EE213 Consider the two-sided exponential decay signal 𝑥a(𝑡) = 𝑒−𝑎|𝑡| for an arbitrary value of a dimensionless parameter 𝑎: introduction to telecommunications, MU, Ireland

Part 2. Sampling and Aliasing (40 marks)

The objective of Part 2 is to give you a feel for the effect that sampling a signal has on its frequency content – as indicated by its frequency spectrum, which is produced by the Fourier Transform.
The prerequisite theory for completing this Part can be found in the Frequency Analysis and Analogue to Digital Conversion sections.

Task 2.1. Frequency Spectrum (15 marks)
Consider the two-sided exponential decay signal 𝑥a(𝑡) = 𝑒−𝑎|𝑡| for an arbitrary value of a dimensionless parameter 𝑎. In this case, the subscript ‘a’ indicates that 𝑥(𝑡) is an analog signal.

One can show that the Fourier Transform of 𝑥a(𝑡) is given by

Plot 𝑋a(𝜔) for a=100 in a range of frequencies [𝜔min, 𝜔max]. Choose the appropriate range and comment on your choice. Hint: a good plot should show all the peculiarities of a function/signal, thus, your plot should clearly show the range where 𝑋a(𝜔) is (i) determined and (ii) has meaningful values.

From the obtained plot, conclude on the bandwidth of 𝑋a(𝜔). You should write the bandwidth 𝐵 down in your report, show it on a graph, and comment on your choice. Hint: see Figure 2 for an example of a bandlimited signal.

How is the chosen bandwidth 𝐵 related to the Nyquist frequency? Explain with numbers.