Fourier Transform for constant and sin

Question

• Given a signal $\,\mathrm{M}\left(\, f\, \right) = A$ for $\left\vert\, f\, \right\vert < B$ and $0$ else, what will be the expression for $$ \mathrm{z}\left(\, t\, \right) = \,\mathrm{m}\left(\, t\, \right) \cos\left(\, 2\pi\,\left[\, 1.9 \times 109\,\right]t + {\pi \over 4}\,\right)\ ?. $$
• So i have used the inverse fourier transform for $\,\mathrm{M}\left(\, f\,\right)$ to find $\,\mathrm{m}\left(\,t\,\right)$. I am not sure but i think it gave me A(delta) $\left[\,\mbox{IFT for constant}\,\right]$.
• The simplifying the $\cos$ term to Euller's formula should give what ?.

I am not sure though. and i am stuck in finding $\,\mathrm{m}\left(\,t\,\right)$. Thanks in advance for the help.

Answer 1

By "the expression" for it you are referring to its Fourier transform right?

First, $M(f)$ is not a constant. It is a rectangular function.

• You have two functions multiplied in time domain. What happens to the Fourier transform of their product?
• What do you know about the inverse Fourier transform of such rectangular function?
• recognize the phase and frequency of the cosine.
• Write the cosine in form of two complex exponentials and find its Fourier transform. Consider the phase shift and use the Fourier transform properties and appropriate trigonometric identities to adjust it according to the given phase shift.
• Note that $\mathcal{F}\{e^{j2\pi f_0 t}\}=\delta(f-f_0)$
• Note that convolution of a function $M(f)$ with $\delta(f-f_0)$ is $M(f-f_0)$.

Following these steps, you will get to $$Z(f)=\frac{A}{2}\left(e^{−j\frac{\pi}{4}}\frac{B}{2}\text{sinc}\frac{B}{2}(f+f_0)+e^{j\frac{\pi}{4}}\frac{B}{2}\text{sinc}\frac{B}{2}(f-f_0)\right)$$ where $f_0=1.9\times109$.