1. A continuous time signal x(t) has the Fourier transform X(ω) = 1/(jω+b) where b is a constant. Determine the Fourier transform for v(t) = t^2 x(t).
2. A continuous time signal x(t) has the Fourier transform X(ω) = 1/(jω+b) where b is a constant. Determine the Fourier transform for v(t) = x(t) * x(t).
3. Compute the Fourier transform for x(t) = te^-t u(t)
4. Compute the inverse Fourier transform for X(ω) = cos 4ω.
5. A signal with the highest frequency component at 10 kHz is to be sampled. To reconstruct the signal, the sampling must be done at a minimum frequency of?© BrainMass Inc. brainmass.com October 25, 2018, 8:23 am ad1c9bdddf
Please see the attached file for detailed solutions.
1. Find the Fourier Transform FT of if the FT of the function is given by
We have the FT
We need to find the time domain function representation of this which can be found from FT look up tables as
A simple proof of this result can be found in the appendix
Thus the FT of becomes the FT given by the integral of
Again splitting the integral into two domains, below and above zero, we get to evaluate
As represents the unit step function which is given by
The first integral disappears (as it equals zero) then we get to evaluate
This is the product of two functions of so we need to evaluate this expression using the method of integration by parts (IBP)
As the IBP formula states that
We make so that
We also make so that
Making these substitutions into our IBP formula we need to solve
Applying the IBP formula again to the integral
we let so that
We also make so that
Then we obtain
Combining all these results we obtain
Applying the limits
Everything in these limits ...
The continuous time signal for Fourier transforms are found. The expert computes the inverse Fourier transform for a function.
Three Fourier analysis of systems problems
Fourier Analysis of Systems:
1. A linear time-invariant continuous-time system has the frequency response function H(Ï?)=5cos(2Ï?), compute the system's impulse response h(t):
2. A signal with the highest frequency component at 10 kHz is to be sampled. To reconstruct the signal, the sampling must be done at a minimum frequency of:
3. An ideal low-pass digital filter has the frequency function H(Ï??) as shown in the figure (see attached doc). Determine the unit-pulse response h[n] of the filter. Note: The discontinuities occur at -Ï?/4 and +Ï?/4.View Full Posting Details