Calculating Fourier Transform with Unit Step Function and Time Shift

[nahanksh]

Homework Statement

Determine the Fourier transform of the following:

u(t) - u(t-T)

where u(t) is a unit step function.

Homework Equations

The Attempt at a Solution

I know that the Fourier transform of u(t) is pi*delta(w) + 1/jw

but when u(t-T) comes into the picture, the time shift would be e^-jwT

But is w inside the formula going to be changed too accordingly?

I am so confused...

I believe this is really simple problem if i could get a little bit of help...

Please someone help me out here..

 

[mighty2000]

Hi there,

Don't worry, this is a common confusion when dealing with Fourier transforms. The time shift does indeed affect the w inside the formula, but not in the way you might think.

First, let's start with the Fourier transform of u(t-T). As you correctly stated, the time shift will result in a term of e^(-jwT) in the formula. However, this term will not affect the w inside the formula. It will simply be multiplied to the entire expression. So, the Fourier transform of u(t-T) will be e^(-jwT) * (pi*delta(w) + 1/jw).

Now, when we consider the function u(t) - u(t-T), we can break it down into two parts: u(t) and u(t-T). The Fourier transform of u(t) is already known to be pi*delta(w) + 1/jw. And as we just determined, the Fourier transform of u(t-T) is e^(-jwT) * (pi*delta(w) + 1/jw). So, the Fourier transform of u(t) - u(t-T) will simply be the difference of these two expressions, which is (pi*delta(w) + 1/jw) - (e^(-jwT) * (pi*delta(w) + 1/jw)).

I hope this helps clear up your confusion. Remember, when dealing with Fourier transforms, it's important to break down the function into simpler parts and then apply the appropriate transformations. Keep practicing and you'll get the hang of it in no time!