2. The linearity property of the Fourier transform is defined as:

3. Determine the exponential Fourier series for:

4. Using complex notation, combine the expressions to form a single sinusoid for: cos(10t+ pi/2) + 2cos(10t - pi/3)

5. The polar notation for the function 1 + e^(j4) is:

6. The duality property of the Fourier transform is defined as:

7. A continuous time signal x(t) has the Fourier transform: x(w) = 1/(jwW +b), where b is a constant. Determine the Fourier transform for v(t) = t^2*x(t).

8. Compute the inverse Fourier transform for X(w) = cos(4w).

9. A continuous time signal x(t) has the Fourier transform: x(w) = 1/(jw +b), where b is a constant. Determine the Fourier transform for v(t) = x(t) * x(t).

10. The polar notation for the function 1 + e^(j4) + e^(j2) is:

I have attached some problems that I think I am working correctly. I wanted to verify the concept is correct.
A signal x(t) = exp(-t)*cos(3t) is turned on at t = 0. What is itsFouriertransform?
Consider the signal in problem 6. What is the Fouriertransform of its derivative with respect to time?
What is the Fourie

Any causal signal x(t) having a Laplace transform with poles in the open-left s-plane (i.e., not including the j? axis) has, as we saw before, a region of convergence that includes the j? axis, and as such itsFouriertransform can be found from its Laplace transform. Consider the following signals:
x1(t) = e^-(2t) * u(t)
X

Please see attachment.
1. What is the FourierTransform for the convolution of sin(2t)*cos(2t).
2. Compute the inverse Fouriertransform for X(w)= sin^2*3w
3. A continuous time signal x(t) has the Fouriertransform
X(w) = 1/jw+b where b is a constant. Determine the Fouriertransform for v(t) = x*(5t-4)

Please see the attached file for details.
1. A continuous time signal x(t) has the Fouriertransform X(w) = 1/(jw + b), where b is a constant. Determine the Fouriertransform for v(t) = x(5t - 4).
2. For a discrete-time signal x[n] with the DTFT X(w) = 1/(e^jw + b), where b is an arbitrary constant compute the DTFT V(Ω

I am trying to get a feel for Fouriertransforms and would like some help.
Please show all work/explanations.
1. Let g(t) = x(2t) - x*(2t- 1/2T0), assume that X(jw) is known
Find G(jw) in terms of X(jw).
2. Let x(t) = 10sin(200t)/t
a) Find X(jw)
b) Let x(t) be the input to a continuous LTI system with impulse response

The problem is from Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform:
Solve using a cosine or sine transform.
u'' - 9u =50e^-3x (0

The expected value of the momentum operator in quantum mechanics can be calculated using the spatial wave function. Using that the momentum wave function is the Fouriertransform of the spatial wave function we obtain an expression for this expected value in term of the momentum wave function, that is, we prove the Equation 15.6

Show that integral from - infinity to + infinity of psi_1(x) times psi_2*(x) dx is equal to integral from - infinity to + infinity of phi_1(k) times phi_2*(k) dk.
* indicates complex conjugate