To find the spectrum and draw the amplitude spectrum for the given function X(t) = 2cos(3pi*t), we need to perform a Fourier transform. The Fourier transform will convert the function from the time domain to the frequency domain, allowing us to analyze its spectral components.
Here are the steps to find the spectrum and draw the amplitude spectrum:
Step 1: Compute the Fourier Transform The Fourier transform of a function f(t) is defined as:
F(ω) = ∫[from -∞ to +∞] f(t)*e^(-jωt) dt
In our case, f(t) = 2cos(3pi*t). Substitute this into the Fourier transform formula and calculate the integral. The result will give you the spectrum in terms of frequency (ω).
Step 2: Calculate the Amplitude Spectrum The amplitude spectrum is the magnitude of the complex numbers obtained from the Fourier transform. For each frequency component in the spectrum, calculate the magnitude as:
|F(ω)| = sqrt(Re(F(ω))^2 + Im(F(ω))^2)
Step 3: Plot the Amplitude Spectrum Plot the amplitude spectrum with frequency (ω) on the x-axis and |F(ω)| on the y-axis. You can use a graphing tool or software like Python with libraries such as NumPy and Matplotlib to plot the spectrum.
Note: In this particular case, since the function X(t) = 2cos(3pit) has only a single frequency component, the amplitude spectrum will have a single peak at ω = 3pi.
Let me know if you would like me to show you a Python code example for calculating the spectrum and plotting the amplitude spectrum.