EE213 introduction to telecommunications

Task 2.3. Discrete Fourier Transform (10 marks)

In Task 2.1, you were asked to plot the Continuous Time Fourier Transform (CTFT) of the analog (or continuous) signal 𝑥a(𝑡). In this task, the objective is to plot the Discrete Time Fourier Transform (DTFT) of the discrete signal 𝑥d[𝑛].
(5 marks)

Recall the definition of the DTFT

In order to observe the periodicity of 𝑋d(𝜔), you should plot 𝑋d(𝜔) for 𝜔 ∈ [−3𝜋, 3𝜋]. Your code will resemble the following:

Plot the obtained DTFT 𝑋d(𝜔). (5 marks)

In your submission, comment on the periodicity of 𝑋d(𝜔).
Hint: To answer this question, you should think about how the DTFT spectrum relates to the CTFT spectrum. Refer to the Analogue to Digital Conversion section for more information.

Task 2.4. Varying the Sampling Rate (5 marks)

Repeat Task 2 and Task 3 for a different sampling rate, namely take it 10 times smaller (𝜔s =𝜔s/10).

Comment on the sampled signal. Comment on the shape of the DTFT of the sampled signal in this case. In particular, compare it with the Fourier Transform of the continuous signal