solution. I will appreciate any help.

  • $\begingroup$ Is there a comma missing in $\: x_1[n]\: x_2[n]\:$???? What is the meaning of the symbol $\times$? Is it multiplication? If so, how is $(x_1\times x_2)[n]$ related to $\: x_1[n]\: x_2[n]\:$ $\endgroup$ – Dilip Sarwate Apr 19 '18 at 16:03

To be linear you must have a function $f$ that satisfy:

$ f(a\overline {x_1} + b \overline{x_2}) = af(\overline{x_1})+ b f(\overline{x_2}) $

I define $ \overline{x}=\left [\begin{matrix}\overline{x}(1) \\ \overline{x}(2) \end{matrix} \right] $ the vector of your two inputs at instant $n$

For each $n$ you have $ y=f(\overline x)=\overline{x}(1) \cdot \overline{x}(2) $

Substituting to linearity definition: $ f(a\overline {x_1} + b \overline{x_2}) = [a\overline{x_1}(1)+b\overline{x_2}(1)] \cdot [a\overline{x_1}(2)+b\overline{x_2}(2)] $

That is not equal to $ af(\overline{x_1})+ b f(\overline{x_2}) = a\overline{x_1}(1)\overline{x_1}(2)+ b\overline{x_2}(1)\overline{x_2}(2)$

It means that this system is not linear


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