Finite dimensionality of a Discrete time system.

[MorrowUoN]

Homework Statement

Determine the finite dimensionality of the following system:

y[n] = nx[n]

Homework Equations

y[n]= f(y[n−1], y[n−2],..., y[n−N],x[n],x[n−1],..., x[n−M],n)

Where N is how many dimensions the system has.

The Attempt at a Solution

I understand that the following system would be 1 dimensional.

y[n]=y[n-1]+nx[n]

However, in the case of my question the output at time n is only a function of the input and not a function of the input and output at some shifted time. Does this mean that the system in question is infinite dimensional?

Thanks.

 

[KataKoniK]

Thank you for your question. To determine the finite dimensionality of a system, we need to look at the number of independent variables that are involved in the system. In this case, we have the input x[n] and the time index n. These two variables are independent of each other and therefore, we can say that the system is 2-dimensional.

The equation you have provided is a recursive equation, which means that the output at time n is dependent on the output at previous times as well as the input at current and previous times. However, in this case, the output at time n is still only a function of the input and the time index, which makes it a 2-dimensional system.

I hope this helps. Let me know if you have any further questions.