Fourier Transform of exp( - abs(t) ) / t?

Question

i have tried math software (maple, matlab, TI-89t) and have gotten different answers. i would prefer that you work the problem out or have a reference to a fourier transform table rather than use software. for clarification on the function: abs() is absolute value, and the exp(...) is divided by t. thanks for any help provided

Answer 1

Hi nascar,

We can solve this two ways. We can observe that we have two signals in time:

f(t) = exp^-| t |

g(t) = 1 / t

In this case we just need to find the fourier transform of each signal. Both of these signals are common and we can look them up on a table.

F(w) = sqrt(2/pi) / [ 1 + w^2]

G(w) = -i sqrt(pi/2) sgn(w)

= -i sqrt(pi/2), ...... w > 0

= 0, ....................... w = 0

= i sqrt(pi/2), ........ w < 0

We know the property:

f(t)g(t) < = > F(w)*G(w)

so we just have to convolve the two signals in frequency to find the fourier transform. Since G(w) is a constant sign function we have.

F(w)*G(w)

= -i / [1 + w^2], ....... w > 0

= 0, .......................... w = 0

= i / [1 + w^2], ......... w < 0

The alternate way to solve is to solve for the fourier transform is to compute the integral as so:

F(w) = 1/2π   ∫[inf, -inf]  (e^-|t| e^-iwt) / t dt

= 1/2π ∫[inf, 1] (e^-t(1+iw) / t dt  + ∫[0,-inf] (e^t(1+iw) / t dt

u = e^-t(1 + iw)

dv = 1/t dt

du = -(1 + iw)e^-t(1 + iw) dt

v = ln(t)

remember ∫udv = uv - ∫vdu so you would then integrate and evaluate the integrals of 1 to inf and -inf to 0.

You will see that the answer will be the same as the table form. In matlab, evaluate the integral from (inf to 1) and (-inf to -1) to see the representation rather than using the fourier command.

Hope that helps!! reply if you get confused / stuck on the analysis.