The Fourier transforms of odd and even functions are very important. The reason is that they are computationally simpler than the Fourier transform. Let x(t) = e-ItI and y(t) = e-tu(t) - etu(-t).
a. Plot x(t) and y(t), and determine whether they are odd or even.
b. Show that the Fourier transform of x(t) is found from x(Ω ) = ∫ (xt )cos (Ωt )dt , which is a real function of Ω , thus its computational importance. Show that X( Ω) is also even as a function of (Ω ).
c. Find x(Ω ) from the above equation (called the cosine transform).
d. Show that the Fourier transform of y(t) is found from y( Ω) = -j ∫ (yt ) sin(Ωt )dt , which is the imaginary function of (Ω ), thus its computational importance. Show that Y( Ω) is also odd as a function of (Ω ).
e. Find Y(Ω ) from the equation above (called the sine transform). Verify that your results are correct by finding the Fourier transform of z(t) = x(t) + y(t) directly and using the above results.
f. What advantages do you see to the sine and cosine transforms> How would you use cosine and the sine transforms to compute the Fourier transform of any signal, not necessarily even or odd, explain.