Question 1

(a) Determine whether the following signals are periodic. If a signal is periodic, then calculate its fundamental period.

(b) A linear system, L, has the following relationship,

between its input 𝑥[𝑛] and its output, 𝑦[𝑛], where ℎ[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 4] and
𝑢[𝑛] is the discrete unit step.

(i) Determine 𝑦[𝑛] when 𝑥[𝑛] = 𝛿[𝑛 − 1].

(ii) Determine 𝑦[𝑛] when 𝑥[𝑛] = 𝛿[𝑛 − 2].

(iii) Is the system linear time-invariant (LTI)?

(iv) Determine and sketch 𝑦[𝑛] when 𝑥[𝑛] = 𝑢[𝑛].

(c) Which of the following impulse responses corresponds to a stable LTI system? Justify your answer.

Question 2

Consider the following three linear time-invariant (LTI) systems connected as shown in Figure Q2 below:

(a) The impulse response of each block is given by:

Describe the overall system impulse response ℎ𝑠 [𝑛].

(b) Analyse the above system’s frequency response by computing the Discrete
Fourier Transform of its impulse response.

(c) An input signal 𝑥[𝑛] = 2 cos ( 2𝜋6 𝑛) is fed into the above system. From the
frequency response obtained in Question 2(b), calculate the output signal, 𝑦[𝑛].

(d) Discuss the stability of the above-given system.