Fourier Transform Integrals are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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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)

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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

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1. Compute the Fouriertransform for x(t) = texp(-t)u(t)
2. The linearity property of the Fouriertransform is defined as:
3. Determine the exponential Fourier series for:
4. Using complex notation, combine the expressions to form a singl

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

A waveform v(t) has a Fouriertransform which extends over the range -F to +F in the frequency domain. The square of the waveform v(t), that is, v(t) v(t), then has a Fouriertransforms which extends over the range:
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Answers:
a) -F to +F
b) -2F to +2F
c) -3F to +3F
d) -4F to +

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1. Let f(x) be a 2pi- periodic function such that f(x) = x^2 −x for x ∈ [−pi,pi].
Find the Fourier series for f(x).
2. Let f(x) be a 2pi- periodic function such that f(x) = x^2 for x ∈ [−1,1]. Using
the complex form, find the Fourier series of the function f(x).
3. See attachmen

Problem attached.
"Eigenvalues and Eigenvectors of the FourierTransform"
Recall that the Fouriertransform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself.
Let .. be an element of L2(?cc,cc).
Let..= , the Fouriertransform of.., be defined by
.....
It is clear that
.....
are square-integrable fu

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2. Let p be a fixed and given square integrable function, i.e.
0 < S g(x)g(x) dx = ||g^2|| < inf
The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x