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$y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$

If I let $(\frac{5\pi}{8}*n+\frac{\pi}{4})=x[n]$, I get the system as non linear. But one of my teacher says it is linear, whereas another teacher says it is non linear.

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  • $\begingroup$ systems are classified as "linear" or "nonlinear". not signals. signals are what they are. $\endgroup$ Sep 13, 2021 at 5:17
  • $\begingroup$ Oh so is it linear system or non linear system? Thanks for the input tho. $\endgroup$ Sep 13, 2021 at 5:34
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    $\begingroup$ there is no system. it's a signal. $\endgroup$ Sep 13, 2021 at 5:48
  • $\begingroup$ So the system's output is always the same, regardless of the input? There's no $x[n]$ in your equation ... $\endgroup$
    – Matt L.
    Sep 13, 2021 at 6:22
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    $\begingroup$ @MattL. What about a system that maps everything to 0 ... ;-). $\endgroup$
    – Jazzmaniac
    Sep 13, 2021 at 9:03

1 Answer 1

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If the input is $$ x[n] = (\frac{5\pi}{8}*n+\frac{\pi}{4}). $$ Then the system is $y[n]=cos(x[n])$, which is a non-linear system.

If the input is $$ x[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4}). $$ Then the system is $y[n] = x[n]$, which is a linear system.

A key criteria to judge if a system is linear or non-linear is to determine if the system is linear additive or not. Which means, if the input signal is sum of two or more sub-input-signals, and the output signal is the sum of the sub-output-signals that is driven by each sub-input-signal individually. Using Math language, a linear system must has the characteristic as below:
Suppose $$ y_1[n] = f(x_1[n]), y_2[n] = f(x_2[n]), $$ then for a linear system, $$ y_1[n]+y_2[n]=f(x_1[n])+f(x_2[n])=f(x_1[n]+x_2[n]) $$ which means, if $x[n] = x_1[n]+x_2[n]$, then $y[n]=f(x[n])=y_1[n]+y_2[n]$.

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