Consider a discrete time system S1 with impulse response𝒉(𝒏) = (𝟏)𝒏 𝒖[𝒏].𝟓(𝒂) 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒈𝒆𝒓 𝑨 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝒉[𝒏] − 𝑨𝒉[𝒏 − 𝟏] = 𝜹[𝒏].(𝒃) 𝑼𝒔𝒊𝒏𝒈 𝒕𝒉𝒆 𝒓𝒆𝒔𝒖𝒍𝒕 𝒇𝒓𝒐𝒎 𝒑𝒂𝒓𝒕 (𝒂), 𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒕𝒉𝒆 𝒊𝒎𝒑𝒖𝒍𝒔𝒆 𝒓𝒆𝒔𝒑𝒐𝒏𝒔𝒆𝒈[𝒏] 𝒐𝒇 𝒂𝒏 𝑳𝑻𝑰 𝒔𝒚𝒔𝒕𝒆𝒎 𝑺𝟐 , 𝒘𝒉𝒊𝒄𝒉 𝒊𝒔 𝒕𝒉𝒆 𝒊𝒏𝒗𝒆𝒓𝒆𝒔𝒆 𝒔𝒚𝒔𝒕𝒆𝒎 𝒐𝒇 𝑺𝟏.

The impulse response of a continuous-time system is denoted by ℎ(𝑡)h(t), while for a discrete-time system, it is denoted by ℎ[𝑛]h[n]. Compare the properties of ℎ(𝑡)h(t) and ℎ[𝑛]h[n] and explain how they differ.

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)  b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)b. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)c. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)d. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)  b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)

A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)