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1.6 Introduction to continuous-time signals: exercises

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  1. Consider the signals shown in [link] . Sketch the following signals Assume that for step discontinuities, the signal takes on the greater of the two values. :
    1. x 1 ( t ) + 2 x 2 ( t ) .
    2. x 1 ( - t ) - x 2 ( t - 1 ) .
    3. x 1 ( - t + 1 ) .
    4. x 2 ( t - 1 ) .
    5. x 1 ( 2 t ) .
    6. x 1 ( t / 2 ) .
    7. x 1 ( t ) x 3 ( t + 1 ) .
    8. x 3 ( 2 t - 4 ) .
    9. x 3 ( - 2 t - 4 ) .
  2. For each of the signals in [link] :
    1. What is the period?
    2. Sketch x ( t - 0 . 25 ) , and x ( t + 1 ) .
    3. Find the power for each signal.
  3. Suppose that z 1 = 3 + j 2 , z 2 = 4 + j 5 . Find:
    • | z 1 | , ∠ z 1 , | z 2 | , ∠ z 2
    • z 1 + z 2 in rectangular coordinates.
    • z 1 z 2 in rectangular and polar coordinates.
    • z 1 / z 2 in polar coordinates.
  4. What is the period, frequency, and power of the sinusoidal signal x ( t ) = 2 cos ( 5 t ) .
  5. Can you find a general formula for the power of the sinusoidal signal x ( t ) = A cos ( Ω t ) ?
  6. Express 2 cos ( 10 t ) + 3 sin ( 10 t ) as a single sinusoidal signal.
  7. Sketch x ( t ) = sin ( t - θ ) for θ = π / 4 , π / 2 , and 3 π / 4 .
  8. Sketch x ( t ) = sin ( 2 t - θ ) for θ = π / 4 , π / 2 , and 3 π / 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 when the second hand is over the 3.
Signals for problem 1.
Signals for problem 2.
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Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
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