# Determine the Given system is linear or nonlinear [closed]

Does the following define a linear or nonlinear system?

$$y(n) -4 y(n)y(2n)=x(n)$$

• What have you tried? Can you write the definitions of linear or nonlinear systems? – Maxtron Aug 6 '19 at 20:27

Using the definition of linear system as one which can be put in the form:

$$y_0 + a_1 y_1 + \dots + a_n y_n = b_0 x_0 + \dots + b_m x_m$$

because of the multiplicative term $$y(n)y(2n)$$ we can conclude that the system is nonlinear.

Your equation does not really define a system, as we don't know what the input/output pair is (is $$x$$ the input, or $$y$$?), and where the values of $$x$$ or $$y$$ dwell. It is not even well-defined: at $$n=0$$, we have the equation $$y[0](1-4y[0])=x[0]$$, which may have two, one or zero solutions.

However, let us suppose that $$x$$ is the input, and $$y$$ the output. The product of two "data" in $$y[n]y[2n]$$ is suspicious with respect to linearity. Because in a such a product, multiplying a variable by $$\lambda$$ turns out to become a product by $$\lambda \times \lambda = \lambda^2$$, hence a risk of loss of linearity. This is just a first intuition, now we shall try to prove it.

When you suspect that a system is "not something", a counter example suffices, and is often easier to find (at least in homework exercices).

So, if $$x$$ is multiplied by $$\lambda$$, what happens if we suppose that the system is linear, and $$y$$ is multiplied by $$\lambda$$ as well? We have two equations:

• $$y[n] - 4y[n]y[2n]= x[n]$$
• $$\lambda y[n] - 4\lambda^2 y[n]y[2n]= \lambda x[n]$$

Can they be fulfilled together? If both terms on the first equation do not vanish, by division, the second one yields $$\lambda -4\lambda^2 = \lambda$$, meaning that $$\lambda=0$$. So, it is not linear (whenever $$x[n] \neq0$$).