half cosine pulse

[logistic_guy]

Find the Fourier transform of the half-cosine pulse shown in the figure.

 

[logistic_guy]

We need to define two functions to get this half-cosine pulse.

\(\displaystyle g(t) = A\cos \omega_c t\)

\(\displaystyle r(t) = \text{rect}\left(\frac{t}{T}\right)\)

Now when we multiply them, we get:

\(\displaystyle h(t) = g(t) \ r(t) = A\cos \omega_c t \ \text{rect}\left(\frac{t}{T}\right) \rightarrow\) half-cosine pulse

where \(\displaystyle \omega_c = 2\pi f_c = \frac{2\pi}{T}\)

 

[logistic_guy]

It is very difficult and even impossible to find the Fourier transform of the function \(\displaystyle h(t)\) by using the Fourier transform integral. Instead of that, the trick is to use the property of multiplication, that is:

\(\displaystyle H(f) = G(f)*R(f)\)

where the symbol \(\displaystyle *\) denotes convolution.

Now it is very easy to find the Fourier transform of \(\displaystyle g(t)\) alone and \(\displaystyle r(t)\) alone. And that's what we'll do in the next post.