Homework 1, ECE438, Fall 2011, Prof. Boutin
Due Wednesday August 31, 2011 (in class)
Before beginning this homework, review the following table of CT Fourier transform pairs and properties, which features the CT Fourier transform in terms of $ \omega $ (in radians per time unit). You should have seen each of these in ECE301. Then compare with the following table of CT Fourier transform pairs and properties, which uses the CT Fourier transform in terms of frequency $ f $ in hertz. Note that we will use the latter in ECE438.
Question 1
In ECE301, you learned that the Fourier transform of a step function $ x(t)=u(t) $ is the following:
$ {\mathcal X} (\omega) = \frac{1}{j \omega} + \pi \delta (\omega ). $
Use this fact to obtain an expression for the Fourier transform $ X(f) $ (in terms of frequency in hertz) of the step function. (Your answer should agree with the one given in this table.) Justify all your steps.
Question 2
What is the Fourier transform of $ x(t)= e^{j \pi t} $? Justify your answer.
Discussion
Please discuss the homework below.
Can someone remind me where does this comes from?
$ {\mathcal X} (\omega) = \frac{1}{j \omega} + \pi \delta (\omega ). $
especially where the delta function $ \pi \delta (\omega ). $ comes from? if directly integrate from definition of Fourier transform then I'll get only the $ \frac{1}{j \omega}. $ Yimin
- Think about this: you will not be able to integrate if $ \omega=0 $ --Rui ,25 August 2011
I think I have same kind of question. Can someone show me how to get the FT of a unit step function?
- Instructor's comment: It is not easy to FT directly. But there is an easier, indirect way to obtain this FT. You just need to observe that u(t) is the integral of a dirac delta from $ -\infty $ to t, then use the properties of the FT. Does that help?-pm
