Linearity and Time Invariance
a.) The system can not be Time Invariant, because the output is scaled by a factor of the delta T that occurs throughout the function.
b.) Assuming the input is linear, an input X[n] that would yield an output of Y[n]=u[n-1] would be:
to get the output that is listed, you need to show how the unit step function is present. In the notes it was mentioned that the unit step function can be shown by a summation of shifted delta functions over a series of - $ \_infty $
The unit step can be written as an infinite sum of shifted delta functions. Since the system is linear, a sum of shifted delta functions as input will yield a sum of shifted delta functions as output. In this system the fact that the desired step function output begins at n = 1 implies that the first delta function in the input should begin at n = 0 because the system always shifts the input by 1 unit. The only remaining task is to counteract the (k + 1)2 term inherent in the system, so that the magnitude of each delta function in the output is 1. Since the system is linear, this couteracting can also be done on each input because the same magnitude will show up in the output. Therefore, the desired input is as follows:
