Task 2:a) Prove the time shifting property for the CTFT using the signal 𝑥 = 𝑡𝑒−𝑡2. The time shiftgiven to this signal is𝑡0 = 3. Plot the signals in time domain using subplot.b) Using the input signal 𝑦 = 𝑒−𝑡2, prove the differentiation property of the CTFT. Plot thesignals in time domain using subplot

To prove the time shifting property for the Continuous-Time Fourier Transform (CTFT) using the signal 𝑥 = 𝑡𝑒−𝑡2, we need to follow these steps:

a) Time shifting property for CTFT:

1. Start with the given signal 𝑥 = 𝑡𝑒−𝑡2.
2. Apply the time shift 𝑡0 = 3 to the signal 𝑥. This means replacing 𝑡 with 𝑡−𝑡0 in the expression for 𝑥. So, the time-shifted signal is 𝑥(𝑡−𝑡0) = (𝑡−𝑡0)𝑒−(𝑡−𝑡0)2.
3. Simplify the expression for the time-shifted signal 𝑥(𝑡−𝑡0) as much as possible.
4. Plot the original signal 𝑥 and the time-shifted signal 𝑥(𝑡−𝑡0) in the time domain using subplot.

b) Differentiation property of CTFT:

1. Consider the input signal 𝑦 = 𝑒−𝑡2.
2. Apply the differentiation property of CTFT to the signal 𝑦. This property states that taking the derivative of a signal in the time domain corresponds to multiplying the CTFT of the signal by 𝑗2𝜋𝑓, where 𝑓 is the frequency variable.
3. Calculate the CTFT of the signal 𝑦, which is denoted as 𝑌(𝑓).
4. Multiply 𝑌(𝑓) by 𝑗2𝜋𝑓 to obtain the CTFT of the differentiated signal.
5. Plot the original signal 𝑦 and the differentiated signal in the time domain using subplot.

Please note that the specific calculations and plots will depend on the mathematical expressions and the software or programming language you are using.