Hello, given a picture of a sawtooth signal f(t) I am suppose to find the Fourier transform. The picture shows that f(t)=t from 0 to 1 otherwise it is 0.
I believe another way to represent this signal is f(t)=t[u(t)-u(t-1)].
The provided solution is below but I don't understand it completely:
Step 1: f'(t)=u(t)-u(t-1)-delta(t-1)
Step 2: f''(t)=delta(t)-delta(t-1)-delta'(t-1)
Step 3: -ω^2F(ω)=1 - e^-jω - jωe^-jω
Step 4: F(ω)=-(1/ω^2)[1 - e^-jω - jωe^-jω]
I don't understand how we get the first derivative in step 1. I understand step 2. I don't understand how the Fourier transform of delta'(t-1) is jωe^-jω.
Please explain step by step.
I believe another way to represent this signal is f(t)=t[u(t)-u(t-1)].
The provided solution is below but I don't understand it completely:
Step 1: f'(t)=u(t)-u(t-1)-delta(t-1)
Step 2: f''(t)=delta(t)-delta(t-1)-delta'(t-1)
Step 3: -ω^2F(ω)=1 - e^-jω - jωe^-jω
Step 4: F(ω)=-(1/ω^2)[1 - e^-jω - jωe^-jω]
I don't understand how we get the first derivative in step 1. I understand step 2. I don't understand how the Fourier transform of delta'(t-1) is jωe^-jω.
Please explain step by step.