Question

For the following signals, find the Fourier transform if T₁=2 :

a) x(t)={ 3 ∣t∣ { 0 ∣t∣>T₁
b) x(t)=sin(T₁t−T₁)u(t−1)

Answer 1

To find the Fourier transform of the given signals, follow the appropriate formulas and split the integral when necessary.

Step-by-step explanation:

To find the Fourier transform of the given signals:

a) For the signal x(t)={ 3 |t| { 0 |t|>T₁, the Fourier transform can be obtained by using the formula:

F(ω) = ∫[−∞,∞] x(t)e^−jωt dt

In this case, we have two piecewise functions: 3|t| for |t|≤T₁ and 0 for |t|>T₁. We can split the integral into two parts and calculate the Fourier transform separately for each function.

b) For the signal x(t)=sin(T₁t−T₁)u(t−1), where u(t) represents the unit step function, we can directly apply the Fourier transform formula to obtain:

F(ω) = ∫[−∞,∞] sin(T₁t−T₁)u(t−1)e^−jωt dt