Exercise 3
[5 points]. 1) Suppose that a discrete-time linear system has outputs \( y[n] \) for the given inputs \( x[n] \), as shown in Fig. 1. Determine the response \( y_{4}[n] \) when the input is as shown in Fig. 2.
a) \( \left[1\right. \) point]. Express \( x_{4}[n] \) as a linear combination of \( x_{1}[n], x_{2}[n] \), and \( x_{3}[n] \).
b) [1 point]. Using the fact that the system is linear, determine \( y_{4}[n] \), the response to \( x_{4}[n] \).
c) [1 point]. From the input-output pairs in Fig. 1, determine whether the system is time-invariant.
2) Determine the discrete-time convolution of \( x[n] \) and \( h[n] \) for the following two cases.
1
a) \( [1 \) point \( ] \). As shown in Fig. 3.
b) \( [1 \) point \( ] \). As shown in Fig. 4.
