Question: The following problems relate to linearity, time-invariance, causality, and stability of discrete-time systems. (a) The output y[n] of a system is related to its input

The following problems relate to linearity, time-invariance, causality, and stability of discrete-time systems.

(a) The output y[n] of a system is related to its input x[n] by y[n] = x[n]x[n ˆ’ 1]. Is this system

i. linear? time-invariant?

ii. causal? bounded-input bounded-output stable?

You may consider x[n] = u[n] as the input to verify your results.

(b) Given the discrete-time system in Figure 9.14

i. Is this system time-invariant?

ii. Suppose that the input is x[n] = cos (Ï€n/4), ˆ’ˆž < n < ˆž, so that the output is y[n] = cos (Ï€n/4) cos (n/4), ˆ’ˆž < n < ˆž. Determine the fundamental period N₀ of x[n]. Is y[n] periodic? If so, determine its fundamental period N₁.

Figure 9.14:

x[n] y[n] (X) cos(n/ 4)

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