Derivative Property

The questions below are due on Thursday October 23, 2025; 02:00:00 PM.
 
You are not logged in.

Please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.mit.edu) to authenticate, and then you will be redirected back to this page.

Part 1a. Determine the Fourier transform F_1(\omega) of f_1(t) defined by

f_1(t)=\begin{cases}1&{\rm if}~|t|\lt1\cr0&{\rm otherwise}\end{cases}\,.

Enter an expression for F_1(\omega). Your expression may contain mathematical constants (pi, j, e), functions (sin, cos) as well as omega.:

Part 1b. Determine the Fourier transform F_2(\omega) of f_2(t) defined by

f_2(t)= \delta(t{+}1)-\delta(t{-}1)

Enter an expression for F_2(\omega). Your expression may contain mathematical constants (pi, j, e), functions (sin, cos) as well as omega.

Part 1c (not graded). Notice that f_2(t) is the derivative of f_1(t). How are their transforms related?

Part 2a. Determine the Fourier transform F_3(\omega) of f_3(t) defined by

f_3(t)=e^{-|t|}

Enter an expression for F_3(\omega). Your expression may contain mathematical constants (pi, j, e), functions (sin, cos) as well as omega.:

Part 2b. Determine the Fourier transform F_4(\omega) of f_4(t) defined by

f_4(t)=e^t u(-t)-e^{-t} u(t)

Enter an expression for F_4(\omega). Your expression may contain mathematical constants (pi, j, e), functions (sin, cos) as well as omega.

Part 2c. Notice that f_4(t) is the derivative of f_3(t). How are their transforms related? Is this the same relation you found in Part 1c? If so, does this relation always hold? Can you prove these relations?

Do the relations found in parts 1c and 2c hold in general?