Find the Fourier Transform of the following functions and draw the spectrum:Part d - Max 400 words.u(t).CosaootSgn(t)

To find the Fourier Transform of the function u(t) * cos(at) * Sgn(t), we can break it down into three separate functions: u(t), cos(at), and Sgn(t).

1. The function u(t) represents the unit step function, which is defined as u(t) = 1 for t >= 0 and u(t) = 0 for t < 0. The Fourier Transform of the unit step function is given by U(f) = 1/(j2πf) + πδ(f), where j is the imaginary unit and δ(f) is the Dirac delta function.

2. The function cos(at) represents a cosine wave with frequency a. The Fourier Transform of cos(at) is given by C(f) = 0.5[δ(f-a) + δ(f+a)], which means that the spectrum consists of two impulses located at frequencies a and -a.

3. The function Sgn(t) represents the signum function, which is defined as Sgn(t) = 1 for t > 0, Sgn(t) = -1 for t < 0, and Sgn(0) = 0. The Fourier Transform of the signum function is given by S(f) = jπδ(f) + 1/(jπf), which means that the spectrum consists of an impulse located at frequency 0 and a sinc function.

Now, to find the Fourier Transform of the given function u(t) * cos(a*t) * Sgn(t), we can use the properties of the Fourier Transform.

First, we can apply the time shifting property to the cosine function. Shifting the cosine function by t = 0 gives us cos(at) = cos(a(t-0)) = cos(at). Therefore, the Fourier Transform of cos(at) remains the same.

Next, we can apply the time scaling property to the unit step function. Scaling the unit step function by a factor of a gives us u(t/a). Therefore, the Fourier Transform of u(t/a) is U(f/a) = 1/(j2πf/a) + πδ(f/a).

Finally, we can multiply the Fourier Transforms of the individual functions to obtain the Fourier Transform of the given function. Multiplying U(f/a), C(f), and S(f) gives us the spectrum of the function.

The spectrum will consist of impulses located at frequencies a and -a, a sinc function centered at frequency 0, and a shifted sinc function centered at frequency a. The exact shape of the spectrum will depend on the specific values of a and the scaling factor used for the unit step function.

To draw the spectrum, you can plot the magnitude of the Fourier Transform on the y-axis and the frequency on the x-axis. The impulses will appear as spikes at the corresponding frequencies, and the sinc functions will have a bell-shaped curve centered at their respective frequencies.