Fourier Transforms: Step-by-Step Guide to Hand Calculations

[Xjac0bmichael]

I have been continuously having problems finding the Fourier transform of signals by hand using the definition. If someone could give me a step by step process for doing this it would be great. Here are some problems from my book iv'e been stuck on. I have tried to set up the integral, but I have problems trying to simplify and get to a final answer.

X(t)=abs(sin(2*pi*t))

X(t)=exp(-abs(t))

 

[jasonRF]

I have been continuously having problems finding the Fourier transform of signals by hand using the definition. If someone could give me a step by step process for doing this it would be great. Here are some problems from my book iv'e been stuck on. I have tried to set up the integral, but I have problems trying to simplify and get to a final answer.

X(t)=abs(sin(2*pi*t))

X(t)=exp(-abs(t))

In general, if you are starting from the definition you have to figure out how to perform the integrals. There is no general step-by-step process of doing integrals of this sort, I'm afraid. Now to your examples:

Your second example is straightforward. Hint: you break up the integral into two pieces, one for negative t and one for positive t. This gets rid of the absolute value, and leaves you with integrals of exponentials, which are about as easy as integration gets.

Your first example is tricky. You can only get a transform of it if you allow Dirac delta functions. Since it is periodic, you can represent X(t) as a Fourier series
$$x(t) = \sum_{n=-\infty}^{\infty} a_n e^{i 2 n \pi t / T}$$
and then use the fact that the Fourier tranform of a complex exponential is proportional to a delta function. You need to work out the details, but for periodic signals this is the general method.

good luck,

jason