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Introduction to Continuous-Time Signals: Exercises

Module by: Carlos E. Davila. E-mail the author

  1. Consider the signals shown in Figure 1. Sketch the following signals1:
    1. x1(t)+2x2(t)x1(t)+2x2(t).
    2. x1(-t)-x2(t-1)x1(-t)-x2(t-1).
    3. x1(-t+1)x1(-t+1).
    4. x2(t-1)x2(t-1).
    5. x1(2t)x1(2t).
    6. x1(t/2)x1(t/2).
    7. x1(t)x3(t+1)x1(t)x3(t+1).
    8. x3(2t-4)x3(2t-4).
    9. x3(-2t-4)x3(-2t-4).
  2. For each of the signals in Figure 2:
    1. What is the period?
    2. Sketch x(t-0.25)x(t-0.25), and x(t+1)x(t+1).
    3. Find the power for each signal.
  3. Suppose that z1=3+j2z1=3+j2, z2=4+j5z2=4+j5. Find:
    • |z1|,∠z1,|z2|,∠z2|z1|,∠z1,|z2|,∠z2
    • z1+z2z1+z2 in rectangular coordinates.
    • z1z2z1z2 in rectangular and polar coordinates.
    • z1/z2z1/z2 in polar coordinates.
  4. What is the period, frequency, and power of the sinusoidal signal x(t)=2cos(5t)x(t)=2cos(5t).
  5. Can you find a general formula for the power of the sinusoidal signal x(t)=Acos(Ωt)x(t)=Acos(Ωt)?
  6. Express 2cos(10t)+3sin(10t)2cos(10t)+3sin(10t) as a single sinusoidal signal.
  7. Sketch x(t)=sin(t-θ)x(t)=sin(t-θ) for θ=π/4,π/2θ=π/4,π/2, and 3π/43π/4.
  8. Sketch x(t)=sin(2t-θ)x(t)=sin(2t-θ) for θ=π/4,π/2θ=π/4,π/2, and 3π/43π/4.
  9. Consider the motion of the second hand of a clock. Assume the length of the second hand is 1 meter. (a) what is the angular frequency of the second hand. (b) find an expression for the horizontal and vertical displacements of the tip of the second hand, assuming the origin is at the clock center and t=0 t 0 when the second hand is over the 3.
Figure 1: Signals for problem 1.
Figure 1 (ch1f1.png)
Figure 2: Signals for problem 2.
Figure 2 (hw1.png)
(1)

Footnotes

  1. Assume that for step discontinuities, the signal takes on the greater of the two values.

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