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Problems on Systems and Signals

Module by: Don Johnson. E-mail the author

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Problem 1

Complex-valued Signals

Complex numbers and phasors play a very important role in electrical engineering. Solving systems for complex exponentials is much easier than for sinusoids, and linear systems analysis is particularly easy.

  1. Find the phasor representation for each, and re-express each as the real and imaginary parts of a phasor. What is the frequency (in Hz) of each? In general are you answers unique? If so, prove it; if not, find an alternative answer for the phasor representation.
    1. 3sin24t 3 24 t
    2. 2cos2π60t+π4 2 2 60 t 4
    3. 2cost+π6+4sint−π3 2 t 6 4 t 3
  2. Show that for linear systems having real-valued outputs for real inputs, that when the input is the real part of a phasor, the output is the real part of the system's output to the phasor. SℜAⅇⅈ2πft=ℜSAⅇⅈ2πft S A 2 f t S A 2 f t

Answer not provided

Problem 2

For each of the indicated voltages, write it as the real part of a complex exponential ( vt=ℜVⅇst v t V s t ). Explicitly indicate the value of the complex amplitude VV and the complex frequency ss. Represent each complex amplitude as a vector in the V-plane, and indicate the location of the frequencies in the complex s-plane.

  1. vt=cos5t v t 5 t
  2. vt=sin8t+π4 v t 8 t 4
  3. vt=ⅇ-t v t -t
  4. vt=ⅇ -3 t sin4t+3π4 v t -3 t 4 t 3 4
  5. vt=5ⅇ 2 t sin8t+2π v t 5 2 t 8 t 2
  6. vt=-2 v t -2
  7. vt=4sin2t+3cos2t v t 4 2 t 3 2 t
  8. vt=2cos100πt+π6−3sin100πt+π2 v t 2 100 t 6 3 100 t 2

Answer not provided.

Problem 3

Express each of the following signals as a linear combination of delayed and weighted step functions and ramps (the integral of a step).

Figure 1
part apart bpart cpart dpart e
(a) (b) (c) (d) (e)
part a (sig1.png)part b (sig2.png)part c (sig3.png)part d (sig4.png)part e (sig5.png)
Answer not provided

Problem 4

Linear, Time-Invariant Systems

When the input to a linear, time-invariant system is the signal xt x t , the output is the signal yt y t

Figure 2
Figure 2 (sig34a.png)

  1. Find and sketch this systems's output when the input is a unit step.
  2. Find and sketch this system's output when the input is a unit step.

Figure 3
Figure 3 (sig34b.png)
Answer not provided

Problem 5

Linear Systems

The depicted input xt x t to a linear, time-invariant system yields the output yt y t .

Figure 4
Figure 4 (sig39.png)

  1. What is the system's output to a unit step input ut u t ?
  2. What will the output be when the input is the depicted square wave?

Figure 5
Figure 5 (sig40.png)
Answer not provided

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