VE216 Homework4 Solved

Problems:

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):

(a)    x(t) = 2rect  

(b)    x(t) = e−3trect  

(c)    x(t) = trect  

(d)   x(t) = cos(4πt)rect  

2.    [5] Find a mathematical expression and sketch or plot the inverse FT of F(ω) = sinc3(ω/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(ω) is purely imaginary and odd. 5. [5] Consider a real signal f(t) and let

F

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

and f(t) = fe(t) + fo(t)

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

                                                             F                                                                                                                 F
                                                  fe(t) ←→ real{F(ω)}                                         fo(t) ←→ j imag{F(ω)}

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 ≤ ω ≤ 7rad/s.

8.       [10] A LTI system has the following frequency response:

 .

(a)    [10] Find the impulse response of the LTI system. Hint: first find the partial differential equation.

(b)    [10] Find the differential equation corresponding to the LTI system. Hint: write H(ω) = Y (ω)/X(ω) 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

(a)    [e−αt cosω0t]u(t),α 0

(b)    e−3|t| sin2t

11.   [10] Determine the continuous-time signal corresponding to the following transform.

(a)    X(jω) = cos(4ω + π/3)

(b)    X(jω) as given by magnitude and phase plots.

 

 

Figure: 0402

12.   [10] Shown in the figure 0403 is the frequency response H(jω) 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).

(a)    x(t) = cos(2πt + θ)

(b)    x(t) = cos(4πt + θ)

(c)    x(t) is a half-wave rectified sine wave of period 1, as sketched in figure 0404.

                                                         |H(jω)|                                                                           ∠H(jω)

 

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 system’s output signal.

 

Figure: 0405

|H(ω)|

 

Figure: 0406