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- Homework Statement
- Determine whether the system is (a) linear, (b) time-invariant:
- Relevant Equations
- y[n] = T{x[n]} = x(t)
y(t) = cos(3t)x(t)
A system is linear if it satisfies the properties of superposition and homogeneity.
Superposition: adding the inputs of two systems results in the addition of the two outputs.
Ex) x1(t) + x2(t) = y1(t) + y2(t)
Homogeneity: multiplying the input by some scalar value is equal to the output multiplied by that same value.
Ex) ax1(t) = ay2(t)
After doing some internet searching, I also found this equation for superposition: f(a+b) = f(a) + f(b)
However, I don't know how to apply these rules to determine whether my equations are linear or not.
My attempt for determining SP:
y1[n] + y2[n] = T{x1[n] + x2[n]} = n(x1[n] + x2[n])
If my attempt at SP is correct, what is the next step? If it is wrong, what is the right way to write it, and then what is the next step?
Superposition: adding the inputs of two systems results in the addition of the two outputs.
Ex) x1(t) + x2(t) = y1(t) + y2(t)
Homogeneity: multiplying the input by some scalar value is equal to the output multiplied by that same value.
Ex) ax1(t) = ay2(t)
After doing some internet searching, I also found this equation for superposition: f(a+b) = f(a) + f(b)
However, I don't know how to apply these rules to determine whether my equations are linear or not.
My attempt for determining SP:
y1[n] + y2[n] = T{x1[n] + x2[n]} = n(x1[n] + x2[n])
If my attempt at SP is correct, what is the next step? If it is wrong, what is the right way to write it, and then what is the next step?