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Fourier transform and its properties

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1. Compute the Fourier transform for x(t) = texp(-t)u(t)

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:

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The solution explains the Fourier transform and its properties in detail. It also shows how to represent a function using polar notation.

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