1. [10] Use the table of FT pairs and the table of properties to find the FT of each of the following signals

(DO NOT USE INTEGRATION):

• x(t) = 2rect
• x(t) = e^3trect
• x(t) = trect
• x(t) = cos(4t)rect

2. [5] Find a mathematical expression and sketch or plot the inverse FT of F() = sinc³(/2). Hint: the inverse FT formula would probably be a hard way to do it.
3. [5] Find the FT of t²e^(t/2)2. Hint: see table of FT pairs.
4. [5] Show that if f(t) is real and odd, then F() is purely imaginary and odd. 5. [5] Consider a real signal f(t) and let

F

f(t) F(), F() = real{F()} + j imag{F()}

and f(t) = f_e(t) + f_o(t)

where f_e(t) and f_o(t) are the even and odd component of f(t) respectively. Show that

F F

f_e(t) real{F()} f_o(t) j imag{F()}

6. [5] Find the energy of the signal x(t) = tsinc²(t) by Fourier methods.
7. [5] What percentage of the total energy in the energy signal f(t) = e^tu(t) is contained in the frequency band 7rad/s 7rad/s.
8. [10] A LTI system has the following frequency response:

.

• [10] Find the impulse response of the LTI system. Hint: first find the partial differential equation.
• [10] Find the differential equation corresponding to the LTI system. Hint: write H() = Y ()/X() and cross multiply.

9. [10] Find the FT of the following signal:

sketch the magnitude of the spectrum.

10. [10] Compute the Fourier transform of each of the following signals
    • [e^t cos₀t]u(t), > 0
    • e3|t| sin2t
11. [10] Determine the continuous-time signal corresponding to the following transform.
    • X(j) = cos(4 + /3)
    • X(j) as given by magnitude and phase plots.

Figure: 0402

12. [10] Shown in the figure 0403 is the frequency response H(j) of a continuous-time filter referred to as a lowpass differentiator. For each of the input signals x(t) below, determine the filter output signal y(t).
    • x(t) = cos(2t + )
    • x(t) = cos(4t + )
    • x(t) is a half-wave rectified sine wave of period 1, as sketched in figure 0404.

|H(j)| H(j)

Figure: 0403

x(t)

Figure: 0404

13. [10] A power signal with the power spectral density shown in figure 0405 is the input of a linear system with the frequency response shown in figure 0406. Calculate and sketch the power spectral density of the systems output signal.

Figure: 0405

|H()|

Figure: 0406